# Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter III

III.
ELEMENTS OF VECTOR ANALYSIS.
[Privately printed, New Haven, pp. 17–50, 1881; pp. 50–90, 1884.]
(The fundamental principles of the following analysis are such as are familiar under a slightly different form to students of quaternions. The manner in which the subject is developed is somewhat different from that followed in treatises on quaternions, since the object of the writer does not require any use of the conception of the quaternion, being simply to give a suitable notation for those relations between vectors, or between vectors and scalars, which seem most important, and which lend themselves most readily to analytical transformations, and to explain some of these transformations. As a precedent for such a departure from quaternionic usage, Clifford's Kinematic may be cited. In this connection, the name of Grassmann may also be mentioned, to whose system the following method attaches itself in some respects more closely than to that of Hamilton.)
CHAPTER I.
concerning the algebra of vectors.
Fundamental Notions.
1. Definition.—If anything has magnitude and direction, its magnitude and direction taken together constitute what is called a vector.
The numerical description of a vector requires three numbers, but nothing prevents us from using a single letter for its symbolical designation. An algebra or analytical method in which a single letter or other expression is used to specify a vector may be called a vector algebra or vector analysis.
Def.—As distinguished from vectors the real (positive or negative) quantities of ordinary algebra are called scalars.
$i,j,k$
$\pi$
​2. Def.—Vectors are said to be equal when they are the same both in direction and in magnitude. This equality is denoted by the ordinary sign, as $\alpha =\beta$. The reader will observe that this vector equation is the equivalent of three scalar equations.
A vector is said to be equal to zero, when its magnitude is zero. Such vectors may be set equal to one another, irrespectively of any considerations relating to direction.
3. Perhaps the most simple example of a vector is afforded by a directed straight line, as the line drawn from ${\text{A}}$ to ${\text{B}}$. We may use the notation ${\overline {\text{AB}}}$ to denote this line as a vector, i.e., to denote its length and direction without regard to its position in other respects. The points ${\text{A}}$ and ${\text{B}}$ may be distinguished as the origin and the terminus of the vector. Since any magnitude may be represented by a length, any vector may be represented by a directed line; and it will often be convenient to use language relating to vectors, which refers to them as thus represented.
Reversal of Direction, Scalar Multiplication and Division.
4. The negative sign ($-$) reverses the direction of a vector. (Sometimes the sign $+$ may be used to call attention to the fact that the vector has not the negative sign.)
Def.—A vector is said to be multiplied or divided by a scalar when its magnitude is multiplied or divided by the numerical value of the scalar and its direction is either unchanged or reversed according as the scalar is positive or negative. These operations are represented by the same methods as multiplication and division in algebra, and are to be regarded as substantially identical with them. The terms scalar multiplication and scalar division are used to denote multiplication and division by scalars, whether the quantity multiplied or divided is a scalar or a vector.
5. Def.—A unit vector is a vector of which the magnitude is unity.
Any vector may be regarded as the product of a positive scalar (the magnitude of the vector) and a unit vector.
The notation $\alpha _{0}$ may be used to denote the magnitude of the vector $\alpha$.
Addition and Subtraction of Vectors.
6. Def.—The sum of the vectors $\alpha ,\beta$, etc. (written $\alpha +\beta +{\text{etc.}}$) is the vector found by the following process. Assuming any point ${\text{A}}$, we determine successively the points ${\text{B, C}}$, etc., so that ${\overline {\text{AB}}}=\alpha ,{\overline {\text{BC}}}=\beta$, etc. The vector drawn from ${\text{A}}$ to the last point thus determined is the sum required. This is sometimes called the geometrical sum, to distinguish it from an algebraic sum or an arithmetical sum. It is also called the resultant, and $\alpha ,\beta$, etc. are called the components. ​When the vectors to be added are all parallel to the same straight line, geometrical addition reduces to algebraic; when they have all the same direction, geometrical addition like algebraic reduces to arithmetical. It may easily be shown that the value of a sum is not affected by changing the order of two consecutive terms, and therefore that it is not affected by any change in the order of the terms. Again, it is evident from the definition that the value of a sum is not altered by uniting any of its terms in brackets, as $\alpha +[\beta +\gamma ]+{\text{etc.,}}$ which is in effect to substitute the sum of the terms enclosed for the terms themselves among the vectors to be added. In other words, the conmiutative and associative principles of arithmetical and algebraic addition hold true of geometrical addition.
7. Def.— A vector is said to be subtracted when it is added after reversal of direction. This is indicated by the use of the sign $-$ instead of $+$.
8. It is easily shown that the distributive principle of arithmetical and algebraic multiplication applies to the multiplication of sums of vectors by scalars or sums of scalars, i.e.,
* ${\begin{aligned}(m+n+{\text{etc.}})[\alpha +\beta +{\text{etc.}}]&=m\alpha +n\beta &+{\text{etc.}}\\&+m\beta +n\beta &+{\text{etc.}}\\&&+{\text{etc.}}\end{aligned}}$
10. Linear relation of four vectors, Coordinates.—If $\alpha ,\beta$, and $\gamma$ are any given vectors not parallel to the same plane, any other vector $\rho$ may be expressed in the form
* $\rho =a\alpha +b\beta +c\gamma .$
$\alpha ,\beta$
$\gamma$
$a,b$
$c$
$\rho$
$\alpha ,\beta$
$\gamma$
$\rho ={\overline {\text{OP}}}$
$\alpha ,\beta ,\gamma$
$a,b$
$c$
${\text{P}}$
${\text{O}}$
$\alpha ,\beta$
$\gamma$
$\rho$
$\alpha ,\beta$
$\gamma$
11. Normal systems of unit vectors.—The letters $i,j,k$ are appropriated to the designation of a normal system of unit vectors, ie., three unit vectors, each of which is at right angles to the other two and determined in direction by them in a perfectly definite manner. We shall always suppose that $k$ is on the side of the $i{\text{-}}j$ plane on which a rotation from $i$ to $j$ (through one right angle) appears counter-clockwise. In other words, the directions of $i,j$, and $k$ are to be so determined that if they be turned (remaining rigidly connected with each other) so that $i$ points to the east, and $j$ to the north, $k$ will point upward. When rectangular axes of ${\text{X}},{\text{Y}}$, and ${\text{Z}}$ are employed, their directions will be conformed to a similar condition, and $i,j,k$ (when the contrary is not stated) will be supposed parallel to these axes respectively. We may have occasion to use more than one such system of unit vectors, just as we may use more than one system of coordinate axes. In such cases, the different systems may be distinguished by accents or otherwise.
12. Numerical computation of a geometrical sum,—If
* $\rho =a\alpha +b\beta +c\gamma ,$
* $\sigma =a'\alpha +b'\beta +c'\gamma ,$
* etc.,
* $\rho +\sigma +{\text{etc.}}=(a+a'+{\text{etc.}})\alpha +(b+b'+{\text{etc.}})\beta +(c+c'+{\text{etc.}})\gamma ,$
Direct and Skew Products of Vectors.
13. Def.—The direct product of $\alpha$ and $\beta$ (written $\alpha .\beta$) is the scalar quantity obtained by multiplying the product of their magnitudes by the cosine of the angle made by their directions.
14. Def—The skew product of $\alpha$ and $\beta$ (written $\alpha \times \beta$) is a vector function of $\alpha$ and $\beta$. Its magnitude is obtained by multiplying the product of the magnitudes of $\alpha$ and $\beta$ by the sine of the angle made by their directions. Its direction is at right angles to $\alpha$ and $\beta$, and on that side of the plane containing $\alpha$ and $\beta$ (supposed drawn from a common origin) on which a rotation from $\alpha$ to $\beta$ through an arc of less than 180° appears counter-clockwise.
$\alpha \times \beta$
$\alpha$
$\beta$
​Again, if $\alpha$ be directed toward the east, and $\beta$ lie in the same horizontal plane and on the north side of $\alpha$, $\alpha \times \beta$ will be directed upward.
15. It is evident from the preceding definitions that
* $\alpha .\beta =\beta .\alpha ,$and$\alpha \times \beta =\beta \times \alpha .$
16. Moreover,$[n\alpha ].\beta =\alpha .[n\beta ],$
and$[n\alpha ]\times \beta =\alpha \times [n\beta ].$
The brackets may therefore be omitted in such expressions.
17. From the definitions of No. 11 it appears that
$i.i=j.j=k.k=1,$ :
| $i.j=j.i=i.k=k.i=j.k=k.j=0,$ |||
| $i\times i=0,$ | $j\times j=0,$ | $k\times k=0,$ |
| $i\times j=k,$ | $j\times k=i,$ | $k\times i=j,$ |
| $j\times i=-k,$ | $k\times j=-i,$ | $i\times k=-j.$ |
18. If we resolve $\beta$ into two components $\beta '$ and $\beta ''$, of which the first is parallel and the second perpendicular to $\alpha$, we shall have
* $\alpha .\beta =\alpha .\beta '$and$\alpha \times \beta =\alpha \times \beta ''.$
19. $\alpha .[\beta +\gamma ]=\alpha .\beta +\alpha .\gamma$ and $\alpha \times [\beta +\gamma ]=\alpha \times \beta +\alpha \times \gamma .$
To prove this, let $\sigma =\beta +\gamma$, and resolve each of the vectors $\alpha ,\beta ,\sigma$ into two components, one parallel and the other perpendicular to $\alpha$. Let these be $\beta ',\beta '',\gamma ',\gamma '',\sigma ',\sigma ''.$ Then the equations to be proved will reduce by the last section to
* $\alpha .\sigma '=\alpha .\beta '+\alpha .\gamma '$and$\alpha \times \sigma ''=\alpha \times \beta ''+\alpha \times \gamma ''.$
$\sigma =\beta +\gamma$
$\beta ,\gamma$
$\sigma .$
$\alpha$
$\beta '',\gamma '',$
$\sigma '',$
$\sigma ''=\beta ''+\gamma ''.$
$\beta ',\gamma ',\sigma '.$
$\sigma '=\beta '+\gamma '.$
$\alpha .\sigma '=\alpha .\beta '+\alpha .\gamma ',$
$\alpha$
$\alpha \times \sigma ''=\alpha \times \beta ''+\alpha \times \gamma '',$
$\alpha .$
20. Hence,
* $[\alpha +\beta ].\gamma =\alpha .\gamma +\beta .\gamma ,$$[\alpha +\beta ]\times \gamma =\alpha \times \gamma +\beta \times \gamma ,$ $[\alpha +\beta ].[\gamma +\delta ]=\alpha .\gamma +\alpha .\delta +\beta .\gamma +\beta .\delta ,$ $[\alpha +\beta ]\times [\gamma +\delta ]=\alpha \times \gamma +\alpha \times \delta +\beta \times \gamma +\beta \times \delta ;$
21. Numerical calculation of direct and skew products.—The properties demonstrated in the last two paragraphs (which may be briefly expressed by saying that the operations of direct and skew multiplication are distributive) afford the rule for the numerical calculation of a direct product, or of the components of a skew product, when the rectangular components of the factors are given numerically. In fact, if
* $\alpha =xi+yj+zk,$and$\beta =x'i+y'j+z'k;$
* $\alpha .\beta =xx'+yy'+zz',$
$\alpha \times \beta =(yz'-zy')i+(zx'-xz')j+(xy'-yx')k.$
22. Representation of the area of a parallelogram by a skew product.—It will be easily seen that $\alpha \times \beta$ represents in magnitude the area of the parallelogram of which $\alpha$ and $\beta$ (supposed drawn from a common origin) are the sides, and that it represents in direction the normal to the plane of the parallelogram on the side on which the rotation from $\alpha$ toward $\beta$ appears coiuter-clockwise.
23. Representation of the volume of a parallelopiped by a triple product.—It will also be seen that $\alpha \times \beta .\gamma$ represents in numerical value the volume of the parallelopiped of which $\alpha ,\beta$, and $\gamma$ (supposed drawn from a common origin) are the edges, and that the value of the expression is positive or negative according as $\gamma$ lies on the side of the plane of $\alpha$ and $\beta$ on which the rotation from $\alpha$ to $\beta$ appears counter-clockwise, or on the opposite side.
24. Hence,
* ${\begin{aligned}\alpha \times \beta .\gamma &=\gamma \times \alpha .\beta =\gamma .\alpha \times \beta =\alpha .\beta \times \gamma \\&=\beta .\gamma \times \alpha =-\beta \times \alpha .\gamma =-\gamma \times \beta .\alpha =-\alpha \times \gamma .\beta \\&=-\gamma .\beta \times \alpha =-\alpha .\gamma \times \beta =-\beta .\alpha \times \gamma .\end{aligned}}$
This kind of product may be called the scalar product of the three vectors. There are two other kinds of products of three vectors, both of which are vectors, viz., products of the type $(\alpha .\beta )\gamma$ or $\gamma (\alpha .\beta )$, and products of the type $\alpha \times [\beta \times \gamma ]$ or $[\gamma \times \beta ]\times \alpha$.
25. $i.j\times k=j.k\times i=k.i\times j=1.$$i.k\times j=k.j\times i=j.i\times k=-1.$
From these equations, which follow immediately from those of No. 17, the propositions of the last section might have been derived, viz., by substituting for $\alpha ,\beta$, and $\gamma$, respectively, expressions of the form $xi+yj+zk,x'i+y'j+z'k$, and $x''i+y''j+z''k$. Such a method, which may be called expansion of terms of $i,j$, and $k$, will on many occasions afford very simple, although perhaps lengthy, demonstrations.
26. Triple products containing only two different letters.—The significance and the relations of $(\alpha .\alpha )\beta ,(\alpha .\beta )\alpha$, and $\alpha \times [\alpha \times \beta ]$ will be most evident, if we consider $\beta$ as made up of two components, $\beta '$ and $\beta ''$, respectively parallel and perpendicular to $\alpha$. Then
| | $\beta$ | $=\beta '+\beta '',$ |
| | $(\alpha .\beta )\alpha$ | $=(\alpha .\beta ')\alpha =(\alpha .\alpha )\beta ',$ |
| | $\alpha \times [\alpha \times \beta ]$ | $=\alpha \times [\alpha \times \beta '']=-(\alpha .\alpha )\beta ''.$ |
| Hence, | $\alpha \times [\alpha \times \beta ]$ | $=(\alpha .\beta )\alpha -(\alpha .\alpha )\beta .$ |
27. General relation of the vector prodAicts of three factors.—In the triple product $\alpha \times [\beta \times \gamma ]$ we may set
* $\alpha =l\beta +m\gamma +n\beta \times \gamma ,$
$\beta$
$\gamma$
* ${\begin{aligned}\alpha \times [\beta \times \gamma ]&=l\beta \times [\beta \times \gamma ]+m\gamma \times [\beta \times \gamma ]\\&=l(\beta .\gamma )-l(\beta .\beta )\gamma -m(\gamma .\beta )\gamma +m(\gamma .\gamma )\beta \\&=(l\beta .\gamma +m\gamma .\gamma )\beta -(l\beta .\beta +m\gamma .\beta )\gamma .\end{aligned}}$
| But | $l\beta .\gamma +m\gamma .\gamma =\alpha .\gamma$,and$l\beta .\beta +m\gamma .\beta =\alpha .\beta .$ |
| Therefore | $\alpha \times [\beta \times \gamma ]=(\alpha .\gamma )\beta -\gamma (\beta .\alpha ),$ |
$\beta$
$\gamma$
* $[\gamma \times \beta ]\times \alpha =\beta (\gamma .\alpha )-\gamma (\beta .\alpha ).$
The student will easily prove the following identical equations, which, although of considerable importance, are here given principally as exercises in the application of the preceding formulæ.
29. $\alpha \times [\beta \times \gamma ]+\beta \times [\gamma \times \alpha ]+\gamma \times [\alpha \times \beta ]=0.$.
| 31. | $[\alpha \times \beta ].[\gamma \times \delta ]$ | $=(\alpha .\gamma )(\beta .\delta )-(\alpha .\delta )(\beta .\gamma ).$ |
| 32. | $[\alpha \times \beta ]\times [\gamma \times \delta ]$ | $=(\alpha .\gamma \times \delta )\beta -(\beta .\gamma \times \delta )\alpha =(\alpha .\beta \times \delta )\gamma -(\alpha .\beta \times \gamma )\delta .$ |
| 32. | $\alpha \times [\beta \times [\gamma \times \delta ]]$ | $=(\alpha .\gamma \times \delta )\beta -(\alpha .\beta )\gamma \times \delta =(\beta .\delta )\alpha \times \gamma -(\beta .\gamma )\alpha \times \delta .$ |
| 33. | $[\alpha \times \beta ].[\gamma \times \delta ]\times [\epsilon \times \zeta ]$ | $=(\alpha .\beta \times \delta )(\gamma .\epsilon \times \zeta )-(\alpha .\beta \times \gamma )(\delta .\epsilon \times \zeta )$ |
| | | $=(\alpha .\beta \times \epsilon )(\zeta .\gamma \times \delta )-(\alpha .\beta \times \zeta )(\epsilon .\gamma \times \delta )$ |
| | | $=(\gamma .\delta \times \alpha )(\beta .\epsilon \times \zeta )-(\gamma .\delta \times \beta )(\alpha .\epsilon \times \zeta ).$ |
| 34. | $[\alpha \times \beta ].[\gamma \times \delta ]\times [\gamma \times \alpha ]$ | $=(\alpha .\beta \times \gamma )^{2}.$ |
35. The student will also easily convince himself that a product formed of any number of letters (representing vectors) combined in any possible way by scalar, direct, and skew multiplications may be reduced by the principles of Nos. 24 and 27 to a sum of products, each of which consists of scalar factors of the forms $\alpha .\beta$ and $\alpha .\beta \times \gamma$, with a single vector factor of the form $\alpha$ or $\alpha \times \beta$, when the original product is a vector.
36. Elimination of scalars from vector equations.—It has already been observed that the elimination of vectors from equations of the form
* $a\alpha +b\beta +c\gamma +d\delta +{\text{etc.}}=0$
* $a\alpha .\lambda +b\beta .\lambda +c\gamma .\lambda +d\delta .\gamma +{\text{etc.}}=0,$
$\lambda$
$\alpha ,\beta ,\gamma ,\delta$
${\text{X-}},{\text{Y-}}$
${\text{Z-}}$
${\text{X-}},{\text{Y-}}$
${\text{Z-}}$
$\lambda$
$a,b,c,d,$
$b$
$c$
$\lambda$
$\beta$
$\gamma$
$\beta \times \gamma$
* $a\alpha .\beta \times \gamma +d\delta .\beta \times \gamma +{\text{etc.}}=0.$
$a,b$
$c$
| $\rho =a\alpha +b\beta +c\gamma ,$ | (1) |
$\rho$
$\beta \times \gamma ,\gamma \times \alpha$
$\alpha \times \beta$
| $\rho .\beta \times \gamma =a\alpha .\beta \times \gamma ,$$\rho .\gamma \times \alpha =b\beta .\gamma \times \alpha ,$$\rho .\alpha \times \beta =c\gamma .\alpha \times \beta ;$ | (2) |
| $a={\frac {\rho .\beta \times \gamma }{\alpha .\beta \times \gamma }},$$b={\frac {\rho .\gamma \times \alpha }{\alpha .\beta \times \gamma }},$$c={\frac {\rho .\alpha \times \beta }{\alpha .\beta \times \gamma }}\cdot$ | (3) |
| $(\alpha .\beta \times \gamma )\rho =(\rho .\beta \times \gamma )\alpha +(\rho .\gamma \times \alpha )\beta +(\rho .\alpha \times \beta )\gamma .$ | (4) |
| $(\alpha .\beta \times \gamma )\rho -(\beta .\gamma \times \rho )\alpha +(\gamma .\rho \times \alpha )\beta -(\rho .\alpha \times \beta )\gamma =0.$ | (5) |
$\rho$
$\alpha$
$\beta ,$
$\beta$
$\gamma$
$\gamma$
$\alpha$
| $\rho =e\beta \times \gamma +f\gamma \times \alpha +g\alpha \times \beta .$ | (6) |
$e,f,g$
$a$
$\beta$
$\gamma$
| $e={\frac {\rho .\alpha }{\beta .\gamma \times \alpha }},$$f={\frac {\rho .\beta }{\gamma .\alpha \times \beta }},$$g={\frac {\rho .\gamma }{\alpha .\beta \times \gamma }}\cdot$ | (7) |
| $(\alpha .\beta \times \gamma )\rho =(\rho .\alpha )\beta \times \gamma +(\rho .\beta )\gamma \times \alpha +(\rho .\gamma )\alpha \times \beta .$ | (8) |
38. Reciprocal systems of vectors.— The results of the preceding section may be more compactly expressed if we use the abbreviations
| $\alpha '={\frac {\beta \times \gamma }{\alpha .\beta \times \gamma }},$$\beta '={\frac {\gamma \times \alpha }{\beta .\gamma \times \alpha }},$ $\gamma '={\frac {\alpha \times \beta }{\gamma .\alpha \times \beta }}\cdot$ | (1) |
| $\rho =(\rho .\alpha ')\alpha +(\rho .\beta ')\beta +(\rho .\gamma ')\gamma ,$ | (2) |
| $\rho =(\rho .\alpha )\alpha '+(\rho .\beta )\beta '+(\rho .\gamma )\gamma '.$ | (3) |
$\alpha ,\beta ,\gamma$
$\alpha ',\beta ',\gamma '$
| $\alpha ={\frac {\beta '\times \gamma '}{\alpha '.\beta '\times \gamma '}},$$\beta ={\frac {\gamma '\times \alpha '}{\beta '.\gamma '\times \alpha '}},$$\gamma ={\frac {\alpha '\times \beta '}{\gamma '.\alpha '\times \beta '}}$ | (4) |
$\alpha ',\beta '$
$\gamma '$
Def.—It will be convenient to use the term reciprocal to designate these relations, i.e., we shall say that three vectors are reciprocals of three others, when they satisfy relations similar to those expressed in equations (1) or (4).
With this underatanding we may say:—
The coefficients by which any vector is expressed in terms of three other vectors are the direct products of that vector with the reciprocals of the three.
Among other relations which are satisfied by reciprocal systems of vectors are the following:
* $\alpha .\alpha '=\beta .\beta '=\gamma .\gamma '=1,$
| $\alpha .\beta =0,$$\alpha .\gamma '=0,$$\beta .\alpha '=0,$$\beta .\gamma '=0,$$\gamma .\alpha '=0,$$\gamma .\beta '=0.$ | (5) |
$\alpha ,\beta ,\gamma$
$\alpha ',\beta ',\gamma '$
| $(\alpha .\beta \times \gamma )(\alpha '.\beta '\times \gamma ')=1.$ | (6) |
| $\alpha \times \alpha '+\beta \times \beta '+\gamma \times \gamma '=0.$ | (7) |
A system of three mutually perpendicular unit vectors is reciprocal to itself, and only such a system. The identical equation
| $\rho =(\rho .i)i+(\rho .j)j+(\rho .k)k$ | (8) |
The system reciprocal to $\alpha \times \beta ,\beta \times \gamma ,\gamma \times \alpha$ is
* $\alpha '\times \beta ',$$\beta '\times \gamma ',$$\gamma '\times \alpha ',$
* ${\frac {\alpha }{\alpha .\beta \times \gamma }},$${\frac {\beta }{\alpha .\beta \times \gamma }},$${\frac {\gamma }{\alpha .\beta \times \gamma }}\cdot$
$\sigma \times \tau$
* $(\alpha .\beta \times \gamma )(\rho .\sigma \times \tau )=\alpha .\rho (\beta .\sigma \,\gamma .\tau -\beta .\tau \,\gamma .\sigma )+\beta .\rho (\gamma .\sigma \,\alpha .\tau -\gamma .\tau \,\alpha .\sigma )+\gamma .\rho (\alpha .\sigma \,\beta .\tau -\alpha .\tau \,\beta .\sigma ),$
* $(\alpha .\beta \times \gamma )\sigma \times \tau =\alpha (\beta .\sigma \,\gamma .\tau -\beta .\tau \,\gamma .\sigma )+\beta (\gamma .\sigma \,\alpha .\tau -\gamma .\tau \,\alpha .\sigma )+\gamma (\alpha .\sigma \,\beta .\tau -\alpha .\tau \,\beta .\sigma )$
From these transformations, with those already given, it follows that a product formed of any number of letters (representing vectors and scalars), combined in any possible way by scalar, direct, and skew multiplications, may be reduced to a sum of products, containing each the sign $\times$ once and only once, when the original product contains it an odd number of times, or entirely free from the sign, when the original product contains it an even number of times.
39. Scalar equations of the first degree with respect to an unknown vector.—It is easily shown that any scalar equation of the first degree with respect to an unknown vector $\rho$, in which all the other quantities are known, may be reduced to the form
* $\rho .\alpha =a,$
$\alpha$
$a$
$\rho$
$\rho$
When we have four scalar equations of the first degree with respect to $\rho$, the elimination may be performed most symmetrically by substituting the values of $\rho .\alpha$, etc., in the equation
* $(\rho .\alpha )(\beta .\gamma \times \delta )-(\rho .\beta )(\gamma .\delta \times \alpha )+(\rho .\gamma )(\delta .\alpha \times \beta )-(\rho .\delta )(\alpha .\beta \times \gamma )=0,$
$\delta$
$\delta$
$\rho$
$\rho$
40. Solution of a vector equation of the first degree with respect to the unknown vector.—It is now easy to solve an equation of the form
| $\delta =\alpha (\lambda .\rho )+\beta (\mu .\rho )+\gamma (\nu .\rho ),$ | (1) |
$\alpha ,\beta ,\gamma ,\delta ,\lambda ,\mu$
$\nu$
$\beta \times \gamma$
$\gamma \times \alpha$
$\alpha \times \beta$
* $\beta .\gamma \times \delta =(\beta .\gamma \times \alpha )(\lambda .\rho ),$$\gamma .\alpha \times \delta =(\gamma .\alpha \times \beta )(\mu .\rho ),$
* $\alpha .\beta \times \delta =(\alpha .\beta \times \delta )(\nu .\rho );$
| or | $\alpha '.\delta =\lambda .\rho ,$$\beta '.\delta =\mu .\rho ,$$\gamma '.\delta =\nu .\rho ,$ |
$\alpha ',\beta ',\gamma '$
$\alpha ,\beta ,\gamma$
* $\rho =\lambda '(\lambda .\rho )+\mu '(\mu .\rho )+\nu '(\nu .\rho ),$
$\lambda ',\mu ',\nu '$
$\lambda ,\mu ,\nu$
| $\rho =\lambda '(\alpha '.\delta )+\mu '(\beta '.\delta )+\nu '(\gamma '.\delta ),$ | (2) |
​It results from the principle stated in No. 85, that any vector equation of the first degree with respect to $\rho$ may be reduced to the form
| | $\delta =$ | $\alpha (\lambda .\rho )+\beta (\mu .\rho )+\gamma (\nu .\rho )+a\rho +\epsilon \times \rho .$ |
| But | $a\rho =$ | $a\lambda '(\lambda .\rho )+a\mu '(\mu .\rho )+a\nu '(\nu .\rho ),$ |
| and | $\epsilon \times \rho =$ | $\epsilon \times \lambda '(\lambda .\rho )+\epsilon \times \mu '(\mu .\rho )+\epsilon \times \nu '(\nu .\rho ),$ |
where $\lambda ',\mu ',\nu '$ represent, as before, the reciprocals of $\lambda ,\mu ,\nu$. By substitution of these values the equation is reduced to the form of equation (1), which may therefore be regarded as the most general form of a vector equation of the first degree with respect to $\rho$.
41. Relations between two normal systems of unit vectors.—If $i,j,k$, and $i',j',k'$ are two normal systems of unit vectors, we have
| | $i'=(i.i')i+(j.i')j+(k.i')k,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$(1) |
| | $j'=(i.j')i+(j.j')j+(k.j')k,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$(1) |
| | $k'=(i.k')i+(j.k')j+(k.k')k,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$(1) |
| and | $j=(i.i')i'+(i.j')j'+(i.k')k',$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$(2) |
| | $j=(j.i')i'+(j.j')j+(j.j')k',$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$(2) |
| | $k'=(k.i')i'+(k.j')j'+(k.k')k',$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$(2) |
(See equation (8) of No. 38.)
The nine coefficients in these equations are evidently the cosines of the nine angles made by a vector of one system with a vector of the other system. The principal relations of these cosines are easily deduced. By direct multiplication of each of the preceding equations with itself, we obtain siz equations of the type
| $(i.i')^{2}+(j.i')^{2}+(k.i')^{2}=1.$ | (3) |
| $(i.i')(i.j')+(j.i')(j.j')+(k.i')(k.j')=0.$ | (4) |
* ${\begin{aligned}k'=\{(j.i')(k.j')-(k.i')(j.j')\}i+\{(k.i')(i.j')&-(i.i')(k.j')\}j\\&+\{(i.i')(j.j')-(j.i')(i.j')\}k.\end{aligned}}$
| $i.k'=(j.i')(k.j')-(k.i')(j.j').$ | (5) |
| ${\begin{aligned}(i.i')(j&.j')(k.k')+(i.j')(j.k')(k.i')+(i.k')(j.i')(k.j')\\&-(k.i')(j.j')(i.k')-(k.j')(j.k')(i.i')-(k.k')(j.i')(i.j')=1.\end{aligned}}$ | (6) |
| $(i.j')k'-(i.k')j'=(k.i')j-(j.i')k.$ | (7) |
CHAPTER II.
concerning the differential and integral calculus of vectors.
42. Differentials of vectors.—The differential of a vector is the geometrical difference of two values of that vector which differ infinitely little. It is itself a vector, and may make any angle with the vector differentiated. It is expressed by the same sign ($d$) as the differentials of ordinary analysis.
With reference to any fixed axes, the components of the differential of a vector are manifestly equal to the differentials of the components of the vector, i.e., if $\alpha ,\beta$, and $\gamma$ are fixed unit vectors, and
* $\rho =x\alpha +y\beta +z\gamma ,$
* $d\rho =dx\,\alpha +dy\,\beta +dz\,\gamma .$
44. Differential of a product.—The differential of a product of any kind due to the variation of a single factor is obtained by prefixing the sign of differentiation to that factor in the product. This is evidently true of differentials, since it will hold true even of finite differences.
45. From these principles we obtain the following identical equations:
| $d(\alpha +\beta )=d\alpha +d\beta ,$ | (1) |
| $d(n\alpha )=dn\,\alpha +n\,d\alpha ,$ | (2) |
| $d(\alpha .\beta )=d\alpha .\beta +\alpha .d\beta ,$ | (3) |
| $d[\alpha \times \beta ]=d\alpha \times \beta +\alpha \times d\beta ,$ | (4) |
| $d(\alpha .\beta \times \gamma )=d\alpha .\beta \times \gamma +\alpha .d\beta \times \gamma +\alpha .\beta \times d\gamma ,$ | (5) |
| $d[(\alpha .\beta )\gamma ]=(d\alpha .\beta )\gamma +(\alpha .d\beta )\gamma +(\alpha .\beta )d\gamma .$ | (6) |
If we suppose the quantities occurring in the six equations of the last section to be functions of a scalar $t$, we may substitute ${\frac {d}{dt}}$ for $d$ in those equations since this is only to divide all terms by the scalar $dt$.
47. Successive differentiations.—The differential coefficient of a vector with respect to a scalar is of course a finite vector, of which we may take the differential, or the differential coefficient with respect to the same or any other scalar. We thus obtain differential coefficients of the higher orders, which are indicated as in the scalar calculus.
A few examples will serve for illustration.
If $\rho$ is the vector drawn from a fixed origin to a moving point at any time $t,{\frac {d\rho }{dt}}$ will be the vector representing the velocity of the point, and ${\frac {d^{2}\rho }{dt^{2}}}$ the vector representing its acceleration.
If $\rho$ is the vector drawn from a fixed origin to any point on a curve, and $s$ the distance of that point measured on the curve from any fixed point, ${\frac {d\rho }{ds}}$ is a unit vector, tangent to the curve and having the direction in which $s$ increases; ${\frac {d^{2}\rho }{ds^{2}}}$ is a vector directed from a point on the curve to the center of curvature, and equal to the curvature; ${\frac {d\rho }{ds}}\times {\frac {d^{2}\rho }{ds^{2}}}$ is the normal to the osculating plane, directed to the side on which the curve appears described counter-clockwise about the center of curvature, and equal to the curvature. The tortuosity (or rate of rotation of the osculating plane, considered as positive when the rotation appears counter-clockwise as seen from the direction in which $s$ increases) is represented by
* ${\frac {{\frac {d\rho }{ds}}\cdot {\frac {d^{2}\rho }{ds^{2}}}\times {\frac {d^{3}\rho }{ds^{3}}}}{{\frac {d^{2}\rho }{ds^{2}}}\cdot {\frac {d^{2}\rho }{ds^{2}}}}}\cdot$
48. Integration of an equation between differentials.—If $t$ and $u$ are two single-valued continuous scalar functions of any number of scalar or vector variables, and
| | $dt=$ | $du,$ |
| then | $t=$ | $u+a,$ |
$a$
​Or, if $\tau$ and $\omega$ are two single-valued continuous vector functions of any number of scalar or vector variables, and
| | $d\tau =$ | $d\omega ,$ |
| then | $\tau =$ | $\omega +\alpha ,$ |
$\alpha$
When the above hypotheses are not satisfied in general, but will be satisfied if the variations of the independent variables are confined within certain limits, then the conclusions will hold within those limits, provided that we can pass by continuous variation of the independent variables from any values within the limits to any other values within them, without transgressing the limits.
49. So far, it will be observed, all operations have been entirely analogous to those of the ordinary calculus.
Functions of Position in Space.
60. Def.—If $u$ is any scalar function of position in space (ie., any scalar quantity having continuously varying values in space), $\nabla u$ is the vector function of position in space which has everywhere the direction of the most rapid increase of $u$, and a magnitude equal to the rate of that increase per unit of length. $\nabla u$ may be called the derivative of $u$, and $u$, the primitive of $\nabla u.$
We may also take any one of the Nos. 51, 52, 58 for the definition of $\nabla u.$
51. If $\rho$ is the vector defining the position of a point in space,
* $du=\nabla u.d\rho .$
| 52. | $\nabla u=i{\frac {du}{dx}}+j{\frac {du}{dy}}+k{\frac {du}{dz}}\cdot$ |
| 53. | ${\frac {du}{dx}}=i.\nabla u,$${\frac {du}{dy}}=j.\nabla u,$${\frac {du}{dz}}=k.\nabla u.$ |
54. Def.—If $\omega$ is a vector having continuously varying values in space,
| | $\nabla .\,\omega =$ | $i.{\frac {d\omega }{dx}}+j.{\frac {d\omega }{dy}}+k.{\frac {d\omega }{dz}},$ | (1) |
| and | $\nabla \times \omega =$ | $i\times {\frac {d\omega }{dx}}+j\times {\frac {d\omega }{dy}}+k\times {\frac {d\omega }{dz}}\cdot$ | (2) |
$\nabla .\,\omega$ is called the divergence of $\omega$ and $\nabla \times \omega$ its curl.
If we set
* $\omega ={\text{X}}i+{\text{Y}}j+{\text{Z}}k,$
* $\nabla .\,\omega ={\frac {d{\text{X}}}{dx}}+{\frac {d{\text{Y}}}{dy}}+{\frac {d{\text{Z}}}{dz}}$
| and | $\nabla \times \omega =i\left({\frac {d{\text{Z}}}{dy}}-{\frac {d{\text{Y}}}{dz}}\right)+j\left({\frac {d{\text{X}}}{dz}}-{\frac {d{\text{Z}}}{dx}}\right)+k\left({\frac {d{\text{Y}}}{dx}}-{\frac {d{\text{X}}}{dy}}\right),$ |
$\nabla .\,\omega$
$\nabla \times \omega .$
​55. Surface integrals.—The integral $\iint \omega .d\sigma ,$ in which $d\sigma$ represents an element of some surface, is called the surface-integral of $\omega$ for that surface. It is understood here and elsewhere, when a vector is said to represent a plane surface (or an element of surface which may be regarded as plane), that the magnitude of the vector represents the area of the surface, and that the direction of the vector represents that of the normal drawn toward the positive side of the surface. When the surface is defined as the boundary of a certain space, the outside of the surface is regarded as positive.
The surface-integral of any given space (ie., the surface-integral of the surface bounding that space) is evidently equal to the sum of the surface-integrals of all the parts into which the original space may be divided. For the integrals relating to the surfaces dividing the parts will evidently cancel in such a sum.
The surface-integral of $\omega$ for a closed surface bounding a space $dv$ infinitely small in all its dimensions is
* $\nabla .\,\omega \,dv.$
$\nabla \omega ,$
$i,j,k.$
$dv$
$\nabla .\,\omega dv.$
$\nabla .\,\omega$
It is possible to attribute such a physical signification to the quantities concerned in the above proposition, as shall make it evident almost without demonstration. Let us suppose $\omega$ to represent a flux of any substance. The rate of decrease of the density of that substance at any point will be obtained by dividing the surface-integral of the flux for any infinitely small closed surface about the point by the volume enclosed. This quotient must therefore be independent of the form of the surface. We may define $\nabla .\,\omega$ as representing that quotient, and then obtain equation (1) of No. 54 by applying the general principle to the case of the rectangular parallelopiped.
$\iint d\sigma \times \omega$
$\omega .$
$dv$
$\nabla \times \omega dv,$
​57. Integration.—If $dv$ represents an element of any space, and $d\sigma$ an element of the bounding surface,
* $\iiint \nabla .\omega dv=\iint \omega .d\sigma .$
The principle may also be expressed as follows:
The surface-integral of any vector function of position in space for a closed surface is equal to the volume-integral of the divergence of that function for the space enclosed.
58. Line-integrals.—The integral $\int \omega .d\rho ,$ in which $d\rho$ denotes the element of a line, is called the line-integral of $\omega$ for that line. It is implied that one of the directions of the line is distinguished as positive. When the line is regarded as bounding a surface, that side of the surface will always be regarded as positive, on which the surface appears to be circumscribed counter-clockwise.
59. Integration.—From No. 51 we obtain directly
* $\int \nabla u.d\rho =u''-u',$
In other words,—The line-integral of the derivative of any (continuous and single-valued) scalar function of position in space is equal to the difference of the values of the function at the extremities of the line. For a closed line the integral vanishes.
60. Integration.—The following principle may be used to reduce double integrals of a certain form to simple integrals.
If $d\sigma$ represents an element of any surface, and dp an element of the bounding line,
* $\iint \nabla \omega \times \omega .d\sigma =\int \omega .d\rho .$
To prove this principle, we will consider the variation of the line-integral which is due to a variation in the closed line for which the integral is taken. We have, in the first place,
| | $\delta \int \omega .d\rho =$ | $\int \delta \omega .d\rho +\int \omega .\delta d\rho .$ |
| But | $\omega .\delta \,d\rho =$ | $d(\omega .\delta \rho )-d\omega .\delta \rho .$ |
Therefore, since $\int d(\omega .\delta \rho )=0$ for a closed line,
* $\delta \int \omega .d\rho =\int \delta \omega .d\rho -\int d\omega .\delta \rho .$
| Now | $\delta \omega =$ | $\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}\delta x\right]=$ | $\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}\delta (i.\delta \rho )\right],$ |
| and | $d\omega =$ | $\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}dx\right]=$ | $\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}\delta (i.d\rho )\right],$ |
where the summation relates to the coordinate axes and connected quantities. Substituting these values in the preceding equation,
* $\delta \int \omega .d\rho =\int \textstyle \sum \displaystyle \left((i.\delta \rho )\left({\frac {d\omega }{dx}}.d\rho \right)-(i.d\rho )\left({\frac {d\omega }{dx}}.\delta \rho \right)\right),$
* $\delta \int \omega .d\rho =\int \textstyle \sum \displaystyle \left[i\times {\frac {d\omega }{dx}}\right].[\delta \rho \times d\rho ]=\int \nabla \times \omega .[\delta \rho \times d\rho ].$
$\delta \rho \times d\rho$
$d\rho ,$
$\nabla \times \omega$
$\omega$
$\nabla \times \omega$
61. The line-integral of $\omega$ for a closed line bounding a plane surface $d\sigma$ infinitely small in all its dimensions is therefore
* $\nabla \times \omega .d\sigma .$
$\nabla \times \omega$
$\omega$
$\nabla \times \omega ,$
$\nabla \times \omega .$
$\nabla ,\nabla .,$ and $\nabla \times$ applied to Functions of Functions of Position.
62. A constant scalar factor after $\nabla ,\nabla .,$ or $\nabla \times$ may be placed before the symbol.
63. If $f(u)$ denotes any scalar function of $u$, and $f'(u)$ the derived function,
* $\nabla f(u)=f'(u)\nabla u.$
$u$
$\omega$
$\nabla u$
$\nabla .\omega$
$\nabla \times \omega$
65. By the use of this principle we easily derive the following identical equations:
| $\nabla (t+u)=\nabla t+\nabla u.$ | (1) |
| $\nabla .(\tau +\omega )=\nabla .\tau +\nabla .\omega .$$\nabla \times [\tau +\omega ]=\nabla \times \tau +\nabla \times \omega .$ | (2) |
| $\nabla (tu)=u\nabla t+t\nabla u.$ | (3) |
| $\nabla .(u\omega )=\omega .\nabla u+u\nabla .\omega .$ | (4) |
| $\nabla \times [u\omega ]=u\times \nabla \omega -\omega \nabla \times u.$ | (5) |
| $\nabla .[\tau \times \omega ]=\omega .\nabla \times \tau -\tau .\nabla \times \omega .$ | (6) |
$\nabla$
$i,j$
$k,$
Combinations of the Operators $\nabla ,\nabla .,$ and $\nabla \times .$
66. If $u$ is any scalar function of position in space,
* $\nabla \times \nabla u=0,$
67. Conversely, if $\omega$ is such a vector function of position in space that
* $\nabla \times \omega =0,$
$\omega$
$\int \omega .d\rho$
$\nabla \times \omega .$
$P'$
$P''$
$u$
$P''$
$P'$
$u$
$P'',$
$du=\omega .d\rho ,$
$\nabla u=\omega .$
​If the region for which $\nabla \times \omega =0$ is unlimited, these functions will be angle-valued. If the region is limited, but acyclic, the functions will still be single-valued and satisfy the condition $\nabla u=\omega$ within the same region. If the region is cyclic, we may determine functions satisfying the condition $\nabla u=\omega$ within the region, but they will not necessarily be single-valued. 68. If $\omega$ is any vector function of position in space, $\nabla .\nabla \times \omega =0.$ This may be deduced directly from the definitions of No. 54.
The converse of this proposition will be proved hereafter.
69. If $u$ is any scalar function of position in space, we have by Nos. 52 and 54
* $\nabla .\nabla u=\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)u.$
70. Def.—If $\omega$ is any vector function of position in space, we may define $\nabla .\nabla \omega$ by the equation
* $\nabla .\nabla \omega =\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)\omega ,$
$\nabla .\nabla$
$\nabla$
* $\omega =i{\text{X}}+j{\text{Y}}+k{\text{Z}},$ $\nabla .\nabla \omega =i\nabla .\nabla {\text{X}}+j\nabla .\nabla {\text{Y}}+k\nabla .\nabla {\text{Z}},$
$\nabla .\nabla$
71. From the above definition with those of Nos, 52 and 54 we may easily obtain
* $\nabla .\nabla \omega =\nabla \nabla .\omega -\nabla \times \nabla \times \omega .$
$\nabla .\nabla$
72. The expression $-{\tfrac {1}{6}}a^{2}\nabla .\nabla u$, where $a$ is any infinitesimal scalar, evidently represents the excess of the value of the scalar function $u$ ​at the point considered above the average of its values at six points at the following vector distances: $ai,-ai,aj,-aj,ak,-ak.$ Since the directions of $i,j,$ and $k$ are immaterial (provided that they are at right angles to each other), the excess of the value of $u$ at the central point above its average value in a spherical surface of radius a constructed about that point as the center will be represented by the same expression, $-{\tfrac {1}{6}}a^{2}\nabla .\nabla u.$
Precisely the same is true of a vector function, if it is understood that the additions and subtractions implied in the terms average and excess are geometrical additions and subtractions.
Maxwell has called $-\nabla .\nabla u$ the concentration of $u,$ whether $u$ is scalar or vector. We may call $\nabla .\nabla u$ (or $\nabla .\nabla \omega$), which is proportioned to the excess of the average value of the function in an infinitesimal spherical surface above the value at the center, the dispersion of $u$ (or $\omega$).
Transformation of Definite Integrals.
73. From the equations of No. 65, with the principles of integration of Nos. 57, 59, and 60, we may deduce various transformations of definite integrals, which are entirely analogous to those known in the scalar calculus under the name of integration by parts. The following formulæ (like those of Nos. 67, 59, and 60) are written for the case of continuous values of the quantities (scalar and vector) to which the signs $\nabla ,\nabla .,$ and $\nabla \times$ are applied. It is left to the student to complete the formulæ for cases of discontinuity in these values. The manner in which this is to be done may in each case be inferred from the nature of the formula itself. The most important discontinuities of scalars are those which occur at surfaces: in the case of vectors discontinuities at surfaces, at lines, and at points, should be considered.
74. From equation (3) we obtain
* $\int \nabla (tu).d\rho =t''u''-t'u'=\int u\nabla t.d\rho +\int t\nabla u.d\rho ,$
$\int u\nabla t.d\rho$
$-\int t\nabla u.d\rho$
76. From equation (5) we obtain
* $\iint \nabla \times (u\omega ).d\sigma =\int u\omega .d\rho =\iint u\nabla \times \omega .d\sigma -\iint \omega \times \nabla u.d\sigma ,$
From this, by substitution of $\nabla t$ for $\omega ,$ we may derive as a particular case
* $\iint \nabla u\times \nabla t.d\sigma =\int u\nabla t.d\rho =-\int t\nabla u.d\rho .$
* $\iiint \nabla .[u\omega ]dv=\iint u\omega -d\sigma =\iiint \omega .\nabla udv+\iiint u\nabla .\omega dv,$
From this, by substitution of $\nabla t$ for $\omega ,$ we derive as a particular case
* $\iiint \nabla t.\nabla udv=\iint u\nabla t.d\sigma -\iiint u\nabla .\nabla tdv=\iint t\nabla u.d\sigma -\iiint t\nabla .\nabla udv,$
$s\nabla t$
$\omega$
* ${\begin{aligned}\iiint s\nabla t.\nabla udv&=\iint us\,t.d\sigma -\iiint u\nabla .[s\nabla t]dv\\&=\iint ts\nabla u.d\sigma -\iiint t\nabla .[s\nabla u]dv.\end{aligned}}$
* $\iiint \nabla .[\tau \times \omega ]dv=\iint \tau \times \omega .d\sigma =\iiint \omega .\nabla \times \tau dv-\iiint \tau .\nabla \times \omega dv.$
* $\iiint \nabla u.\nabla \times \omega dv=\iint \omega \times \nabla u.d\sigma .$
Integration of Differential Equations.
78. If throughout any continuous space (or in all space)
* $\nabla u=0,$
* $u={\text{constant,}}$
* $\nabla u=0,$
* $\nabla u=0,$and$u={\text{constant.}}$
If $\nabla u=0$ in any finite part of the space, $u$ is constant in that part.
If $u$ is not constant throughout, let us imagine a sphere situated principally in the part in which $u$ is constant, but projecting slightly into a part in which $u$ has a greater value, or else into a part in which $u$ has a less. The surface-integral of $\nabla u$ for the part of the spherical surface in the region where $u$ is constant will have the value zero: for the other part of the surface, the integral will be either greater than zero, or less than zero. Therefore the whole surface-integral for the spherical surface will not have the value zero, which is required by the general condition, $\nabla .\nabla u=0.$
Again, if $\nabla u=0$ only in a surface in or bounding the space in ​which $\nabla .\nabla u=0,u$ will be constant in this surface, and the surface will be contiguous to a region in which $\nabla .\nabla u=0$ and $u$ has a greater value than in the surface, or else a less value than in the surface. Let us imagine a sphere lying principally on the other side of the surface, but projecting slightly into this region, and let us particularly consider the surface-integral of $\nabla u$ for the small segment cut off by the surface $\nabla u=0.$ The integral for that part of the surface of the segment which consists of part of the surface $\nabla u=0$ will have the value zero, the integral for the spherical part will have a value either greater than zero or else less than zero. Therefore the integral for the whole surface of the segment cannot have the value zero, which is demanded by the general condition, $\nabla .\nabla u=0.$
80. If throughout a certain space (which need not be continuous, and which may extend to infinity)
* $\nabla .\nabla u=0,$
* $u={\text{constant}}=a,$
$u=a,$
* $\nabla u=0,$and$u=a.$
$\nabla u$
$P$
$u$
$b$
$a,$
$u<b.$
$\nabla u$
$\nabla .\nabla u=0.$
$P$
$b>a$
$u$
This proposition may be generalized by substituting the condition $\nabla .[t\nabla u]=0$ for $\nabla .\nabla u=0,t$ denoting any positive (or any negative) scalar function of position in space. The conclusion would be the same, and the demonstration similar.
81. If throughout a certain space (which need not be continuous, and which may extend to infinity)
* $\nabla .\nabla u=0,$
$\nabla u$
$r^{2}{\frac {du}{dr}}=0,$
$r$
* $\nabla u=0,$
* $u={\text{constant.}}$
$\nabla u$
$P,$
$u$
$b.$
$P,$
$r.$
$u<b.$
$\nabla u$
$\nabla .\nabla u=0.$
$4\pi r^{2}\left({\frac {du}{dr}}\right)',$
$\left({\frac {du}{dr}}\right)'$
${\frac {du}{dr}}$
$r$
$u=b,$
$P$
This proposition also may be generalized by substituting $\nabla .[t\nabla u]=0$ for $\nabla .\nabla u=0,$ and $tr^{2}{\frac {du}{dr}}$ for $r^{2}{\frac {du}{dr}}=0.$
82. If throughout any continuous space (or in all space)
* $\nabla t=\nabla u,$
* $t=u+{\text{const.}}$
$t-u.$
83. If throughout any continuous space (or in all space)
* $\nabla .\nabla t=\nabla .\nabla u,$
* $\nabla t=\nabla u,$
* $\nabla t=\nabla u,$and$t=u+{\text{const.}}$
* $\nabla .\nabla t=\nabla .\nabla u,$
* $t=u,$
* $t=u,$
* $t=u.$
86. If throughout a certain space (which need not be continuous, and which may extend to infinity)
* $\nabla .\nabla t=\nabla .\nabla u,$
$\nabla t$
$\nabla u$
$r^{2}\left({\frac {dt}{dr}}-{\frac {du}{dr}}\right)=0,$
$r$
* $\nabla t=\nabla u,$
* $t-u={\text{constant.}}$
86. If throughout any continuous space (or in all space)
* $\nabla \times \tau =\nabla \times \omega$and$\nabla .\tau =\nabla .\omega ,$
* $\tau =\omega ,$
* $\tau =\omega .$
$\nabla \times (\tau -\omega )=0,$
$\nabla u=\tau -\omega ,$
$u$
$\nabla .\nabla u=0,$
$\nabla u=0.$
$\nabla u=\tau -\omega =0.$
87. If throughout an aperiphractic space contained within finite boundaries but not necessarily continuous
* $\nabla \times \tau =\nabla \times \omega$and$\nabla .\tau =\nabla .\omega ,$
$\tau$
$\omega$
* $\tau =\omega .$
$\nabla u=\tau -\omega ,$
$\nabla .\nabla u=0$
$u={\text{constant}}$
* $\nabla u=\tau -\omega =0.$
* $\nabla \times \tau =\nabla \times \omega$and$\nabla .\tau =\nabla .\omega ,$
$\tau$
$\omega$
* $\tau =\omega .$
$\nabla u=\tau -\omega ,$
$\nabla .\nabla u=0,$
$\nabla u$
$\nabla u=\tau -\omega =0.$
89. If throughout a certain space (which need not be continuoas, and which may extend to infinity)
* $\nabla .\nabla \tau =\nabla .\nabla \omega$
* $\tau =\omega ,$
* $\tau =\omega ,$
* $\tau =\omega .$
$\tau$
$\omega .$
Minimum values of the Volume-integral $\iiint u\omega .\omega dv.$
(Thomson's Theorems.)
90. Let it be required to determine for a certain space a vector function of position $\omega$ subject to certain conditions (to be specified hereafter), so that the volume-integral
* $\iiint u\omega .\omega dv$
$u$
$\omega$
$\nabla .\omega$
$\omega$
$\omega$
$\int \nabla .\omega dv.$
$\nabla \times (u\omega )=0,$
$u\omega$
​The existence of the minimum requires that
* $\iiint u\omega .\delta \omega dv=0,$
$\delta \omega$
* $\nabla .\delta \omega =0,$
$u\omega$
$dz,$
$\nabla .\delta \omega =0$
$\delta \omega =0$
$\delta \omega \,dz=\delta a{\frac {d\rho }{ds}}$
$\delta a$
$\rho$
$ds$
* $\iiint u\omega .\delta \omega \,dv=\int u\omega .\delta \omega \,dz\,ds=\int u\omega .d\rho \,\delta a=\delta a\int u\omega .d\rho =0,$
$\int u\omega .d\rho =0.$
$u\omega$
If for certain parts of the surface the normal component of $\omega$ is not given for each point, but only the surface-integral of $\omega$ for each such part, then the above reasoning will apply not only to closed curves, but also to curves commencing and ending in such a part of the surface. The primitive of $u\omega$ will then have a constant value in each such part.
If the space extends to infinity and there is no special condition respecting the value of $\omega$ at infinite distances, the primitive of $u\omega$ will have a constant value at infinite distances within the space or within each separate continuous part of it.
If we except those cases in which the problem has no definite meaning because the data are such that the integral $\int u\omega .\omega \,dv$ must be infinite, it is evident that a minimum must always exist, and (on account of the quadratic form of the integral) that it is unique. That the conditions just found are sufficient to insure this minimum, is evident from the consideration that any allowable values of $\delta \omega$ may be made up of such values as we have supposed. Therefore, there will be one and only one vector function of position in space which satisfies these conditions together with those enumerated at the banning of this number.
b. In the second place, let the vector $\omega$ be subject to the conditions that $\nabla \times \omega$ is given throughout the space, and that the tangential component of $\omega$ is given at the bounding surface. The solution is that
* $\nabla .[u\omega ]=0,$
$u\omega$
The existence of the minimum requires that
* $\iiint u\omega .\delta \omega \,dv=0,$
$\delta \omega$
* $\nabla \times \delta \omega =0,$
$\delta \omega$
* $\delta \omega =\nabla \delta q,$
$\delta q$
* $\iiint u\omega .\nabla \,\delta q\,dv=0,$
* $\iint u\omega .d\sigma \,\delta q-\iiint \nabla .[u\omega ]\delta q\,dv=0.$
$\delta q$
* $\nabla .[u\omega ]=0;$
$\delta q$
* $\iint u\omega .d\sigma =0.$
Potentials, Newtonians, Laplacians.
91. Def.—If $u'$ is the scalar quantity of something situated at a certain point $\rho ',$ the potential of $u'$ for any point $\rho$ is a scalar function of $\rho ,$ defined by the equation
* ${\text{pot }}u'={\frac {u'}{[\rho '-\rho ]_{0}}},$
$u'$
$\rho$
$\rho$
* ${\text{new }}u'={\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}u'.$
$\omega '$
$\rho ',$
$\omega '$
$\rho$
$\rho$
* ${\begin{aligned}{\text{pot }}\omega '&={\frac {\omega '}{[\rho '-\rho ]_{0}}},\\{\text{lap }}\omega '&={\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\times \omega '.\end{aligned}}$
$u$
$\omega$
${\text{Pot }}u,{\text{New }}u,{\text{Pot }}\omega ,{\text{Lap }}\omega$
${\text{pot }}u',$
$\rho ';$
| ${\text{Pot }}u$ | $=\iiint {\text{pot }}u'dv'$ | $=\iiint {\frac {u'}{[\rho '-\rho ]_{0}}}dv',$ |
| ${\text{New }}u$ | $=\iiint {\text{new }}u'dv'$ | $=\iiint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}u'\,dv',$ |
| ${\text{Pot }}\omega$ | $=\iiint {\text{pot }}\omega 'dv'$ | $=\iiint {\frac {\omega '}{[\rho '-\rho ]_{0}}}dv',$ |
| ${\text{Lap }}\omega$ | $=\iiint {\text{lap }}\omega 'dv'$ | $=\iiint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\times \omega '\,dv',$ |
$\rho$
$u'$
$\omega '$
${\frac {d{\text{ Pot }}u}{dx}}={\text{Pot }}{\frac {du}{dx}},$
${\frac {d{\text{ Pot }}\omega }{dx}}={\text{Pot }}{\frac {d\omega }{dx}}\cdot$
$x$
$dv'$
${\text{Pot}}$
Since we may substitute $y$ and $z$ for $x$ in the preceding formula, and since a constant factor of any kind may be introduced under the sign of integration, we have
* ${\begin{aligned}\nabla {\text{ Pot }}u&={\text{ Pot }}\nabla u,\\\nabla .{\text{ Pot }}\omega &={\text{ Pot }}\nabla .\omega ,\\\nabla \times {\text{ Pot }}\omega &={\text{ Pot }}\times \omega ,\\\nabla .\nabla {\text{ Pot }}u&={\text{ Pot }}\nabla .\nabla u,\\\nabla .\nabla {\text{ Pot }}\omega &={\text{ Pot }}\nabla .\nabla \omega ,\end{aligned}}$
$\nabla ,\nabla .,\nabla \times ,\nabla .\nabla$
${\text{Pot}}.$
Yet a certain restriction is to be observed. When the operation of taking the (integral) potential does not give a definite finite value, the first members of these equations are to be regarded as entirely indeterminate, but the second members may have perfectly definite values. This would be the case, for example, if $u$ or $\omega$ had a constant value throughout all space. It might seem harmless to set an indefinite expression equal to a definite, but it would be dangerous, since ​we might with equal right set the indefinite expression equal to other definite expressions, and then be misled into supposing these definite expressions to be equal to one another. It will be safe to say that the above equations will hold, provided that the potential of $u$ or $\omega$ has a definite value. It will be observed that whenever ${\text{Pot }}u$ or ${\text{Pot }}\omega$ has a definite value in general (i.e., with the possible exception of certain points, lines, and surfaces), the first members of all these equations will have definite values in general, and therefore the second members of the equation, being necessarily equal to the first members, when these have definite values, will also have definite values in general. 94. Again, whenever Potu has a definite value we may write
* $\nabla {\text{Pot }}u=\nabla \iiint {\frac {u'}{r}}dv'=\iiint \nabla {\frac {1}{r}}u'\,dv',$
$r$
$[\rho '-\rho ]_{0}.$
* $\nabla {\frac {1}{r}}={\frac {\rho '-\rho }{r^{3}}},$
$\nabla {\text{Pot }}u={\text{New }}u.$
${\text{New }}u$
${\text{Pot }}u$
95. In like manner, whenever ${\text{Pot }}\omega$ has a definite value,
* $\nabla \times {\text{Pot }}\omega =\nabla \times \iiint {\frac {\omega '}{r}}dv'=\iiint \nabla \times {\frac {\omega '}{r}}dv'=\iiint \nabla {\frac {1}{r}}\times \omega '\,dv'.$
$\nabla {\frac {1}{r}}$
* $\nabla \times {\text{Pot }}\omega ={\text{Lap }}\omega .$
${\text{Lap }}\omega$
${\text{Pot }}\omega$
96. Hence, with the aid of No. 93, we obtain
* ${\begin{aligned}\nabla \times {\text{Lap }}\omega &={\text{Lap }}\nabla \times \omega ,\\\nabla .{\text{Lap }}\omega &=0,\end{aligned}}$
${\text{Pot }}\omega$
97. By the method of No. 93 we obtain
* $\nabla .{\text{New }}u=\nabla .\iiint {\frac {\rho '-\rho }{r^{3}}}u'\,dv'=\iiint \nabla u'.{\frac {\rho '-\rho }{r^{3}}}dv'.$
$\rho ,$
$r$
$\rho ',$
* $dv'=r^{2}\,dq\,dr,$
$r^{2}dq$
$\rho$
$r.$
* $\nabla u'.{\frac {\rho '-\rho }{r}}={\frac {du'}{dr}}\cdot$
* $\nabla .{\text{New }}u=\iiint {\frac {du'}{dr}}dq\,dr=4\pi \int {\frac {d{\bar {u}}'}{dr}}dr=4\pi {\bar {u}}'_{r=\infty }-4\pi {\bar {u}}'_{r=0},$
${\bar {u}}$
$u$
$r$
$\rho$
Now if ${\text{Pot }}u$ has in general a definite value, we must have ${\bar {u}}'=0$ for $r=\infty .$ Also, $\nabla .{\text{New }}u$ will have in general a definite value. For $r=0,$ the value of ${\bar {u}}'$ is evidently $u.$ We have, therefore,
| $\nabla .{\text{New }}u$ | $=-4\pi u,$ |
| $\nabla .\nabla {\text{Pot }}u$ | $=-4\pi u.$ |
${\text{Pot }}\omega$
* ${\begin{aligned}\nabla .\nabla {\text{Pot }}\omega &=\nabla .\nabla {\text{Pot }}[ui+vj+wk]\\&=\nabla .\nabla {\text{Pot }}ui+\nabla .\nabla {\text{Pot }}vj+\nabla .\nabla {\text{Pot }}wk\\&=-4\pi ui-4\pi vj-4\pi wk\\&=-4\pi \omega .\end{aligned}}$
* $\nabla \times \nabla \times {\text{Pot }}\omega -\nabla \nabla .{\text{Pot }}\omega =4\pi \omega .$
${\text{Lap }}\nabla \times \omega -{\text{New }}\nabla .\omega =4\pi \omega .$
* $\omega _{1}={\frac {1}{4\pi }}{\text{Lap }}\nabla \times \omega ,$$\omega _{2}={\frac {-1}{4\pi }}{\text{New }}\nabla .\omega ,$
$\omega =\omega _{1}+\omega _{2},$
$\omega _{1}$
$\omega _{2}$
$\nabla .\omega _{1}=0,$
$\nabla \times \omega _{2}=0.$
$\omega _{1}$
$\omega _{2}$
${\text{Pot }}\omega _{1}$
${\text{Pot }}\omega _{2},$
${\text{Pot }}\omega ,$
It is worth while to notice that there is only one way in which a vector function of position in space having a definite potential can be thus divided into solenoidal and irrotational parts having definite potentials. For if $\omega _{1}+\epsilon ,\,\omega _{2}-\epsilon$ are two other such parts,
* $\nabla .\epsilon =0$and$\nabla \times \epsilon =0.$
${\text{Pot }}\epsilon$
$\epsilon ={\frac {1}{4\pi }}{\text{Lap }}\nabla \times \epsilon -{\frac {1}{4\pi }}{\text{New }}\nabla .\epsilon =0.$
Let $\omega _{1}$ be any solenoidal vector function of position in space, $\omega _{2}$ any irrotational vector function, and $u$ any scalar function, satisfying the conditions that their potentials have in general definite values.
With respect to the solenoidal function $\omega _{1},{\frac {1}{4\pi }}{\text{Lap}}$ and $\nabla \times$ are inverse operators; i.e.,
* ${\frac {1}{4\pi }}{\text{Lap }}\nabla \times \omega _{1}=\nabla \times {\frac {1}{4\pi }}{\text{Lap }}\omega _{1}=\omega _{1}.$
$\omega _{2},$
* ${\text{Lap }}\omega _{2}=0,$$\nabla \times \omega _{2}=0.$
$\omega _{2},$
$u,{\frac {1}{4\pi }}{\text{New}}$
$-\nabla .$
* $-{\frac {1}{4\pi }}{\text{New }}\nabla .\omega _{2}=\omega _{2},$$-\nabla .{\frac {1}{4\pi }}{\text{New }}u=u.$
$\omega _{1},$
$\nabla .$
* $\nabla .\omega _{1}=0.$
$\omega _{1}+\omega _{2},$
With respect to the solenoidal function $\omega _{1},{\frac {1}{4\pi }}{\text{Pot }}$ and $\nabla \times \nabla \times$ are inverse operators; i.e.,
* ${\frac {1}{4\pi }}{\text{Pot }}\nabla \times \nabla \times \omega _{1}=\nabla \times {\frac {1}{4\pi }}{\text{Pot }}\nabla \times \omega _{1}=\nabla \times \nabla \times {\frac {1}{4\pi }}{\text{Pot }}\omega _{1}=\omega _{1}.$
$\omega _{2},{\frac {1}{4\pi }}{\text{Pot }}$
$-\nabla \nabla .$
* $-{\frac {1}{4\pi }}{\text{Pot }}\nabla \nabla .\omega _{2}=-\nabla {\frac {1}{4\pi }}{\text{Pot }}\nabla .\omega _{2}=-\nabla \nabla .{\frac {1}{4\pi }}{\text{Pot }}\omega _{2}=\omega _{2}.$
${\frac {1}{4\pi }}{\text{Pot}}$
$-\nabla .\nabla$
* $-{\frac {1}{4\pi }}{\text{Pot }}\nabla .\nabla u=-\nabla {\frac {1}{4\pi }}{\text{Pot }}\nabla u=-\nabla .\nabla {\frac {1}{4\pi }}{\text{Pot }}u=u,$ $-{\frac {1}{4\pi }}{\text{Pot }}\nabla .\nabla [\omega _{1}+\omega _{2}]=-\nabla .\nabla {\frac {1}{4\pi }}{\text{Pot }}[\omega _{1}+\omega _{2}]=\omega _{1}+\omega _{2}.$
$\omega _{1},-\nabla .\nabla$
$\nabla \times \nabla \times$
$\omega _{2},\nabla .\nabla$
$\nabla \nabla .$
* $-\nabla .\nabla \omega _{1}=\nabla \times \nabla \times \omega _{1},$$\nabla .\nabla \omega _{2}=\nabla \nabla .\omega _{2}.$
The only cases which we shall here consider in detail are those of surfaces at which the functions of position ($u$ or $\omega$) are discontinuous, and the values of $\nabla u,\nabla \times \omega ,\nabla .\omega$ thus become infinite. Let the function $u$ have the value $u_{1}$ on the side of the surface which we regard as the negative, and the value $u_{2}$ on the positive side. Let $\Delta u=u_{2}-u_{1}.$ If we substitute for the surface a shell of very small thickness $a,$ within which the value of $u$ varies uniformly as we pass through the shell, we shall have $\nabla u=\nu {\frac {\nabla u}{a}}$ within the shell, $\nu$ denoting a unit normal on the positive side of the surface. The elements of volume which compose the shell may be expressed by $a[d\sigma ]_{0},$ where $[d\sigma ]_{0}$ is the magnitude of an element of the surface, $d\sigma$ being the vector element. Hence,
* $\nabla udv=\nu \Delta u[d\sigma ]_{0}=\Delta u\,d\sigma .$
$u$
${\text{Pot }}\nabla u$
* $\iint {\frac {\Delta u'}{[\rho '-\rho ]_{0}}}d\sigma '$
$\Delta u'$
$d\sigma '$
$\rho '.$
In the case of a vector function which is discontinuous at a surface, the expressions $\nabla .\omega \,dv$ and $\nabla \times \omega \,dv,$ relating to the element of the shell which we substitute for the surface of discontinuity, are easily transformed by the principle that these expressions are the direct and skew surface-integrals of $\omega$ for the element of the shell. (See Nos. 65, 56.) The part of the surface-integrals relating to the edge of the element may evidently be neglected, and we shall have
* ${\begin{aligned}\nabla .\omega \,dv&=\omega _{2}.d\sigma -\omega _{1}.d\sigma =\Delta \omega .d\sigma ,\\\nabla \times \omega \,dv&=d\sigma \times \omega _{2}-d\sigma \times \omega _{1}=d\sigma \times \Delta \omega .\end{aligned}}$
$\omega$
${\text{Pot }}\nabla .\omega$
${\text{New }}\nabla .\omega$
* $\iint {\frac {1}{[\rho '-\rho ]_{0}}}\Delta \omega '.d\sigma '$and$\iint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\Delta \omega '.d\sigma '$
${\text{Pot }}\nabla \times \omega$
${\text{Lap }}\nabla \times \omega$
* $\iint {\frac {1}{[\rho '-\rho ]_{0}}}d\sigma '\times \Delta \omega '$and$\iint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\times [d\sigma '\times \Delta \omega ']$
101. We have already seen that if $\omega$ is the curl of any vector function of position, $\nabla .\omega =0.$ (No. 68.) The converse is evidently true, whenever the equation $\nabla .\omega =0$ holds throughout all space, and $\omega$ has in general a definite potential; for then
* $\omega =\nabla \times {\frac {1}{4\pi }}{\text{Lap }}\omega .$
$\nabla .\omega =0$
${\text{A}},$
${\text{B}}$
${\text{A}}.$
$\omega ',$
$\omega '=\omega$
${\text{A}},\omega '=0$
${\text{B}}'$
$\iiint \omega '.\omega 'dv$
${\text{B}}$
$\omega '$
$\omega ,$
$\nabla .\omega '=0$
$\nabla .\omega '=0$
$\omega '$
* $\omega '=\nabla \times {\frac {1}{4\pi }}{\text{Lap }}\omega ',$
$\omega$
${\text{A}}.$
102. Def.—If $\omega$ is a vector function of position in space, the Maxwellian of $\omega$ is a scalar function of position, defined by the equation
* ${\text{Max }}\omega =\iiint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}.\omega '\,dv'.$
$\omega$
$u$
* ${\begin{aligned}{\text{Max }}\omega &=\nabla .{\text{Pot }}\omega ={\text{Pot }}\nabla .\omega ,\\\nabla {\text{Max }}\omega &=\nabla \nabla .{\text{Pot }}\omega ={\text{New }}\nabla .\omega ,\\{\text{Max }}\nabla u&=-4\pi u,\\4\pi u&=\nabla \times {\text{Lap }}\omega -\nabla {\text{Max }}\omega .\end{aligned}}$
${\text{Lap Lap }}\omega ,{\text{New Max }}\omega ,$
${\text{Max New }}u$
* ${\begin{aligned}4\pi \,{\text{Pot }}\omega &={\text{Lap Lap }}\omega -{\text{New Max }}\omega ,\\4\pi \,{\text{Pot }}u&=-{\text{Max New }}u.\end{aligned}}$
$-{\frac {\text{Max}}{4\pi }}$
$\nabla$
$-{\frac {1}{4\pi }}\nabla {\text{Max}}$
${\frac {-1}{4\pi }}{\text{Max New}}$
${\frac {-1}{4\pi }}{\text{New Max}}$
${\frac {1}{4\pi }}{\text{Lap Lap}}$
${\frac {-1}{4\pi }}{\text{New Max}}$
${\frac {1}{4\pi }}{\text{Lap Lap}}$
103. Def.—The following double volume-integrals are of frequent occurrence in physical problems. They are all scalar quantities, and none of them functions of position in space, as are the single volume-integrals which we have been considering. The integrations extend over all space, or as far as the expression to be integrated has values other than zero.
The mutual potential, or potential product, of two scalar functions of position in space is defined by the equation
* ${\text{Pot}}(u,w)=\iiint \iiint {\frac {uw'}{r}}dv\,dv'=\iiint u{\text{Pot }}w\,dv=\iiint w{\text{Pot }}u\,dv.$
$r$
$u$
$dv$
$w'$
$dv'.$
The mutual potential, or potential product, of two vector functions of position in space is defined by the equation
* ${\text{Pot}}(\phi ,\omega )=\iiint \iiint {\frac {\phi .\omega '}{r}}dv\,dv'=\iiint \phi .{\text{Pot }}w\,dv=\iiint \omega .{\text{Pot }}\phi \,dv.$
* ${\text{Lap}}(\phi ,\omega )=\iiint \iiint \omega .{\frac {\rho '-\rho }{r^{3}}}\times \phi 'dv\,dv'=\iiint \omega .{\text{Lap }}\phi \,dv=\iiint \phi .{\text{Lap }}\omega \,dv.$
* ${\text{New}}(u,\omega )=\iiint \iiint \omega .{\frac {\rho '-\rho }{r^{3}}}u'\,dv\,dv'=\iiint \omega .{\text{New }}u\,dv.$
* ${\text{Max }}(\omega ,u)=\iiint \iiint u{\frac {\rho '-\rho }{r^{3}}}.\omega '\,dv\,dv'=\iiint {\text{Max }}\omega \,dv=-{\text{New}}(u,\omega ).$
$u,w,\phi ,\omega ,$
104. By No. 97,
* $4\pi u{\text{Pot }}w=-\nabla .{\text{New }}u{\text{ Pot }}w=-\nabla .[{\text{New }}u{\text{ Pot }}w]+{\text{New }}u.{\text{ New }}w.$
* $4\pi {\text{Pot}}(u,w)=\iiint {\text{New }}u.{\text{New }}w\,dv,$
* $\iint d\sigma .{\text{New }}u{\text{Pot }}w,$
$u$
$w$
Again, by No. 102,
| $4\pi \omega .{\text{Pot }}\phi$ | $=\nabla \times {\text{Lap }}\omega .{\text{Pot }}\phi -\nabla {\text{Max }}\omega .{\text{Pot }}\phi$ |
| | $=\nabla .[{\text{Lap }}\omega \times {\text{Pot }}\phi ]+{\text{Lap }}\omega .{\text{Lap }}\phi$ |
| | $-\nabla .[{\text{Max }}\omega {\text{ Pot }}\phi ]+{\text{Max }}\omega {\text{ Max }}\phi .$ |
The volume-integral of this equation gives
* $4\pi {\text{ Pot}}(\phi ,\omega )=\iiint {\text{Lap }}\phi .{\text{Lap }}\omega \,dv+\iiint {\text{Max }}\phi {\text{Max }}\omega \,dv,$
* $\iint d\sigma .{\text{Lap }}\omega \times {\text{Pot }}\phi ,$$\iint d\sigma .{\text{Pot }}\phi {\text{ Max }}\omega ,$
$\phi$
$\omega$
CHAPTER III.
concerning linear vector functions.
105. Def.—A vector function of a vector is said to be linear, when the function of the sum of any two vectors is equal to the sum of the functions of the vectors. That is, if
* ${\text{func.}}[\rho +\rho ']={\text{func.}}[\rho ]+{\text{func.}}[\rho ']$
$\rho$
$\rho ',$
* ${\text{func.}}[a\rho +b\rho '+c\rho ''+{\text{etc.}}]=a{\text{ func.}}[\rho ]+b{\text{ func.}}[\rho ']+c{\text{ func.}}[\rho '']+{\text{etc.}}$
* $\alpha \,\lambda .\rho +\beta \,\mu .\rho +{\text{etc.}}$
$\rho ,$
* $\{\alpha \lambda +\beta \mu +{\text{etc.}}\}.\rho .$
$\rho .\alpha \lambda +\rho .\beta \mu +{\text{etc.,}}$
or$\rho .\{\alpha \lambda +\beta \mu +{\text{etc.}}\},$
also represents a linear function of $\rho ,$ which is, in general, different from the preceding, and will be called its conjugate.
107. Def.—An expression of the form $\alpha \lambda$ or $\beta \mu$ will be called a dyad. An expression consisting of any number of dyads united by the signs $+$ or $-$ will be called a dyadic binomial, trinomial, etc, as the case may be, or more briefly, a dyadic. The latter term will be used so as to include the case of a single dyad. When we desire to express a dyadic by a single letter, the Greek capitals will be used, except such as are like the Roman, and also $\Delta$ and $\mathrm {E} .$ The letter ${\text{I}}$ will also be used to represent a certain dyadic, to be mentioned hereafter.
Since any linear vector function may be expressed by means of a dyadic (as we shall see more particularly hereafter, see No. 110), the study of such functions, which is evidently of primary importance in the theory of vectors, may be reduced to that of dyadics.
108. Def.—Any two dyadics $\Phi$ and $\Psi$ are equal,
| when | $\Phi .\rho =\Psi .\rho$ | for all values of $\rho ,$ |
| or, when | $\rho .\Phi =\rho .\Psi$ | for all values of $\rho ,$ |
| or, when | $\sigma .\Phi .\rho =\sigma .\Psi .\rho$ | for all values of $\sigma$ and of $\rho .$ |
The third condition is easily shown to be equivalent both to the first and to the second. The three conditions are therefore equivalent.
It follows that $\Phi =\Psi ,$ if $\Phi .\rho =\Psi .\rho ,$ or $\rho .\Phi =\rho .\Psi ,$ for three non-complanar values of $\rho .$
109. Def.—We shall call the vector $\Phi .\rho$ the (direct) product of $\Phi$ and $\rho ,$ the vector $\rho .\Phi$ the (direct) product of $\rho$ and $\Phi ,$ and the scalar $\sigma .\Phi .\rho$ the (direct) product of $\sigma ,\Phi ,$ and $\rho .$
In the combination $\Phi .\rho ,$ we shall say that $\Phi$ is used as a prefactor, in the combination $\rho .\Phi ,$ as a postfactor.
110. If $\tau$ is any linear function of $\rho ,$ and for $\rho =i,\rho =j,\rho =k,$ the values of $\tau$ are respectively $\alpha ,\beta ,$ and $\gamma ,$ we may set
* $\tau =\{\alpha i+\beta j+\gamma k\}.\rho ,$
* $\tau =\rho .\{i\alpha +j\beta +k\gamma \}.$
​111. Def.—We shall say that a dyadic is multiplied by a scalar, when one of the vectors of each of its component dyads is multiplied by that scalar. It is evidently immaterial to which vector of any dyad the scalar factor is applied. The product of the dyadic $\Phi$ and the scalar $a$ may be written either $a\Phi$ or $\Phi a.$ The minus sign before a dyadic reverses the signs of all its terms.
112. The sign $+$ in a dyadic, or connecting dyadics, may be regarded as expressing addition, since the combination of dyads and dyadics with this sign is subject to the laws of association and commutation.
113. The combination of vectors in a dyad is evidently distributive. That is,
* $[\alpha +\beta +{\text{etc.}}][\lambda +\mu +{\text{etc.}}]=\alpha \lambda +\alpha \mu +\beta \lambda +\beta \mu +{\text{etc.}}$
114. Since any vector may be expressed as a sum of $i,j,$ and $k$ with scalar coefficients, every dyadic may be reduced to a sum of the nine dyads
* $ii,ij,ik,ji,jj,jk,ki,kj,kk,$
On this account, we may regard the dyad as the most general form of product of two vector& We shall call it the indeterminate product. The complete determination of a single dyad involves five independent scalars, of a dyadic, nine.
115. It follows from the principles of the last paragraph that if
* $\textstyle \sum \displaystyle \alpha \beta =\textstyle \sum \displaystyle \kappa \lambda ,$
* $\textstyle \sum \displaystyle \alpha \times \beta =\textstyle \sum \displaystyle \kappa \times \lambda ,$
* $\textstyle \sum \displaystyle \alpha .\beta =\textstyle \sum \displaystyle \kappa .\lambda .$
We shall write $\Phi _{\text{X}}$ and $\Phi _{\text{S}}$ to indicate the vector and the scalar thus obtained
* $\Phi _{\text{X}}=(j.\Phi .k-k.\Phi .j)i+(k.\Phi .i-i.\Phi .k)j+(i.\Phi .j-j.\Phi .i)k,$ $\Phi _{\text{S}}=i.\Phi .i+j.\Phi .j+k.\Phi .k,$
$\Phi$
$ii,ij,$
116. Def.—The (direct) product of two dyads (indicated by a dot) is the dyad formed of the first and last of the four factors, multiplied by the direct product of the second and third. That is,
* $\{\alpha \beta \}.\{\gamma \delta \}=\alpha \beta .\gamma \delta =\beta .\gamma \,\alpha \delta .$
* $\alpha .\Phi .\Theta .\Psi .\beta ,$$\alpha .\Phi .\Theta ,$$\Phi .\Theta .\Psi .\beta ,$$\Phi .\Theta .\Psi$
117. Def.—The expressions $\Phi \times \rho$ and $\rho \times \Phi$ represent dyadics which we shall call the skew products of $\Phi$ and $\rho .$ If
* $\Phi =\alpha \lambda +\beta \mu +{\text{etc.,}}$
* ${\begin{aligned}\Phi \times \rho &=\alpha \,\lambda \times \rho +\beta \,\mu \times \rho +{\text{etc.,}}\\\rho \times \Phi &=\rho \times \alpha \,\lambda +\rho \times \beta \,\mu +{\text{etc.,}}\end{aligned}}$
| $\{\rho \times \Phi \}.\Psi$ | $=\rho$ | $\Psi .\{\Phi \times \rho \}$ | $=\{\Psi .\Phi \}\times \rho ,$ |
| $\{\rho \times \Phi \}.\alpha$ | $=\rho \times [\Phi .\alpha ],$ | $\alpha .\{\Phi \times \rho \}$ | $=[\alpha .\Phi ]\times \rho ,$ |
| $\{\rho \times \Phi \}\times \alpha$ | $=\rho \times \{\Phi \times \alpha \}.$ |
We may therefore write without ambiguity
* $\rho \times \Phi .\Psi ,$$\Psi .\Phi \times \rho ,$$\rho \times \Phi .\alpha ,$$\alpha .\Phi \times \rho ,$$\rho \times \Phi .\alpha .$
Moreover,
* $\alpha .\rho \times \Phi =[\alpha \times \rho ].\Phi$and$\Phi \times \rho .\alpha =\Phi .[\rho \times \alpha ].$
* $\Psi .\{\rho \times \Phi \}=\{\Psi \times \rho \}.\Phi .$
​118. Since all the antecedents or all the consequents in any dyadic may be expressed in parts of any three non-complanar vectors, and since the sum of any number of dyads having the same antecedent or the same consequent may be expressed by a single dyad, it follows that any dyadic may be expressed as the sum of three dyads, and so, that either the antecedents or the consequents shall be any desired non-complanar vectors, but only in one way when either the antecedents or the consequents are thus given.
In particular, the dyadic
* ${\begin{aligned}aii&+bij+cik\\+a'ji&+b'jj+c'jk\\+a'ki&+b''kj+c''kk,\end{aligned}}$
| $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | $a$ | $b$ | $c$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$, |
| $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | $a'$ | $b'$ | $c'$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$, |
| $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | $a''$ | $b''$ | $c''$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$, |
is equal to
* $\alpha i+\beta j+\gamma k,$
* ${\begin{aligned}\alpha &=ai+a'j+a''k,\\\beta &=bi+b'j+b''k,\\\gamma &=c'+c'j+c''k,\end{aligned}}$
* $i\lambda +j\mu +k\nu ,$
* $\alpha i+\beta j+\gamma k,$
* ${\begin{aligned}\lambda &=ai+bj+ck,\\\mu &=a'i+b'j+c'k,\\\nu &=a''+b''j+c''k,\end{aligned}}$
* $\alpha \lambda +\beta \mu +\gamma \nu$
* $\{\alpha \lambda +\beta \mu +\gamma \nu \}.\rho$
$\rho$
$\alpha$
$\rho$
$\mu$
$\nu ,$
$\beta$
$\rho$
$\lambda$
$\mu ,$
$\gamma$
$\rho$
$\lambda$
$\mu .$
$\rho .$
​120. In like manner, the sum of two dyads may be reduced to a single dyad, if either the antecedents or the consequents are parallel, and only in such cases.
A sum of three dyads cannot be reduced to a single dyad, unless either their antecedents or consequents are parallel, or both antecedents and consequents are (separately) complanar. In the first case the reduction can always be made, in the second, occasionally.
121. Def.—A dyadic which cannot be reduced to the sum of less than three dyads will be called complete.
A dyadic which can be reduced to the sum of two dyads will be called planar. When the plane of the antecedents coincides with that of the consequents, the dyadic will be called uniplanar. These planes are invariable for a given dyadic, although the dyadic may be so expressed that either the two antecedents or the two consequents may have any desired values (which are not parallel) within their planes.
A dyadic which can be reduced to a single dyad will be called linear. When the antecedent and consequent are parallel, it will be called unilinear.
A dyadic is said to have the value zero when all its terms vanish.
122. If we set
* $\sigma =\Phi .\rho ,$$\tau =\rho .\Phi ,$
$\rho$
$\sigma$
$\tau$
$\Phi$
$\sigma$
$\tau$
$\Phi$
$\Phi$
$\sigma$
$\tau$
$\Phi$
$\Phi$
123. The products of complete dyadics are complete, of complete and planar dyadics are planar, of complete and linear dyadics are linear.
The products of planar dyadics are planar, except that when the plane of the consequents of the first dyadic is perpendicular to the plane of the antecedents of the second dyadic, the product reduces to a linear dyadic.
The products of linear dyadics are linear, except that when the consequent of the first is perpendicular to the antecedent of the second, the product reduces to zero.
The products of planar and linear dyadics are linear, except when, the planar preceding, the plane of its consequents is perpendicular to the antecedent of the linear, or, the linear preceding, its consequent is perpendicular to the plane of the antecedents of the planar. In these cases the product is zero.
All these cases are readily proved, if we set
* $\sigma =\Phi .\Psi .\rho ,$
$\sigma$
$\rho$
The products $\Psi \times \rho$ and $\rho \times \Phi$ are evidently planar dyadics.
124. Def.—A dyadic $\Phi$ is said to be an idemfactor, when
* $\Phi .\rho =\rho$ for all values of $\rho ,$
* $\rho .\Phi =\rho$ for all values of $\rho .$
$\Phi$
* $ii+jj+kk.$
$I.$
The direct product of an idemfactor with another dyadic is equal to that dyadic. That is,
* $I.\Phi =\Phi ,$$\Phi .I=\Phi ,$
$\Phi$
A dyadic of the form
* $\alpha \alpha '+\beta \beta '+\gamma \gamma ',$
$\alpha ',\beta ',\gamma '$
$\alpha ,\beta ,\gamma ,$
125. If one of the direct products of two dyadics is an idemfactor, the other is also. For, if $\Phi .\Psi =I,$
* $\sigma .\Phi .\Psi =\sigma$
$\sigma ,$
$\Phi$
* $\sigma .\Phi .\Psi .\Phi =\sigma .\Phi$
$\sigma ,$
$\sigma .\Phi ,$
$\Psi .\Phi =I.$
Def. — In this case, either dyadic is called the reciprocal of the other.
It is evident that an incomplete dyadic cannot have any (finite) reciprocal.
Reciprocals of the same dyadic are equal. For if $\Phi$ and $\Psi$ are both reciprocals of $\Omega ,$
* $\Phi =\Phi .\Omega .\Psi =\Psi .$
126. The reciprocal of any complete dyadic
* $\alpha \lambda +\beta \mu +\gamma \nu$
* $\lambda '\alpha '+\mu '\beta '+\nu '\gamma ',$
$\alpha ',\beta ',\gamma '$
$\alpha ,\beta ,\gamma ,$
$\lambda ',\mu ',\nu '$
$\lambda ,\mu ,\nu .$
​127. Def.—We shall write $\Phi ^{-1}$ for the reciprocal of any (complete) dyadic $\Phi ,$ also $\Phi ^{2}$ for $\Phi .\Phi$ etc., and $\Phi ^{-2}$ for $\Phi ^{-1}.\Phi ^{-1}$ etc. It is evident that $\Phi ^{-n}$ is the reciprocal of $\Phi ^{n}$,
128. In the reduction of equations, if we have
* $\Phi .\Psi =\Phi .\Omega ,$
$\Phi$
$\Phi ^{-1}$
$\Phi$
* $\Phi .\sigma =\Phi .\rho ,$$\Psi .\Phi =\Omega .\Phi ,$$\rho .\Phi =\sigma .\Phi .$
129. Def.—If in any dyadic we transpose the factors in each term, the dyadic thus formed is said to be conjugate to the first. Thus
* $\alpha \lambda +\beta \mu +\gamma \nu$and$\lambda \alpha +\mu \beta +\nu \gamma$
$\Phi$
$\Phi _{\text{C}}$
* $\rho .\Phi =\Phi _{\text{C}}.\rho$and$\Phi .\rho =\rho .\Phi _{\text{C}}.$
$\Phi _{\text{C}}.\rho$
$\Phi .\rho$
$\rho .$
$\{\Phi _{\text{C}}\}^{2}=\{\Phi ^{2}\}_{\text{C}},$
$\Phi _{\text{C}}^{2},$
130. The reciprocal of the product of any number of dyadics is equal to the product of their reciprocals taken in inverse order. Thus
* $\{\Phi .\Psi .\Omega \}_{\text{C}}=\Omega _{\text{C}}.\Psi _{\text{C}}.\Phi _{\text{C}}\$
* $\{\Phi .\Psi .\Omega \}_{\text{C}}=\Omega _{\text{C}}.\Psi _{\text{C}}.\Phi _{\text{C}}.$
* $\Phi _{\text{C}}.\{\Phi ^{-1}\}_{\text{C}}=\{\Phi ^{-1}.\Phi \}_{\text{C}}=I,$ $\{\Phi ^{-1}\}_{\text{C}}=\{\Phi _{\text{C}}\}^{-1},$
$\Phi _{\text{C}}^{-1}$
131. It is sometimes convenient to be able to express by a dyadic taken in direct multiplication the same operation which would be effected by a given vector ($\alpha$) in skew multiplication. The dyadic $I\times \alpha$ will answer this purpose. For, by No. 117,
| $\{I\times \alpha \}.\rho =\alpha \times \rho ,$ | $\rho .\{I\times \alpha \}=\rho \times \alpha ,$ |
| $\{I\times \alpha \}.\Phi =\alpha \times \Phi ,$ | $\Phi .\{I\times \alpha \}=\Phi \times \alpha .$ |
The same is true of the dyadic $\alpha \times I,$ which is indeed identical with $I\times \alpha ,$ as appears from the equation $I.\{\alpha \times I\}=\Phi .\{I\times \alpha \}.I.$ ​If $\alpha$ is a unit vector,
* ${\begin{aligned}\{{\text{I }}\times \alpha \}^{2}&=-\{{\text{I}}-\alpha \alpha \},\\\{{\text{I }}\times \alpha \}^{3}&=-{\text{I }}\times \alpha ,\\\{{\text{I }}\times \alpha \}^{4}&={\text{I}}-\alpha \alpha ,\\\{{\text{I }}\times \alpha \}^{5}&={\text{I }}\times \alpha ,\\{\text{etc.}}\end{aligned}}$
$i,j,k$
* ${\begin{aligned}{\text{I }}\times {\text{ I}}&={\text{I }}\times {\text{ I}}=kj-jk,\\{\text{I }}\times j&=j\times {\text{ I}}=ik-ki,\\{\text{I }}\times k&=k\times {\text{ I}}=ji-ij.\end{aligned}}$
$\alpha$
$\beta$
* $[\alpha \times \beta ]\times {\text{ I}}={\text{I }}\times [\alpha \times \beta ]=\beta \alpha -\alpha \beta .$
$\alpha \times \beta$
$\{\beta \alpha -\alpha \beta \}$
* ${\begin{aligned}[\alpha \times \beta ]\times \rho &=\{\beta \alpha -\alpha \beta \}.\rho ,\\\rho \times [\alpha \times \beta ]&=\rho .\{\beta \alpha -\alpha \beta \}.\end{aligned}}$
132. The equation
* $\alpha \,\beta \times \gamma +\beta \,\gamma \times \alpha +\gamma \,\alpha \times \beta =\alpha .\beta \times \gamma \,{\text{I}}$
$\rho ,$
* $\rho .\alpha \,\beta \times \gamma +\rho .\beta \,\gamma \times \alpha +\rho .\gamma \,\alpha \times \beta =\alpha .\beta \times \gamma \,\rho .$
* $\alpha \alpha '+\beta \beta '\gamma \gamma '={\text{I}},$
$\alpha ,\beta ,$
$\gamma$
* $\beta \times \gamma \,\alpha +\rho .\gamma \times \alpha \,\beta +\alpha \times \beta \,\gamma =\alpha .\beta \times \gamma \,{\text{I}}.$
$\rho$
$\rho$
* $\rho .\beta \times \gamma \,\alpha +\rho .\gamma \times \alpha \,\beta +\rho .\alpha \times \beta \,\gamma =\alpha .\beta \times \gamma \,\rho .$
* $\alpha \,\beta \times \gamma +\beta \,\gamma \times \alpha +\gamma \,\alpha \times \beta =0.$
$\nu ,$
$\alpha ,\beta ,$
$\gamma$
* $\alpha \,\beta \times \gamma .\nu +\beta \,\gamma \times \alpha .\nu +\gamma \,\alpha \times \beta .\nu =0.$
138. Def.—If two dyadics $\Phi$ and $\Psi$ are such that
* $\Phi .\Psi =\Psi .\Phi ,$
If any number of dyadics are homologous to one another, and any other dyadics are formed from them by the operations of taking multiples, sums, differences, powers, reciprocals, or products, such dyadics will be homologous to each other and to the original dyadics. This requires demonstration only in regard to reciprocals. Now if
* ${\begin{aligned}\Phi .\Psi &=\Psi .\Phi ,\\\Psi .\Phi ^{-1}=\Phi ^{-1}.\Phi .\Psi .\Phi ^{-1}&=\Phi ^{-1}.\Psi .\Phi .\Phi ^{-1}=\Phi ^{-1}.\Psi .\end{aligned}}$
That is, $\Phi ^{-1}$ is homologous to $\Psi ,$ if $\Phi$ is.
134. If we call $\Psi .\Phi ^{-1}$ or $\Phi ^{-1}.\Psi$ the quotient of $\Psi$ and $\Phi ,$ we may say that the rules of addition, subtraction, multiplication and division of homologous dyadics are identical with those of arithmetic or ordinary algebra, except that limitations analogous to those respecting zero in algebra must be observed with respect to all incomplete dyadics.
It foUows that the algebraic and higher analysis of homologous dyadics is substantially identical with that of scalars.
135. It is always possible to express a dyadic in three terms, so that both the antecedents and the consequents shall be perpendicular among themselves.
To show this for any dyadic $\Phi ,$ let us set
* $\rho '=\Phi .\rho ,$
$\rho$
$\rho$
$\rho .$
$i$
$\rho$
$i,$
$\rho '$
$\rho .$
$j$
$\rho$
$j,$
$\rho '$
$\rho$
$i.$
$k$
$i$
$j.$
$\Phi$
* $\alpha i+\beta j+\gamma k.$
* $\rho '=\{\alpha i+\beta j+\gamma k\}.\rho ,$
* $d\rho '=\{\alpha i+\beta j+\gamma k\}.d\rho .$
$i$
$\rho$
$i$
$d\rho$
$i,d\rho '$
$\rho ',$
$\alpha .$
$d\rho$
$j$
$k,$
$i,$
$d\rho '$
$\beta$
$\gamma ,$
$\beta$
$\gamma$
$\alpha .$
$j$
$\gamma$
$\beta .$
* $\Phi =ai'i+bj'j+ck'k,$
$i',j',k',$
$i,j,k,$
$a,b,c$
It makes an important difference whether the number of these scalars which are negative is even or odd. If two are negative, say $a$ and $b,$ we may make them positive by reversing the directions of $i'$ and $j'.$ The vectors $i',j',k'$ will still constitute a normal system. But if we should reverse the directions of an odd number of these vectors, they would cease to constitute a normal system, and to be superposable upon the system $i,j,k.$ We may, however, always set either
| | $\Phi$ | $=ai'i+bj'j+ck'k,$ |
| or | $\Phi$ | $=-\{ai'i+bj'j+ck'k\},$ |
with positive values of $a,b,$ and $c.$ At the limit between these cases are the planar dyadics, in which one of the three terms vanishes, and the dyadic reduces to the form
* $ai'i+bj'j,$
$a$
$b$
$i'$
$j'.$
If the numerical values of $a,b,c$ are all unequal, there will be only one way in which the value of $\Phi$ may be thus expressed. If they are not all unequal, there will be an infinite number of ways in which $\Phi$ may be thus expressed, in all of which the three scalar coefficients will have the same values with exception of the changes of signs mentioned above. If the three values are numerically identical, we may give to either system of normal vectors an arbitrary position.
136. It follows that any self-conjugate dyadic may be expressed in the form
* $aii+bjj+ckk,$
$i,j,k$
$a,b,c$
137. Any dyadic may be divided into two parts, of which one shall be self-conjugate, and the other of the form ${\text{I}}\times \alpha .$ These parts are found by taking half the sum and half the difference of the dyadic and its conjugate. It is evident that
* $\Phi ={\tfrac {1}{2}}\{\Phi +\Phi _{\text{C}}\}+{\tfrac {1}{2}}\{\Phi -\Phi _{\text{C}}\}.$
${\tfrac {1}{2}}\{\Phi +\Phi _{\text{C}}\}$
* ${\tfrac {1}{2}}\{\Phi +\Phi _{\text{C}}\}={\text{I}}\times [-{\tfrac {1}{2}}\Phi _{\text{X}}].$
Rotations and Strains.
138. To illustrate the use of dyadics as operators, let us suppose that a body receives such a displacement that
* $\rho '=\Phi .\rho ,$
$\rho$
$\rho _{1},\rho _{2}$
$\rho _{1}',\rho _{2}'$
* $\rho _{1}'=\Phi .\rho _{1},$$\rho _{2}'=\Phi .\rho _{2},$
* $\rho _{2}'-\rho _{1}'=\Phi .[\rho _{2}-\rho _{1}].$
139. The vectors (\sigma, \sigma ') which represent any plane surface in the body in its initial and final positions will be linear functions of each other. (This will appear, if we consider the four sides of a tetrahedron in the body.) To find the relation of the dyadics which express $\sigma '$ as a function of $\sigma ,$ and $\rho '$ as a function of $\rho ,$ let
* $\rho '=\{\alpha \lambda +\beta \mu +\gamma \nu \}.\rho .$
$\lambda ',\mu ',\nu '$
$\lambda ,\mu ,\nu$
$\lambda ',\mu ',\nu '$
$\alpha ,\beta ,\gamma$
$\mu '\times \ nu',\nu '\times \lambda ',\lambda '\times \mu '$
$\beta \times \gamma ,\gamma \times \alpha ,\alpha \times \beta .$
$\mu '\times \nu ',\nu '\times \lambda ',\lambda '\times \mu '$
$\mu \times \nu ,\nu \times \lambda ,\lambda \times \mu .$
* $\sigma '=\{\beta \times \lambda \,\mu \times \nu +\gamma \times \alpha \,\nu \times \lambda +\alpha \times \beta \,\lambda \times \mu \}.\sigma .$
$\lambda '.\mu '\times \nu '$
$\alpha .\beta \times \gamma .$
$(\alpha .\beta \times \gamma )(\lambda .\mu \times \nu ).$
Def.—It follows that the scalar product of the three antecedents multiplied by the scalar product of the three consequents of a dyadic expressed as a trinomial is independent of the particular form in which the dyadic is thus expressed. This quantity is the determinant of the coefficients of the nine terms of the form
* $aii+bij+{\text{etc.,}}$
* $\left\vert \Phi \right\vert$
If a dyadic is incomplete, its determinant is zero, and conversely.
The determinant of the product of any number of dyadics is equal to the product of their determinants. The determinant of the reciprocal of a dyadic is the reciprocal of the determinant of that dyadic. The determinants of a dyadic and its conjugate are equal.
The relation of the surfaces $\sigma '$ and $\sigma$ may be expressed by the equation
* $\sigma '=\left\vert \Phi \right\vert \Phi _{\text{C}}^{-1}.\sigma .$
$\Phi .$
If $\Phi$ is reducible to the form
* $i'i+j'j+k'k,$
$i,j,k,i',j',k'$
* $\rho =xi+yj+zk,$
* $\rho '=xi'+yj'+zk'.$
* $i'i+j'j+k'k$
142. The conjugate operator evidently produces the reverse rotation. A versor, therefore, is the reciprocal of its conjugate.
Conversely, if a dyadic is the reciprocal of its conjugate, it is either a versor, or a versor multiplied by -1. For the dyadic may be expressed in the form
* $\alpha i+\beta j+\gamma k.$
* $i\alpha +j\beta +k\gamma .$
* $\{\alpha i+\beta j\gamma k\}.\{i\alpha +j\beta +k\gamma \}=\alpha \alpha +\beta \beta +\gamma \gamma ={\text{I}}.$
$\alpha ,\beta ,\gamma$
* $\alpha i+\beta j+\gamma k$and$-\alpha i-\beta j-\gamma k$
* $\Phi .\Phi _{\text{C}}={\text{I}},$and$\left\vert \Phi \right\vert =1.$
* $\left\vert \Phi \right\vert >0,$or$\left\vert \Phi \right\vert \gtrless -1.$
143. If we take the axis of the rotation for the direction of $i,i'$ will have the same direction, and the versor reduces to the form
* $ii+j'j+k'k,$
$i,j,k$
$i,j',k'$
We may set
* ${\begin{aligned}j'&=\cos q\,j+\sin q\,k,\\k'&=\cos q\,k-\sin q\,j,\end{aligned}}$
* $ii+\cos q\,\{jj+kk\}+\sin q\,\{kj-jk\},$
* $ii+\cos q\,\{{\text{I}}-ii\}+\sin q\,{\text{I}}\times i,$
$q$
$j$
$k,$
144. When any versor $\Phi$ is used as a pref actor, the vector $-\Phi _{\text{X}}$ will be parallel to the axis of rotation, and equal in magnitude to twice the sine of the angle of rotation measured counter-clockwise as seen from the direction in which the vector points. (This will appear if we suppose $\Phi$ to be represented in the form given in the last paragraph.) The scalar $\Phi _{\text{S}}$ will be equal to unity increased by twice the cosine of the same angle. Together, $-\Phi _{\text{X}}$ and $\Phi _{\text{S}}$ determine the versor without ambiguity. If we set
* $\theta ={\frac {-\Phi _{\text{X}}}{1+\Phi _{\text{S}}}},$
$\theta$
* ${\frac {2\sin q}{2+2\cos q}}$or$\tan {\tfrac {1}{2}}q,$
$q$
$\theta$
$\theta ,$
145. The versor $\Phi$ may be expressed in terms of $\theta$ in various ways. Since $\Phi$ (as pref actor) changes $\alpha -\theta \times \alpha$ into $\alpha +\theta \times \alpha$ ($\alpha$ being any vector), we have
* $\Phi =\{{\text{I}}+{\text{I}}\times \theta \}.\{{\text{I}}-{\text{I}}\times \theta \}^{-1}.$
* $\Phi ={\frac {\theta \theta +\{{\text{I}}+{\text{I}}\times \theta \}^{2}}{1+\theta .\theta }}={\frac {(1-\theta .\theta ){\text{I}}+2\theta \theta +2{\text{I}}\times \theta }{1+\theta .\theta }},$
$\Phi .\rho$
$\theta ,$
* $ii+\cos q(jj+kk)+\sin q(kj-jk)$
$\cos q$
$\sin q$
$\tan {\tfrac {1}{2}}q.$
If we set, in either of these equations,
* $\theta =ai+bj+ck,$
* $\Phi ={\frac {\begin{Bmatrix}(1+a^{2}-b^{2}-c^{2})ii+(2ab-2c)ij+(2ac+2b)ik\\+(2ab+2c)ji+(1-a^{2}+b^{2}-c^{2})jj+(2bc-2a)jk\\+(2ac-2b)ki+(2bc+2a)kj+(1-a^{2}-b^{2}+c^{2})kk\end{Bmatrix}}{1+a^{2}+b^{2}+c^{2}}},$
146. If $\alpha ,\beta ,\gamma$ are unit vectors, expressions of the form
* $2\alpha \alpha -{\text{I}}$$2\beta \beta -{\text{I}},$$2\gamma \gamma -{\text{I}},$
* $\{2\beta \beta -{\text{I}}\}.\{2\alpha \alpha -{\text{I}}\}$
$\alpha$
$\beta ,$
$\alpha$
$\beta .$
$\alpha$
$\beta$
* $\Phi =\{2\beta \beta -{\text{I}}\}.\{2\alpha \alpha -{\text{I}}\},$and$\Psi =\{2\gamma \gamma -{\text{I}}\}.\{2\beta \beta -{\text{I}}\},$
* $\Psi .\Phi =\{2\gamma \gamma -{\text{I}}\}.\{2\alpha \alpha -{\text{I}}\}.$
$\beta$
$\Phi ,\Psi ,$
$\Psi .\Phi .$
* $\theta _{1}={\frac {-\Phi _{\text{X}}}{1+\Phi _{\text{S}}}},$$\theta _{2}={\frac {-\Psi _{\text{X}}}{1+\Psi _{\text{S}}}},$$\theta _{3}={\frac {-\{\Psi .\Phi \}_{\text{X}}}{1+\{\Psi .\Phi \}_{\text{S}}}}$
* ${\begin{aligned}\Phi &=4\alpha .\beta \beta \alpha -2\alpha \alpha -2\beta \beta +{\text{I}},\\\theta _{1}&={\frac {\alpha \times \beta }{\alpha .\beta }},\end{aligned}}$
* $\theta _{2}={\frac {\beta \times \gamma }{\beta .\gamma }}$$\theta _{3}={\frac {\alpha \times \gamma }{\alpha .\gamma }}\cdot$
* ${\begin{aligned}\theta _{1}\times \theta _{2}&={\frac {[\alpha \times \beta ]\times [\beta \times \gamma ]}{\alpha .\beta \,\beta .\gamma }}={\frac {\alpha \times \beta .\gamma \,\beta }{\alpha .\beta \,\beta .\gamma }}\\&={\frac {\beta .\alpha \,\beta \times \gamma +\beta .\beta \,\gamma \times \alpha +\beta .\gamma \,\alpha \times \beta }{\alpha .\beta \,\beta .\gamma }}\cdot \end{aligned}}$
* $\theta _{1}\times \theta _{2}=\theta _{2}-{\frac {\alpha .\gamma }{\alpha .\beta \,\beta .\gamma }}\theta _{3}+\theta _{1}.$
* $\theta _{1}.\theta _{2}={\frac {\alpha \times \beta .\beta \times \gamma }{\alpha .\beta \,\beta .\gamma }}=1-{\frac {\alpha .\gamma }{\alpha .\beta \,\beta .\gamma }}\cdot$
* ${\begin{aligned}\theta _{1}\times \theta _{2}&=\theta _{2}-(1-\theta _{1}.\theta _{2})\theta _{3}+\theta _{1},\\\theta _{3}&={\frac {\theta _{1}+\theta _{2}+\theta _{2}\times \theta _{1}}{1-\theta _{1}.\theta _{3}}},\end{aligned}}$
147. The versors just described constitute a particular class under the more general form
* $\alpha \alpha '+\cos q\{\beta \beta '+\gamma \gamma '\}+\sin q\{\gamma \beta '-\beta \gamma '\},$
$\alpha ,\beta ,\gamma$
$\alpha ',\beta ',\gamma '$
$\alpha .$
$\beta {\text{-}}\gamma$
$\beta$
$\gamma$
$\beta ,\gamma ,$
$q,$
$\beta {\text{-}}\gamma$
$\beta$
$\gamma$
$\beta$
$\gamma ,$
$q$
$2\pi .$
Def.—Such dyadics we shall call cyclic.
The Nth power (N being any whole number) of such a dyadic is obtained by multiplying $q$ by N. If $q$ is of the form $2\pi$ N/M (N and M being any whole numbers) the Mth power of the dyadic will be an idemfactor. A cyclic dyadic, therefore, may be regarded as a root of ​${\text{I}}$, or at least capable of expression with any required degree of accuracy as a root of ${\text{I}}.$
It should be observed that the value of the above dyadic will not be altered by the substitution for $\alpha$ of any other parallel vector, or for $\beta$ and $\gamma$ of any other conjugate semi-diameters (which succeed one another in the same angular direction) of the same or any similar and similarly situated ellipse, with the changes which these substitutions require in the values of $\alpha ',\beta ',\gamma '.$ Or, to consider the same changes from another point of view, the value of the dyadic will not be altered by the substitution for $\alpha '$ of any other parallel vector or for $\beta '$ and $\gamma '$ of any other conjugate semi-diameters (which succeed one another in the same angular direction) of the same or any similar and similarly situated ellipse, with the changes which these substitutions require in the values of $\alpha ,\beta ,$ and $\gamma ,$ defined as reciprocals of $\alpha ',\beta ',\gamma '.$
148. The strain represented by the equation
* $\rho '=\{aii+bjj+ckk\}.\rho$
$a,b,c$
$i,j,k,$
$a,b,c$
* $aii+jj+kk,$$ii+bjj+kk,$$ii+jj+ckk,$
Def.—A dyadic which is reducible to this form we shall call a right tensor. The displacement represented by a right tensor is called a pure strain. A right tensor is evidently self-conjugate.
149. We have seen (No. 135) that eveiy dyadic may be expressed in the form
* $\pm \{ai'i+bj'j+ck'k\},$
$a,b,c$
* $\pm \{ai'i'+bj'j'+ck'k'\}$
* $\pm \{i'i+j'j+k'k\}.\{aii+bjj+ckk\}.$
​Def.—The displacement represented by the equation
* $\rho '=-\rho$
150. If
| | $\Phi$ | $=ai'i+bj'j+ck'k,$ |
| | $\Phi .\Phi _{\text{C}}$ | $=a^{2}i'i'+b^{2}j'j'+c^{2}k'k',$ |
| and | $\Phi _{\text{C}}.\Phi$ | $=a^{2}ii+b^{2}jj+c^{2}kk.$ |
The general problem of the determination of the principal ratios and axes of strain for a given dyadic may thus be reduced to the case of a right tensor.
151. Def.—The effect of a prefactor of the form
* $a\alpha \alpha '+b\beta \beta '+c\gamma \gamma ',$
$a,b,c$
$\alpha ,\beta ,\gamma$
$\alpha ',\beta ',\gamma '$
$\alpha$
$a\alpha ,\beta$
$b\beta ,$
$\gamma$
$c\gamma .$
$\alpha '$
$a\alpha ',\beta '$
$b\beta ',$
$\gamma$
$c\gamma '.$
$a,b,c.$
The value of the dyadic is evidently not affected by substituting vectors of different lengths but the same or opposite directions for $\alpha ,\beta ,\gamma ,$ with the necessaiy changes in the values of $\alpha ',\beta ',\gamma ',$ defined as reciprocals of $\alpha ,\beta ,\gamma .$ But, except this change, if $a,b,c$ are unequal, the dyadic can be expressed only in one way in the above form. If, however, two of these coefficients are equal, say $\alpha$ and $\beta ,$ any two non-collinear vectors in the $\alpha {\text{-}}\beta$ plane may be substituted for $\alpha$ and $\beta ,$ or, if the three coefficients are equal, any three non-complanar vectors may be substituted for $\alpha ,\beta ,\gamma .$
152. Tonics having the same axes (determined by the directions of $\alpha ,\beta ,\gamma$) are homologous, and their multiplication is effected by multiplying their coefficients. Thus,
* ${\begin{aligned}\{a_{1}\alpha \alpha '+b_{1}\beta \beta '+c_{1}\gamma \gamma '\}.\{a_{2}\alpha \alpha '+b_{2}\beta \beta '+c_{2}\gamma &\gamma '\}\\&=\{a_{1}a_{2}\alpha \alpha '+b_{1}b_{2}\beta \beta '+c_{1}c_{2}\gamma \gamma '\}.\end{aligned}}$
$a,b,c$
153. The effect of a prefactor of the form
* $a\alpha \alpha '+b\{\beta \beta '+\gamma \gamma '\}+c\{\gamma \beta '-\beta \gamma '\},$
* $a\alpha \alpha '+p\,\cos q\{\beta \beta '+\gamma \gamma '\}+p\,\sin q\{\gamma \beta '-\beta \gamma '\},$
$\alpha ',\beta ',\gamma '$
$\alpha ,\beta ,\gamma ,$
$a,b,c,p,$
$q$
$p$
* $a\alpha \alpha '+\beta \beta '+\gamma \gamma ',$ $\alpha \alpha '+p\beta \beta '+p\gamma \gamma ',$ $\alpha \alpha '+\cos q\{\beta \beta '+\gamma \gamma \}+\sin q\{\gamma \beta '-\beta \gamma '\},$
$\alpha ,$
$\beta {\text{-}}\gamma$
$a$
$\alpha ,$
$p$
$\beta {\text{-}}\gamma$
$\beta {\text{-}}\gamma$
The effect of the same dyadic as a postfactor is of the same nature.
The value of the dyadic is not affected by the substitution for a of another vector having the same direction, nor by the substitution for $\beta$ and $\gamma$ of two other conjugate semi-diameters of the same or a similar and similarly situated ellipse, and which follow one another in the same angular direction.
Def. — Such dyadics we shall call cyclotonic.
154. Cyclotonics which are reducible to the same form except with respect to the values of $a,p,$ and $q$ are homologous. They are multiplied by multiplying the values of $a,$ and also those of $p,$ and adding those of $q.$ Thus, the product of
* $a_{1}\alpha \alpha '+p_{1}\,\cos q_{1}\{\beta \beta '+\gamma \gamma '\}+p_{1}\,\sin q_{1}\{\gamma \beta '-\beta \gamma '\}$
* $a_{2}\alpha \alpha '+p_{2}\,\cos q_{2}\{\beta \beta '+\gamma \gamma '\}+p_{2}\,\sin q_{2}\{\gamma \beta '-\beta \gamma '\}$
* ${\begin{aligned}a_{1}a_{2}\alpha \alpha '+p_{1}p_{2}\cos(q_{1}&+q_{2})\{\beta \beta '+\gamma \gamma '\}\\&+p_{1}p_{2}\sin(q_{1}+q_{2})\{\gamma \beta '-\beta \gamma '\}\end{aligned}}$
A dyadic of this form, in which the value of $q$ is not zero, or the product of $\pi$ and a positive or negative integer, is homologous only with such dyadics as are obtained by varying the values of $a,p,$ and $q.$
156. In general, any dyadic may be reduced to the form either of a tonic or of a cyclotonic. (The exceptions are such as are made by the limiting cases.) We may show this, and also indicate how the reduction may be made, as follows. Let $\Phi$ be any dyadic. We have first to show that there is at least one direction of $\rho$ for which
* $\Phi .\rho =a\rho .$
* $\Phi .\rho -a\rho =0,$
* $\{\Phi -a{\text{I}}\}.\rho =0.$
$\Phi -a{\text{I}}$
* $\left\vert \Phi -a{\text{I}}\right\vert =0.$
* $\Phi =\lambda i+\mu j+\nu k;$
* $\left\vert [\lambda -ai]i+[\mu -aj]j+[\nu -ak]k\right\vert =0,$
* $[\lambda -ai]\times [\mu -aj].[\nu -ak]=0$
* $a^{3}-(i.\lambda +j.\mu +k.\nu )a^{2}+(i.\mu \times \nu +j.\nu \times \lambda +k.\lambda \times \mu )a-\lambda \times \mu .\nu =0.$
* $a^{3}-\Phi _{\text{S}}a^{2}+\{\Phi ^{-1}\}_{\text{S}}\left\vert \Phi \right\vert a-\left\vert \Phi \right\vert =0.$
$\Phi$
* $\Phi _{\text{S}},$$\{\Phi ^{-1}\}_{\text{S}},$$\left\vert \Phi \right\vert$
$a,$
$a,$
* $\left\vert \Phi -a{\text{I}}\right\vert =0.$
$\Phi -a{\text{I}}$
$\alpha$
* ${\begin{aligned}\{\Phi -a{\text{I}}\}.\alpha &=0,\\\Phi .\alpha &=a\alpha .\end{aligned}}$
$\Phi$
* $\Phi .\alpha =a\alpha ,$$\Phi .\beta =b\beta ,$$\Phi .\gamma =c\gamma ,$
$\alpha ,\beta ,\gamma$
* $\Phi =a\alpha \alpha '+b\beta \beta '+c\gamma \gamma ',$
$\alpha ',\beta ',\gamma '$
$\alpha ,\beta ,\gamma .$
In any case, we may suppose $a$ to have the same sign as $\left\vert \Phi \right\vert ,$ since the cubic equation must have such a root. Let $\alpha$ (as before) be normal to the plane of the consequents of the planar $\Phi -a{\text{I}},$ and $\alpha '$ normal to the plane of the antecedents, the lengths of $\alpha$ and $\alpha '$ being such that $\alpha .\alpha '=1.$ Let $\beta$ be any vector normal to $\alpha ',$ and such that $\Phi .\beta$ is not parallel to $\beta .$ (The case in which $\Phi .\beta$ is always parallel to $\beta ,$ if $\beta$ is perpendicular to $\alpha ',$ is evidently that of a tonic, and needs no farther discussion.) $\{\Phi -a{\text{I}}\}.\beta$ and therefore $\Phi .\beta$ will be perpendicular to $\alpha '.$ The same will be true of $\Phi ^{2}.\beta .$ Now (by No. 140)
* $[\Phi .\alpha ].[\Phi ^{2}.\beta ]\times [\Phi .\beta ]=\left\vert \Phi \right\vert \,\alpha .[\Phi .\beta ]\times \beta ,$
* $a\alpha .[\Phi ^{2}.\beta ]\times [\Phi .\beta ]=\left\vert \Phi \right\vert \,\alpha .[\Phi .\beta ]\times \beta .$
$[\Phi ^{2}.\beta ]\times [\Phi .\beta ]$
$[\Phi .\beta ]\times \beta$
* $a[\Phi ^{2}.\beta ]\times [\Phi .\beta ]=\left\vert \Phi \right\vert [\Phi .\beta ]\times \beta .$
$a^{-1}\left\vert \Phi \right\vert$
* $p^{2}=a^{-1}\left\vert \Phi \right\vert .$
| $\beta _{1}$ | $=p^{-1}\Phi .\beta ,$ | $\beta _{2}$ | $=p^{-2}\Phi ^{2}.\beta ,$ | etc., |
| $\beta _{-1}$ | $=p\Phi ^{-1}.\beta ,$ | $\beta _{-2}$ | $=p^{2}\Phi ^{-2}.\beta ,$ | etc., |
the vectors $\beta ,\beta _{1},\beta _{2},$ etc., $\beta _{-1},\beta _{-2},$ etc., will all lie in the plane perpendicular to $\alpha ',$ and we shall have
* $\beta _{2}\times \beta _{1}=\beta _{1}\times \beta ,$ $[\beta _{2}+\beta ]\times \beta _{1}=0.$
* $\beta _{2}+\beta =2n\beta _{1}.$
$p^{-1}\Phi ,$
$p\Phi ^{-1},$
| $\beta _{3}+\beta _{1}$ | $=2n\beta _{2},$ | $\beta _{4}+\beta _{2}$ | $2n\beta _{3},$ | etc., |
| $\beta _{1}+\beta _{-1}$ | $=2n\beta ,$ | $\beta +\beta _{-2}$ | $=2n\beta _{-1},$ | etc., |
Now, if $n>1,$ and we lay off from a common origin the vectors
* $\beta ,\beta _{1},\beta _{2}$etc.,$\beta _{-1},\beta _{-2}$etc.,
$\beta _{\infty }$
$\beta _{-\infty }.$
$\Phi .$
$\Phi$
$n<-1$
* $\beta ,-\beta _{1},\beta _{2},-\beta _{3},\beta _{4},$etc.,$-\beta _{-1},\beta _{-2},-\beta _{-3},$etc.,
$\Phi ,$
If $1>n>-1,$ we may set
* $n=\cos q.$
* $\beta _{-1}+\beta _{1}=2\cos q\,\beta .$
$\gamma$
* $\beta _{1}=\cos q\,\beta +\sin q\,\gamma .$
* $\beta _{-1}=\cos q\,\beta -\sin q\,\gamma .$
$\alpha '$
$\alpha ,\beta ,$
$\gamma .$
$\beta '$
$\gamma '$
* $\Psi =\cos q\{\beta \beta '+\gamma \gamma '\}+\sin q\{\gamma \beta '-\beta \gamma '\},$
* $\Psi .\alpha =0,$$\Psi .\beta =\beta _{1},$$\Psi .\beta _{-1}=\beta .$
* $\{a\alpha \alpha '+p\Psi \}.\alpha =a\alpha =\Phi .\alpha ,$ $\{a\alpha \alpha '+p\Psi \}.\beta =p\beta _{1}=\Phi .\beta ,$ $\{a\alpha \alpha '+p\Psi \}.\beta _{-1}=p\beta =\Phi .\beta _{-1},$
* $\Phi =a\alpha \alpha '+p\Psi =a\alpha \alpha '+p\,\cos q\{\beta \beta '+\gamma \gamma '\}+p\,\sin q\{\gamma \beta '-\beta \gamma '\}.$
If $n=\pm 1,$ (page 72), the dyadic may be reduced to the form
* $a\alpha \alpha '+b\{\beta \beta '+\gamma \gamma '\}+bc\beta \gamma ',$
$\alpha ,\beta ,\gamma$
$\alpha ',\beta ',\gamma '$
$a,b,c$
* ${\begin{aligned}a\alpha \alpha '&+\beta \beta '+\gamma \gamma ',\\\alpha \alpha '&+b\{\beta \beta '+\gamma \gamma '\},\\\alpha \alpha '&+\beta \beta '+\gamma \gamma '+c\beta \gamma '.\end{aligned}}$
$\beta ,$
$\alpha {\text{-}}\beta$
This dyadic is homologous with such as are obtained by varying the values of $a,b,c,$ and only with such, when the values of $a$ and $b$ are different, and that of $c$ other than zero.
157. If the planar $\Phi -a{\text{I}}$ (page 71) has perpendicular planes, there may be another value of $a,$ of the same sign as $\left\vert \Phi \right\vert ,$ which will give a planar which has not perpendicular planes. When this is not the case, the dyadic may always be reduced to the form
* $a\{\alpha \alpha '+\beta \beta '+\gamma \gamma '\}+ab\{\alpha \beta '+\beta \gamma '\}+ae\alpha \gamma ',$
$\alpha ,\beta ,\gamma$
$\alpha ',\beta ',\gamma ',$
$a,b,c,$
* $a{\text{I}}$and${\text{I}}+b\{\alpha \beta '+\beta \gamma '\}+c\alpha \gamma '.$
$\alpha {\text{-}}\gamma$
$b\beta +c\alpha$
$\alpha {\text{-}}\beta$
​This dyadic is homologous with snch as are obtained by varying the values of $a,b,c,$ and only such, unless $b=0.$
It is always possible to take three mutually perpendicular vectors for $\alpha ,\beta ,$ and $\gamma ;$ or, if it be preferred, to take such values for these vectors as shall make the term containing $c$ vanish.
158. The dyadics described in the two last paragraphs may be called shearing dyadics.
The criterion of a shearer is
* $\{\Phi -{\text{I}}\}^{3}=0,$$\Phi -{\text{I}}\neq 0.$
* $\{\Phi -{\text{I}}\}^{2}=0,$$\Phi -{\text{I}}\neq 0.$
* $\{\Phi -{\text{I}}\}^{2}=0,$$\{\Phi -{\text{I}}\}^{2}\neq 0.$
Note.—If a dyadic $\Phi$ is a linear function of a vector $p$ (the term linear being used in the same sense as in No. 105), we may represent the relation by an equation of the form
* $\Phi =\alpha \beta \,\gamma .\rho +e\zeta \,\eta .\rho +{\text{etc.,}}$
* $\Phi =\{\alpha \beta \gamma +e\zeta \eta +{\text{etc.,}}\}.\rho ,$
$\alpha \beta \gamma$
$\alpha ,\beta ,\gamma .$
It may sometimes be convenient to use notations like
* ${\frac {\lambda ,\mu ,\nu }{\vert \alpha ,\beta ,\gamma }}$and${\frac {\lambda ,\mu ,\nu }{\alpha ,\beta ,\gamma \vert }}$
$\alpha ,\beta ,\gamma$
$\lambda ,\mu ,\nu ,$
* $\lambda \alpha '+\mu \beta '+\nu \gamma '$and$\alpha '\lambda +\beta '\mu +\gamma '\nu$
$\alpha ',\beta ',\gamma '$
$\alpha ,\beta ,\gamma .$
$\tau$
$\rho ,$
$\rho$
$\tau$
* ${\frac {\tau }{\vert \rho }}$and${\frac {\tau }{\rho \vert }}\cdot$
$\tau$
$\rho ,$
$d\rho$
$d\tau$
* ${\frac {d\tau }{\vert d\rho }}$and${\frac {d\tau }{d\rho \vert }}\cdot$
$\rho$
$\nabla \tau .$
$d\rho$
${\frac {d\tau }{\vert d\rho }}$
${\frac {d^{2}\tau }{\vert d\rho ^{2}}},$
$d\rho$
${\frac {d\tau }{d\rho \vert }}$
${\frac {d^{2}\tau }{d\rho ^{2}\vert }}\cdot$
$\nabla \nabla \tau$
CHAPTER IV.
(Supplementary to Chapter II.)
concerning the differential and integral calculus of vectors.
169. If $\omega$ is a vector having continuously varying values in space, and $\rho$ the vector determining the position of a point, we may set
* ${\begin{aligned}\rho &=xi+yj+zk,\\d\rho &=dx\,i+dy\,j+dz\,k,\end{aligned}}$
$\omega$
$\rho$
$x,y,$
$z.$
* $d\omega =dx{\frac {d\omega }{dx}}+dy{\frac {d\omega }{dy}}+dz{\frac {d\omega }{dz}},$
* $d\omega =d\rho .\left\lbrace i{\frac {d\omega }{dx}}+j{\frac {d\omega }{dy}}+k{\frac {d\omega }{dz}}\right\rbrace \cdot$
* $\nabla \omega =i{\frac {d\omega }{dx}}+j{\frac {d\omega }{dy}}+k{\frac {d\omega }{dz}},$ $d\omega =d\rho .\nabla \omega .$
$\nabla$
* $i{\frac {d}{dx}}+j{\frac {d}{dy}}+k{\frac {d}{dz}},$
$\nabla \omega$
$x,y,$
$z,$
$\nabla u$
$u$
$\rho$
$u$
$x,y,$
$z.$
It is evident that the expressions $\nabla .\omega$ and $\nabla \times \omega$ already defined (No. 54) are equivalent to $\{\nabla \omega \}_{\text{S}}$ and $\{\nabla \omega \}_{\text{X}}.$
160. An important case is that in which the vector operated on is of the form $\nabla u.$ We have then
* $d\nabla u=d\rho .\nabla \nabla u,$
* $\nabla \nabla u={\begin{Bmatrix}{\frac {d^{2}u}{dx^{2}}}ii&+{\frac {d^{2}u}{dx\,dy}}ij&+{\frac {d^{2}u}{dx\,dz}}ik\\+{\frac {d^{2}u}{dy\,dx}}ji&+{\frac {d^{2}u}{dy^{2}}}jj&+{\frac {d^{2}u}{dy\,dz}}jk\\+{\frac {d^{2}u}{dz\,dx}}ki&+{\frac {d^{2}u}{dz\,dy}}kj&+{\frac {d^{2}u}{dz^{2}}}kk.\end{Bmatrix}}$
$u$
$x,y,$
$z.$
161. The operators $\nabla \times$ and $\nabla .$ may be applied to dyadics in a manner entirely analogous to their use with scalars. Thus we may define $\nabla \times \Phi$ and $\nabla .\Phi$ by the equations
* ${\begin{aligned}\nabla \times \Phi &=i\times {\frac {d\Phi }{dx}}+j\times {\frac {d\Phi }{dy}}+k\times {\frac {d\Phi }{dz}}\\\nabla .\Phi &=i.\,{\frac {d\Phi }{dx}}+j.\,{\frac {d\Phi }{dy}}+k.\,{\frac {d\Phi }{dz}}\end{aligned}}$
* ${\begin{aligned}\Phi &=\alpha i+\beta j+\gamma k,\\\nabla \times \Phi &=\nabla \times \alpha i+\nabla \times \beta j+\nabla \times \gamma k.\\\nabla .\Phi &=\nabla .\alpha i+\nabla .\beta j+\nabla .\gamma k.\end{aligned}}$
* ${\begin{aligned}\Phi &=i\alpha +j\beta +k\gamma ,\\\nabla \times \Phi &=i\left[{\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right]+j\left[{\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}\right]+k\left[{\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}\right],\\\nabla .\Phi &={\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}\cdot \end{aligned}}$
$\nabla .\nabla$
$\nabla .\nabla \omega$
* ${\frac {d^{2}\omega }{dx^{2}}}+{\frac {d^{2}\omega }{dy^{2}}}+{\frac {d^{2}\omega }{dz^{2}}}$
$\nabla .\nabla \Phi$
* ${\frac {d^{2}\Phi }{dx^{2}}}+{\frac {d^{2}\Phi }{dy^{2}}}+{\frac {d^{2}\Phi }{dz^{2}}},$
$\nabla .\nabla$
$\nabla \Phi .$
That $\nabla .\nabla \Phi =\nabla \nabla .\Phi -\nabla \times \nabla \times \Phi$ will be evident if we suppose $\Phi$ to be expressed in the form $\alpha i+\beta j+\gamma k.$ (See No. 71.)
163. We have already seen that
* $u''-u'=\int d\rho .\nabla u,$
$u'$
$u''$
$u$
* $\omega ''-\omega '=\int d\rho .\nabla \omega .$
| $\iint d\sigma \,u=\iiint dv\,\nabla u,$ | (1) |
| $\iint d\sigma \,\omega =\iiint dv\,\nabla \omega ,$ | (2) |
| $\iint d\sigma .\omega =\iiint dv\,\nabla .\omega ,$ | (3) |
| $\iint d\sigma .\Phi =\iiint dv\,\nabla .\Phi ,$ | (4) |
| $\iint d\sigma \times \omega =\iiint dv\,\nabla \times \omega ,$ | (5) |
| $\iint d\sigma \times \Phi =\iiint dv\,\nabla \times \Phi .$ | (6) |
$\iiint dv\,\nabla$
$\iint d\sigma .$
165. The following equations between line-integrals for a closed line and surface-integrals for any surface bounded by the line, may also be mentioned. (One of these has already been given. See No. 60.)
| $\int d\rho \,u=\iint d\sigma \times \nabla u,$ | (1) |
| $\int d\rho \,\omega =\iint d\sigma \times \nabla \omega ,$ | (2) |
| $\int d\rho .\omega =\iint d\sigma .\nabla \times \omega ,$ | (3) |
| $\int d\rho .\Phi =\iint d\sigma .\nabla \times \Phi ,$ | (4) |
| $\int d\rho \times \omega =\iint \nabla \omega .d\sigma -\iint d\sigma \,\nabla .\omega .$ | (5) |
$\iint [d\sigma \times \nabla ]$
$\int d\rho .$
$d\sigma$
$i,j,k$
$\nabla$
166. To the equations in No. 65 may be added many others, as,
| $\nabla [u\omega ]=\nabla u\omega +u\nabla \omega ,$ | (1) |
| $\nabla [\tau \times \omega ]=\nabla \tau \times \omega -\nabla \omega \times \tau ,$ | (2) |
| $\nabla \times [\tau \times \omega ]=\omega .\nabla \tau -\nabla .\tau \omega -\tau .\nabla \omega +\nabla .\omega \tau ,$ | (3) |
| $\nabla (\tau .\omega )=\nabla \tau .\omega +\nabla \omega .\tau ,$ | (4) |
| $\nabla .\{\tau \omega \}=\nabla .\tau \omega +\tau .\nabla \omega ,$ | (5) |
| $\nabla \times \{\tau \omega \}=\nabla \times \tau \omega -\tau \times \nabla \omega ,$ | (6) |
| $\nabla .\{u\Phi \}=\nabla u.\Phi +u\nabla .\Phi ,$ | (7) |
* ${\text{etc.}}$
$\nabla ,\nabla .,\nabla \times$
$\nabla$
$\nabla$
167. From the relations indicated in the last four paragraphs, may be obtained directly a great number of transformations of definite integrals similar to those given in Nos. 74–77, and corresponding to those known in the scalar calculus by the name of integration by parts.
168. The student will now find no difficulty in generalizing the integrations of differential equations given in Nos. 78–89 by applying to vectors those which relate to scalars, and to dyadics those which relate to vectors.
169. The propositions in No. 90 relating to minimum values of the volume-integral $\iiint u\omega .\omega \,dv$ may be generalized by substituting $\omega .\Phi .\omega$ for $u\omega .\omega ,\Phi$ being a given dyadic function of position in space.
170. The theory of the integrals which have been called potentials, Newtonians, etc (see Nos. 91–102) may be extended to cases in which the operand is a vector instead of a scalar or a dyadic instead of a vector. So far as the demonstrations are concerned, the case of a vector may be reduced to that of a scalar by considering separately its three components, and the case of a dyadic may be reduced to that of a vector, by supposing the dyadic expressed in the form $\phi i+\chi j+\omega k$ and considering each of these terms separately.
CHAPTER V.
concerning transcendental functions of dyadics.
171. Def.—The exponential function, the sine and the cosine of a dyadic may be defined by infinite series, exactly as the corresponding functions in scalar analysis, viz.,
* ${\begin{aligned}e^{\Phi }&={\text{I}}+\Phi +{\tfrac {1}{2}}\Phi ^{2}+{\tfrac {1}{2.3}}\Phi ^{3}+{\text{etc.,}}\\\sin \Phi &=\Phi -{\tfrac {1}{2.3}}\Phi ^{3}+{\tfrac {1}{2.3.4.5}}\Phi ^{5}-{\text{etc.,}}\\\cos \Phi &={\text{I}}-{\tfrac {1}{2}}\Phi ^{2}+{\tfrac {1}{2.3.4}}\Phi ^{4}-{\text{etc.}}\end{aligned}}$
$\Phi$
* $\left({\text{I}}-{\frac {\Phi }{\text{N}}}\right)^{\text{N}},$
${\text{N}},$
These functions of $\Phi$ are homologous with $\Phi .$
172. We may define the logarithm as the function which is the inverse of the exponential, so that the equations
* ${\begin{aligned}e^{\Psi }&=\Phi ,\\\Psi &=\log \Phi ,\end{aligned}}$
173. It follows at once from the second definition of the exponential function that, if $\Phi$ and $\Psi$ are homologous,
* $e^{\Phi }.e^{\Psi }=e^{\Phi +\Psi },$
${\text{T}}$
* $\{e^{\Phi }\}^{\text{T}}=e^{{\text{T}}\Phi }.$
$\Xi$
$\Phi$
* $\Xi ^{2}.\Phi =-\Phi ,$
* ${\begin{aligned}e^{\Xi .\Phi }&=\cos \Phi +\Xi \,\sin \Phi ,\\e^{-\Xi .\Phi }&=\cos \Phi -\Xi \,\sin \Phi ,\\\end{aligned}}$
* ${\begin{aligned}\cos \Phi &={\tfrac {1}{2}}\{e^{\Xi \Phi }+e^{-\Xi \Phi }\},\\\sin \Phi &=-{\tfrac {1}{2}}\Xi \{e^{\Xi \Phi }-e^{-\Xi \Phi }\}.\end{aligned}}$
$\Phi .\Psi =\Psi .\Phi =0,$
* $\{\Phi +\Psi \}^{2}=\Phi ^{2}+\Psi ^{2},$$\{\Phi +\Psi \}^{3}=\Phi ^{3}+\Psi ^{3},{\text{ etc.}}$
* ${\begin{aligned}e^{\Phi +\Psi }&=e^{\Phi }+e^{\Psi }-{\text{I}},\\\cos\{\Phi +\Psi \}&=\cos \Phi +\cos \Psi -{\text{I}},\\\sin\{\Phi +\Psi \}&=\sin \Phi +\sin \Psi .\end{aligned}}$
* $\left\vert e^{\Phi }\right\vert =e^{\Phi _{\text{S}}}.$
* $\left\vert \{{\text{I}}+{\text{N}}^{-1}\Phi \}^{\text{N}}\right\vert ,$that is, of$\left\vert {\text{I}}+{\text{N}}^{-1}\Phi \right\vert ^{\text{N}}.$
$\Phi =\alpha i+\beta j+\gamma k,$
* $(1+{\text{N}}^{-1}\alpha .i+{\text{N}}^{-1}\beta .j+{\text{N}}^{-1}\gamma .k)^{\text{N}},$or$(1+{\text{N}}^{-1}\Phi _{\text{S}})^{\text{N}},$
177. By the definition of exponentials, the expression
* $e^{q\{kj-jk\}}$
* $\{{\text{I}}+q{\text{N}}^{-1}\{kj-jk\}\}^{\text{N}}.$
${\text{I}}+q{\text{N}}^{-1}\{kj-jk\}$
$i$
$q{\text{N}}^{-1}.$
$q.$
$qi=\omega ,$
* $e^{{\text{I}}\times \omega }.$
$\omega ,$
$\omega .$
$\omega$
$2\pi .$
178. If, as in No. 151,
* $\Phi =a\alpha \alpha '+b\beta \beta '+c\gamma \gamma ',$
* ${\begin{aligned}e^{\Phi }&=e^{a}\alpha \alpha '+e^{b}\beta \beta '+e^{c}\gamma \gamma ',\\\cos \Phi &=\cos a\,\alpha \alpha '+\cos b\,\beta \beta '+\cos c\,\gamma \gamma ',\\\sin \Phi &=\sin a\,\alpha \alpha '+\sin b\,\beta \beta '+\sin c\,\gamma \gamma ',\end{aligned}}$
$a,b,c$
* $\log \Phi =\log a\alpha \alpha '+\log b\beta \beta '+\log c\gamma \gamma '.$
* $\Phi =a\alpha \alpha '+b\{\beta \beta '+\gamma \gamma '\}+c\{\gamma \beta '-\beta \gamma '\}=a\alpha \alpha '+p\,\cos b\,\beta \beta '\{\beta \beta '+\gamma \gamma '\}+p\,\sin \,q\{\gamma \beta '-\beta \gamma '\},$
* $e^{\Phi }=e^{a\alpha \alpha '}.e^{b\{\beta \beta '+\gamma \gamma '\}}.e^{c\{\gamma \beta '-\beta \gamma '\}}.$
* ${\begin{aligned}e^{a\alpha \alpha '}&=e^{a}\alpha \alpha '+\beta \beta '+\gamma \gamma ',\\e^{b\{\beta \beta '+\gamma \gamma '\}}&=\alpha \alpha '+e^{b}\{\beta \beta '+\gamma \gamma '\},\\e^{c\{\gamma \beta '-\beta \gamma '\}}&=\alpha \alpha '+\cos \,c\{\beta \beta '+\gamma \gamma '\}+e^{b}\sin \,c\{\gamma \beta '-\beta \gamma '\}.\end{aligned}}$
* $e^{\Phi }=e^{a}\alpha \alpha '+e^{b}\cos \,c\{\beta \beta '+\gamma \gamma '\}+e^{b}\sin \,c\{\gamma \beta '-\beta \gamma '\}.$
$a$
* $\log \Phi =\log a\,\alpha \alpha '+\log p\,\{\beta \beta '+\gamma \gamma '\}+q\{\gamma \beta '-\beta \gamma '\}.$
$\Phi$
$q$
$2\pi ,$
$\log \Phi$
To find the value of $\cos \Phi$ and $\sin \Phi ,$ let us set
* ${\begin{aligned}\Theta &=b\{\beta \beta '+\gamma \gamma '\}+c\{\gamma \beta '-\beta \gamma '\},\\\Xi &=\gamma \beta '-\beta \gamma '.\end{aligned}}$
* $\cos \Phi =\cos\{a\alpha \alpha '\}+\cos \Theta -{\text{I}}.$
* $\cos\{a\alpha \alpha '\}-{\text{I}}=\cos a\alpha \alpha '-\alpha \alpha '.$
* $\cos \Phi =\cos a\alpha \alpha '-\alpha \alpha '+\cos \Theta .$
* $\cos \Theta ={\tfrac {1}{2}}\{e^{\Xi .\Theta }+e^{-\Xi .\Theta }\}.$
* ${\begin{aligned}\Xi .\Theta &=-c\{\beta \beta '+\gamma \gamma '\}+b\{\gamma \beta '-\beta \gamma '\},\\e^{\Xi .\Theta }&=\alpha \alpha '+e^{-c}\cos b\{\beta \beta '+\gamma \gamma '\}+e^{-c}\sin b\{\gamma \beta '-\beta \gamma '\},\\e^{-\Xi .\Theta }&=\alpha \alpha '+e^{c}\cos b\{\beta \beta '+\gamma \gamma '\}-e^{c}\sin b\{\gamma \beta '-\beta \gamma '\}.\end{aligned}}$
* $\cos \Theta =\alpha \alpha '+{\tfrac {1}{2}}(e^{c}+e^{-c})\cos b\{\beta \beta '+\gamma \gamma '\}-{\tfrac {1}{2}}(e^{c}-e^{-c})\sin b\{\gamma \beta '-\beta \gamma '\}.$
* $\cos \Phi =\cos a\,\alpha \alpha '+{\tfrac {1}{2}}(e^{c}+e^{-c})\sin b\{\beta \beta '+\gamma \gamma '\}-{\tfrac {1}{2}}(e^{c}-e^{-c})\cos b\{\gamma \beta '-\beta \gamma '\}.$
* $\sin \Phi =\sin a\,\alpha \alpha '+{\tfrac {1}{2}}(e^{c}+e^{-c})\sin b\{\beta \beta '+\gamma \gamma '\}+{\tfrac {1}{2}}(e^{c}-e^{-c})\cos b\{\gamma \beta '-\beta \gamma '\}.$
$\alpha ,\beta ,\gamma$
$\alpha ',\beta ',\gamma '$
* $\Phi =a\,\alpha \alpha '+b\{\beta \beta '+\gamma \gamma '\}+c\,\beta \gamma ',$
${\text{N}}$
* $\Phi ^{\text{N}}=a^{\text{N}}\,\alpha \alpha '+b^{\text{N}}\{\beta \beta '+\gamma \gamma '\}+{\text{N}}b^{{\text{N}}-1}c\,\beta \gamma '.$
* ${\begin{aligned}e^{\Phi }&=e^{a}\alpha \alpha '+e^{b}\{\beta \beta '+\gamma \gamma '\}+e^{b}\,c\,\beta \gamma ',\\\cos \Phi &=\cos a\,\alpha \alpha '+\cos b\,\{\beta \beta '+\gamma \gamma '\}-c\sin b\,\beta \gamma ',\\\sin \Phi &=\sin a\,\alpha \alpha '+\sin b\,\{\beta \beta '+\gamma \gamma '\}-c\cos b\,\beta \gamma '.\end{aligned}}$
$a$
$b$
$c$
* $\log \Phi =\log a\,\alpha \alpha '+\log b\,\{\beta \beta '+\gamma \gamma '\}+cb^{-1}\beta \gamma .$
$\alpha ,\beta ,\gamma ,$
$\alpha ',\beta ',\gamma '$
* $\Phi =a{\text{I}}+b\{\alpha \beta '+\beta \gamma '\}+c\,\alpha \gamma ',$
${\text{N}}$
* $\Phi ^{\text{N}}=a^{\text{N}}{\text{I}}+{\text{N}}a^{{\text{N}}-1}b\{\alpha \beta '+\beta \gamma '\}+({\text{N}}a^{{\text{N}}-1}c+{\tfrac {1}{2}}{\text{N}}({\text{N}}-1)a^{{\text{N}}-2}b^{2})\alpha \gamma '.$
* ${\begin{aligned}e^{\Phi }&=e^{a}{\text{I}}+e^{a}b\{\alpha \beta '+\beta \gamma '\}+e^{a}({\tfrac {1}{2}}b^{2}+c)\alpha \gamma ',\\\cos \Phi &=\cos a\,{\text{I}}-b\,\sin a\,\{\alpha \beta '+\beta \alpha '\}-({\tfrac {1}{2}}b^{2}\cos a+c\sin a)\alpha \gamma ',\\\sin \Phi &=\sin a\,{\text{I}}+b\,\cos a\,\{\alpha \beta '+\beta \alpha '\}-({\tfrac {1}{2}}b^{2}\sin a-c\cos a)\alpha \gamma '.\end{aligned}}$
$b=0,$
* $\log \Phi =\log a{\text{ I}}+ba^{-1}\{\alpha \beta '+\beta \alpha '\}+(ca^{-1}-{\tfrac {1}{2}}b^{2}a^{-2})\alpha \gamma '.$
$\Phi$
| $d\{e^{\Phi }\}=e^{\Phi }.d\Phi ,$ | (1) |
| $d\sin \Phi =\cos \Phi .d\Phi ,$ | (2) |
| $d\cos \Phi =-\sin \Phi .d\Phi ,$ | (3) |
| $d\log \Phi =\Phi ^{-1}.d\Phi ,$ | (4) |
$\Phi$
183. If, however, $\Gamma$ is any constant dyadic, the variations of $t\Gamma$ will necessarily be homologous with $t\Gamma$ and we may write without other limitation than that $\Gamma$ is constant,
| ${\frac {d\{e^{t\Gamma }\}}{dt}}=\Gamma .e^{t\Gamma }$ | (1) |
| ${\frac {d\sin\{t\Gamma \}}{dt}}=\Gamma .\cos\{t\Gamma \},$ | (2) |
| ${\frac {d\cos\{t\Gamma \}}{dt}}=-\Gamma .\sin\{t\Gamma \},$ | (3) |
| ${\frac {d\log\{t\Gamma \}}{dt}}={\frac {I}{t}}\cdot$ | (4) |
| ${\frac {d^{2}\{e^{t\Gamma }\}}{dt^{2}}}=\Gamma ^{2}.e^{t\Gamma }$ | (5) |
| ${\frac {d^{2}\sin\{t\Gamma \}}{dt^{2}}}=-\Gamma ^{2}.\sin\{t\Gamma \},$ | (6) |
| ${\frac {d^{2}\cos\{t\Gamma \}}{dt^{2}}}=-\Gamma ^{2}.\cos\{t\Gamma \}.$ | (7) |
* ${\frac {d\rho }{dt}}=\Gamma .\rho ,$
* $\rho =e^{t\Gamma }.\rho ',$
$\rho '$
$\rho$
$t=0.$
* ${\frac {d\rho }{dt}}=\Gamma .e^{t\Gamma }.\rho '=\Gamma .\rho ,$
$\rho$
$t=0.$
185. Def.—A flux which is a linear function of the position-vector is called a homogeneous-strain-flux from the nature of the strain which it produces. Such a flux may evidently be represented by a dyadic.
In the equations of the last paragraph, we may suppose $\rho$ to represent a position-vector, $t$ the time, and $\Gamma$ a homogeneous-strain-flux. Then $e^{t\Gamma }$ will represent the strain produced by the flux $\Gamma$ in the time $t.$
$\Lambda$
$\{\log \Lambda \}/t$
$\Lambda$
$t.$
​186. If we have
* ${\frac {d^{2}\rho }{dt^{2}}}=\Gamma ^{2}.\rho ,$
$\Gamma$
* $\rho =e^{t\Gamma }.\alpha +e^{-t\Gamma }.\beta .$
* ${\frac {d\rho }{dt}}=\Gamma .e^{t\Gamma }.\alpha -\Gamma .e^{-t\Gamma }.\beta ,$
* ${\frac {d^{2}\rho }{dt^{2}}}=\Gamma ^{2}.e^{t\Gamma }.\alpha -\Gamma ^{2}.e^{-t\Gamma }.\beta =\Gamma ^{2}.\rho ,$
$\alpha$
$\beta$
* ${\begin{aligned}\rho _{t=0}&=\alpha +\beta ,\\\left[{\frac {d\rho }{dt}}\right]_{t=0}=\Gamma .\{\alpha -\beta \}.\end{aligned}}$
* ${\frac {d^{2}\rho }{dt^{2}}}=-\Gamma ^{2}.\rho$
* $\rho =\cos\{t\Gamma \}.\alpha +\sin\{t\Gamma \}.\beta ,$
* ${\frac {d\rho }{dt}}=-\Gamma .\sin\{t\Gamma \}.\alpha +\Gamma .\cos\{t\Gamma \}.\beta ,$
* ${\frac {d^{2}\rho }{dt^{2}}}=-\Gamma ^{2}.\cos\{t\Gamma \}.\alpha -\Gamma ^{2}.\sin\{t\Gamma \}.\beta .=-\Gamma ^{2}.\rho .$
$\Gamma$
$\alpha$
$\beta$
* ${\begin{aligned}\rho _{t=0}&=\alpha ,\\\left[{\frac {d\rho }{dt}}\right]_{t=0}&=\Gamma .\beta .\end{aligned}}$
${\frac {d^{2}\rho }{dt^{2}}}=\{\Gamma ^{2}-\Lambda ^{2}\}.\rho ,$
$\Gamma ^{2}-\Lambda ^{2}$
* $\Gamma .\Lambda =\Lambda .\Gamma =0,$
* $\rho =\{{\tfrac {1}{2}}e^{t\Gamma }+{\tfrac {1}{2}}e^{-t\Gamma }+\cos\{t\Lambda \}-{\text{I}}.\alpha +\{{\tfrac {1}{2}}e^{t\Gamma }-{\tfrac {1}{2}}e^{-t\Gamma }+\sin\{t\Lambda \}\}.\beta$
* ${\begin{aligned}{\frac {d\rho }{dt}}&=\{{\tfrac {1}{2}}\Gamma .e^{t\Gamma }-{\tfrac {1}{2}}\Gamma .e^{-t\Gamma }-\Lambda .\sin\{t\Lambda \}\}.\alpha +\{{\tfrac {1}{2}}\Gamma .e^{t\Gamma }+{\tfrac {1}{2}}\Gamma .e^{-t\Gamma }+\Lambda .\cos\{t\Lambda \}\}.\beta ,\\{\frac {d^{2}\rho }{dt^{2}}}&=\{{\tfrac {1}{2}}\Gamma ^{2}.e^{t\Gamma }+{\tfrac {1}{2}}\Gamma ^{2}.e^{-t\Gamma }-\Lambda ^{2}.\cos\{t\Lambda \}\}.\alpha +\{{\tfrac {1}{2}}\Gamma ^{2}.e^{t\Gamma }-{\tfrac {1}{2}}\Gamma ^{2}.e^{-t\Gamma }-\Lambda ^{2}.\sin\{t\Lambda \}\}.\beta \\&=\{\Gamma ^{2}-\Lambda ^{2}\}.\rho .\end{aligned}}$
$\alpha$
$\beta$
* ${\begin{aligned}\rho _{t=0}&=\alpha ,\\\left[{\frac {d\rho }{dt}}\right]_{t=0}&=\{\Gamma +\Lambda \}.\beta ,\end{aligned}}$
* ${\frac {d^{2}\rho }{dt^{2}}}=\Theta .\rho$
$\Theta$
note on bivector analysis.
1. A vector is determined by three algebraic quantities. It often occurs that the solution of the equations by which these are to be detenmined gives imaginary values, i.e., instead of scalars we obtain biscalars, or expressions of the form $a+\iota \,b,$ where $a$ and $b$ are scalars, and $\iota ={\sqrt {-1}}.$ It is most simple, and always allowable, to consider the vector as determined by its components parallel to a normal system of axes. In other words, a vector may be represented in the form
* $xi+yj+zk.$
$x,y,$
$z,$
* ${\begin{aligned}x&=x_{1}+\iota x_{2},\\y&=y_{1}+\iota y_{2},\\z&=z_{1}+\iota z_{2},\end{aligned}}$
$x_{1},x_{2},y_{1},y_{2},z_{1},z_{2}$
* $xi+yj+zk,$
* $(x_{1}+\iota x_{2})i+(y_{1}+\iota y_{2})j+(z_{1}+\iota z_{2})k;$
* ${\begin{aligned}\rho _{1}&=x_{1}i+y_{1}j+z_{1}k,\\\rho _{2}&=x_{2}i+y_{2}j+z_{2}k,\end{aligned}}$
* $\rho _{1}+\iota \rho _{2}.$
* ${\mathfrak {r}}=\rho _{1}+\iota \rho _{2}.$
$\rho _{1}$
$\rho _{2}$
$(a+\iota b)\rho ,$
* $(x_{1}+\iota x_{2})i+(y_{1}+\iota y_{2})j+(z_{1}+\iota z_{2})k$
$i,j,$
$k,$
2. The consideration that operations upon bivectors may be regarded as operations upon their biscalar x-, y- and z-components is sufficient to show the possibility of a bivector analysis and to indicate what its rules must be. But this point of view does not afford the most simple conception of the operations which we have to perform upon bivectors. It is desirable that the definitions of the fundamental operations should be independent of such extraneous considerations as any system of axes.
The various signs of our analysis, when applied to bivectors, may therefore be defined as follows, viz.,
The bivector equation
* $\mu '+\iota \nu '=\mu ''+\iota \nu ''$
* $\mu '=\mu ''$and$\nu '=\nu ''.$
| $-[\mu +\iota \nu ]$ | $=-\mu +\iota [-\nu ].$ |
| $[\mu '+\iota \nu ']+[\mu ''+\iota \nu '']$ | $=[\mu '+\mu '']+\iota [\nu '+\nu ''].$ |
| $[\mu '+\iota \nu '].[\mu ''+\iota \nu '']$ | $=[\mu '.\mu ''-\nu '.\nu '']+\iota [\mu '.\nu ''+\nu '.\mu ''].$ |
| $[\mu '+\iota \nu ']\times [\mu ''+\iota \nu '']$ | $=[\mu '\times \mu ''-\nu '\times \nu '']+\iota [\mu '\times \nu ''+\nu '\times \mu ''].$ |
3. Def.—Two bivectors, or two biscalars, are said to be conjugate, when their real parts are the same, and their imaginary parts difler in sign, and in sign only.
Hence, the product of the conjugates of any number of bivectors and biscalars is the conjugate of the product of the bivectors and biscalars. This is true of any kind of product.
The products of a vector and its conjugate are as follows:
| $[\mu +\iota \nu ].[\mu -\iota \nu ]$ | $=\mu .\mu +\nu .\nu$ |
| $[\mu +\iota \nu ]\times [\mu -\iota \nu ]$ | $=2\iota \nu \times \mu$ |
| $[\mu +\iota \nu ][\mu -\iota \nu ]$ | $=\{\mu \mu +\nu \nu \}+\iota \{\nu \mu -\mu \nu \}.$ |
$\mu$
$\nu$
* $\mu .\mu +\nu .\nu ,$$\mu \times \nu ,$$\mu \mu +\nu \nu ,$$\nu \mu -\mu \nu ,$
$a+\iota b,$
$a^{2}+b^{2}=1,$
* $\mu '+\iota \nu '=(a+\iota b)[\mu +\iota \nu ],$
* $\mu '-\iota \nu '=(a-\iota b)[\mu -\iota \nu ],$
* $[\mu '+\iota \nu '].[\mu '-\iota \nu ']=[\mu +\iota \nu ].[\mu -\iota \nu ].$
* $\mu '.\mu '+\nu '.\nu '=\mu .\mu +\nu .\nu ;$
4. Def.—In biscalar analysis, the product of a biscalar and its conjugate is a positive scalar. The positive square root of this scalar is called the modulus of the biscalar. In bivector analysis, the direct product of a bivector and its conjugate is, as seen above, a positive scalar. The positive square root of this scalar may be called the modulus of the bivector. When this modulus vanishes, the bivector vanishes, and only in this case. If the bivector is multiplied by a biscalar, its modulus is multiplied by the modulus of the biscalar. The conjugate of a (real) vector is the vector itself, and the modulus of the vector is the same as its magnitude.
5. Def.—If between two vectors, $\alpha$ and $\beta ,$ there subsists a relation of the form
* $\alpha =n\beta ,$
$n$
* ${\mathfrak {a}}=m{\mathfrak {b}},$
$m$
To aid us in comprehending the geometrical signification of this relation, we may regard the biscalar as consisting of two factors, one of which is a positive scalar (the modulus of the biscalar), and the other may be put in the form $\cos q+\iota \sin q.$ The effect of multiplying a bivector by a positive scalar is obvious. To understand the effect of a multiplier of the form $\cos q+\iota \sin q$ upon a bivector $\mu +\iota \nu ,$ let us set
* $\mu '+\iota \nu '=(\cos q+\iota \sin q)[\mu +\iota \nu ].$
* ${\begin{aligned}\mu '&=\cos q\mu -\sin q\nu ,\\\nu '&=\cos q\nu +\sin q\mu .\end{aligned}}$
$\mu$
$\nu$
$q,$
$\nu$
$\mu .$
$\mu$
$\nu$
$\mu '$
$\nu '$
$q.$
$\mu$
$\nu$
$\mu '$
$\nu '$
$\mu$
$\mu '$
$\nu$
$\nu ',$
$q$
$2\pi ,$
$\nu$
$\nu '$
$\nu$
$\mu ,$
$\mu$
$\mu '$
$\mu$
$-\nu .$
It follows that any bivector $\mu +\iota \nu$ may be put in the form
* $(\cos q+\iota \sin q)[\alpha +\iota \beta ],$
$\alpha$
$\beta$
$\mu$
$\nu$
​6. To reduce a given bivector ${\mathfrak {r}}$ to the above form, we may set
* ${\begin{aligned}{\mathfrak {r}}.{\mathfrak {r}}&=(\cos q+\iota \sin q)^{2}[\alpha +\iota \beta ].[\alpha +\iota \beta ]\\&=(\cos 2q+\iota \sin 2q)(\alpha .\alpha -\beta .\beta )\\&=a+\iota b,\end{aligned}}$
$a$
$b$
$q$
* $\tan 2q={\frac {b}{a}},$
$2q$
$\sin 2q$
$\cos 2q$
$b$
$a.$
$\alpha$
$\beta$
* $\alpha +\iota \beta =(\cos q-\iota \sin q){\mathfrak {r}}.$
* ${\mathfrak {r}}.{\mathfrak {r}}=0.$
* ${\mathfrak {r}}=0,$
${\mathfrak {r}}$
$xi+yj+zk,$
$x,y,z$
* $x^{2}+y^{2}+z^{2}=0,$
$x,y,$
$z$
7. Def.—We call two vectors $\rho$ and $\sigma$ perpendicular when $\rho .\sigma =0.$ Allowing the same analogy, we shall call two bivectors ${\mathfrak {r}}$ and ${\mathfrak {s}}$ perpendicular, when
* ${\mathfrak {r}}.{\mathfrak {s}}=0.$
${\mathfrak {r}}$
${\mathfrak {s}}$
${\mathfrak {r}}$
${\mathfrak {s}}$
* $m[\alpha +\iota \beta ],$$m'[\alpha '+\iota \beta '],$
$m$
$n'$
$\alpha$
$\beta$
$\alpha '$
$\beta .$
${\mathfrak {r}}.{\mathfrak {s}}=0$
* $\beta .\beta '=0,$and$\alpha .\beta '+\beta .\alpha '=0.$
${\mathfrak {r}}$
${\mathfrak {s}}$
It will be observed that every circular bivector is perpendicular to itself, and to every parallel bivector.
If two bivectors, $\mu +\iota \nu ,\,\mu '+\iota \nu ',$ which do not lie in the same plane are perpendicular, we may resolve $\mu$ and $\nu$ into components parallel and perpendicular to the plane of $\mu '$ and $\nu '.$ The components perpendicular to the plane evidently contribute nothing to the value of
* $[\mu +\iota \nu ].[\mu '+\iota \nu ']$
$\mu$
$\nu$
$\mu ',\nu ',$
$\mu '+\iota \nu '.$
8. A bivector may be divided in one and only one way into parts parallel and perpendicular to another, provided that the second is not circular. If ${\mathfrak {a}}$ and ${\mathfrak {b}}$ are the bivectors, the parts of ${\mathfrak {a}}$ will be
* ${\frac {\mathfrak {a.b}}{\mathfrak {bb}}}{\mathfrak {b}}$and${\mathfrak {a}}-{\frac {\mathfrak {b.a}}{\mathfrak {b.b}}}{\mathfrak {b}}.$
${\mathfrak {b}}$
${\mathfrak {a}}$
${\mathfrak {b}}.$
9. Since ${\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {a}}=0,$ and ${\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {b}}=0,\,{\mathfrak {a}}\times {\mathfrak {b}}$ is perpendicular to ${\mathfrak {a}}$ and ${\mathfrak {b}}.$ We may regard the plane of the product as determined by the condition that the directional ellipses of the factors projected upon it become similar and similarly situated. The directional ellipse of the product is similar to these projections, but its orientation is different by 90°. It may easily be shown that ${\mathfrak {a}}\times {\mathfrak {b}}$ vanishes only with ${\mathfrak {a}}$ or ${\mathfrak {b}},$ or when ${\mathfrak {a}}$ and ${\mathfrak {b}}$ are parallel.
10. The bivector equation
* $({\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {c}}){\mathfrak {d}}-({\mathfrak {b}}.{\mathfrak {c}}\times {\mathfrak {d}}){\mathfrak {a}}+({\mathfrak {c}}.{\mathfrak {d}}\times {\mathfrak {a}}){\mathfrak {b}}-({\mathfrak {d}}.{\mathfrak {a}}\times {\mathfrak {b}}){\mathfrak {c}}=0$
$xi+yj+zk$
$x,y,z$
${\mathfrak {a}}\times {\mathfrak {b}}$
${\mathfrak {a}}.{\mathfrak {b}}\times {\mathfrak {c}}$
${\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {c}}$
${\mathfrak {a}},{\mathfrak {b}}$
${\mathfrak {c}}$
11. The equation
* $({\mathfrak {r}}.{\mathfrak {a}}){\mathfrak {b}}\times {\mathfrak {c}}+({\mathfrak {r}}.{\mathfrak {b}}){\mathfrak {c}}\times {\mathfrak {a}}+({\mathfrak {r}}.{\mathfrak {c}}){\mathfrak {a}}\times {\mathfrak {b}}=({\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {c}}){\mathfrak {r}}$
* ${\mathfrak {c}}={\mathfrak {a}}\times {\mathfrak {b}},$
* ${\mathfrak {r}}.{\mathfrak {a}}=0,$${\mathfrak {r}}.{\mathfrak {b}}=0,$
* $({\mathfrak {r}}.{\mathfrak {a}}\times {\mathfrak {b}}){\mathfrak {a}}\times {\mathfrak {b}}=({\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {a}}\times {\mathfrak {b}}){\mathfrak {r}}.$
${\mathfrak {r}}$
${\mathfrak {a}}$
${\mathfrak {b}},$
${\mathfrak {r}}$
${\mathfrak {a}}\times {\mathfrak {b}}.$
[Note by Editors.—The notation $\left\vert \Phi \right\vert \Phi _{\text{C}}^{-1}$ used on page 64, was later improved by the author by the introduction of his Double Multiplication, aooording to which the above expression is represented by $\Phi _{2},$ and $\left\vert \Phi \right\vert$ by $\Phi _{3}.$ See this volume, pages 112, 160, and 181. For an extended treatment of Professor Gibbs's researches on Double Multiplication in their application to Vector Analysis see pp. 906–321, and 333 of "Vector Analysis," by E. B. Wilson, Chas. Scribner's Sons, New York, 1901.]
* ↑ The imaginaries of ordinary algebra may be called biscalars, and that which corresponds to them in the theory of vectors, bivectors. But we shall have no occasion to consider either of these. [See, however, footnote on p. 84.]
* ↑ Since the sign $\times$ is only used between vectors, the skew multiplication in expressions of this kind is evidently to be performed first. In other words, the above expression must be interpreted as $[\alpha \times \beta ].\gamma$.
* ↑ The student who is familiar with the nature of determinants will not fail to observe that the triple product $\alpha .\beta \times \gamma$ is the determinant formed by the nine rectangular components of $\alpha ,\beta$, and $\gamma$, nor that the rectangular components of $\alpha \times \beta$ are determinants of the second order formed from the components of $\alpha$ and $\beta$. (See the last equation of No. 21.)
* ↑ If every closed line within a given region can oontraot to a single point, without breaking its continuity, or passing out of the region, the region is called acyclic, otherwise cyclic. A cyclic region may be made acyclic by diaphragms, which must then be regarded as forming part of the surface bounding the region, each diaphragm contributing its own area twice to that surface. This process may be used to reduce many-valued functions of position in space, having single-valued derivatives, to single-valued functions. When functions are mentioned or implied in the notation, the reader will always understand single-valued functions, unless the contrary is distinctly intimated, or the case is one in which the distinction is obviously immaterial. Diaphragms may be applied to bring functions naturally many-valued under the application of some of the following theorems, as Nos. 74 ff.
* ↑ If a space encloses within itself another space, it is called periphractict, otherwise aperiphractic.
* ↑ Whenever it is said that a function of position in space has a definite value in general, this phrase is to be understood as explained ahove. The term definite is intended to exclude both indeterminate and infinite values.
* ↑ Better thus: ${\begin{aligned}\nabla .\nabla {\text{Pot }}u=\iiint {\frac {1}{r}}\nabla .\nabla u\,dv=\iiint \nabla \left({\frac {1}{r}}\nabla u\right)dv-\iiint \nabla .\left(u\nabla {\frac {1}{r}}\right)dv\\+\iiint u\nabla .\nabla {\frac {1}{r}}dv=-\iint u\nabla {\frac {1}{r}}.d\sigma =-4\pi u.\end{aligned}}$ [MS. note by author.]
* ↑ [The foregoing portion of this paper was printed in 1881, the rest in 1884.]
* ↑ The frequent occurrence of the integral in Maxwell's Treatise on Electricity and Magnetism has suggested this name.
* ↑ [See note on p. 90.]
* ↑ [See note on p. 90.]
* ↑ In particular cases, $\Phi -a{\text{I}}$ may reduce to a linear dyadic, or to zero. These, however, will present no difficulties to the student.
* ↑ For the case in which the two planes are perpendicular to each other, see No. 157.
* ↑ The termini of the vectors will in fact lie on a hyperbola.
* ↑ For the limiting cases, in which $n=1,$ or $n=-1,$ see No. 156.
* ↑ We might proceed to higher steps in differentiation by means of the triadics $\nabla \nabla \omega ,\nabla \nabla \nabla \omega ,$ the tetradios $\nabla \nabla \nabla \omega ,\nabla \nabla \nabla \nabla \omega ,$ etc. See note on page 74. In like manner a dyadic function of position in space ($\Phi$) might be differentiated by means of the triadic $\nabla \Phi ,$ the tetradic $\nabla \nabla \Phi ,$ etc.
* ↑ See footnote to No. 160.
* ↑ Thus far, in accordanoe with the purpoee expressed in the footnote on page 17, we have considered only real values of soalars and vectors. The object of this limitation has been to present the subject in the most elementary manner. The limitation is however often inconvenient, and does not allow the most symmetrical and complete development of the subject in many important directions. Thus in Chapter V, and the latter part of Chapter III, the exclusion of imaginary values has involved a considerable sacrifice of simplicity both in the enunciation of theorems and in their demonstration. The student will find an interesting and profitable exercise in working over this part of the subject with the aid of imaginary values, especially in the discussion of the imaginary roots of the cubic equation on page 71, and in the use of the formula $e^{\iota \Phi }=\cos \Phi +\iota \sin \Phi$ in developing the properties of the sines, cosines, and exponentials of dyadics.
* ↑ $(a+\iota b)[\mu +\iota \nu ]$ $=a\mu -b\nu +\iota [a\nu +b\mu ].$ $[\mu +\iota \nu ](a+\iota b)$ $=\mu a-\nu b+\iota [\mu b+\nu a].$ Therefore the position of the scalar factor is indifferent. [MS. note by author.]