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# Schwere, Elektricität und Magnetismus:387

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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-clock-wise 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-clock-wise as seen from the direction in which $s\,$ increases,) is represented by
* ${\frac {{\frac {d\rho }{ds}}\cdot {\frac {d^{2}\rho }{ds^{2}}}\times {\frac {d^{3}\rho }{ds^{3}}}}{{\frac {d\rho }{ds}}\cdot {\frac {d\rho }{ds}}}}$
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 | $t=u+a,\,$ |
where $a\,$ is a scalar constant.
# Schwere, Elektricität und Magnetismus:386

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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
* $\rho =x\alpha +y\beta +z\gamma ,\,$
* $d\rho =dx\alpha +dy\beta +dz\gamma .\,$
43. Differential of a function of several variables.—The differential of a vector or scalar function of any number of vector or scalar variables is evidently the sum (geometrical or algebraic, according as the function is vector or scalar,) of the differentials of the function due to the separate variation of the several variables.
44. Differetntial 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) |
| $d(\alpha \cdot \beta )=d\alpha \cdot \beta +\alpha \cdot d\beta ,\,$ | (3) |
| $d\lbrack \alpha \times \beta \rbrack =d\alpha \times \beta +\alpha \times d\beta ,\,$ | (4) |
| $d(\alpha \cdot \beta \times \gamma )=d\alpha \cdot \beta \times \gamma +\alpha \cdot d\beta \times \gamma +\alpha \cdot \beta \times d\gamma ,\,$ | (5) |
| $d\lbrack (\alpha \cdot \beta )\gamma \rbrack =(d\alpha \cdot \beta )\gamma +(\alpha \cdot d\beta )\gamma +(\alpha \cdot \beta )d\gamma .\,$ | (6) |
46. Differential coeffcient with respect to a scalar.—The quotient obtained by dividing the differential of a vector due to the variation of any scalar of which it is a function by the differential of that scalar is called the differential coefficient of the vector with respect to the scalar, and is indicated in the same manner as the differential coefficients of ordinary analysis.
# Schwere, Elektricität und Magnetismus:388

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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 | $\tau =\omega +\alpha ,\,$ |
where $\alpha \,$ is a vector constant.
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.
50. Def.—If $u\,$ is any scalar function of position in space, (i. e., 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, 53 for the definition of $\nabla u\,$.
51. If $\rho \,$ is the vector defining the position of a point in space,
* $du=\nabla u\cdot d\rho .\,$
| 52. | $\nabla u=i{\frac {du}{dx}}+j{\frac {du}{dy}}+k{\frac {du}{dz}}.\,$ |
| 53. | ${\frac {du}{dx}}=i\cdot \nabla u,\quad {\frac {du}{dy}}=j\cdot \nabla u,\quad {\frac {du}{dz}}=k\cdot \nabla u.\,$ |
54. Def.—If $\omega \,$ is a vector having continuously varying values in space,
| $\nabla \cdot \omega =i\cdot {\frac {d\omega }{dx}}+j\cdot {\frac {d\omega }{dy}}+k\cdot {\frac {d\omega }{dz}},\,$ | (1) |
and
| $\nabla \times \omega =i\times {\frac {d\omega }{dx}}+j\times {\frac {d\omega }{dy}}+k\times {\frac {d\omega }{dz}}.\,$ | (2) |
$\nabla \cdot \omega \,$ is called the divergence of $\omega \,$ and $\nabla \times \omega \,$ its curl.
If we set
* $\omega =Xi+Yj+Zk,\,$
# Schwere, Elektricität und Magnetismus:393

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$\omega \,$ is the derivative of a scalar function of position in space. This will appear from the following considerations:
The line-integral $\int \omega \cdot d\rho \,$ will vanish for any closed line, since it may be expressed as the surface-integral of $\nabla \times \omega \,$. (No. 60.) The line-integral taken from one given point $P'\,$ to another given point $P''\,$ is independent of the line between the points for which the integral is taken. (For, if two lines joining the same points gave different values, by reversing one we should obtain a closed line for which the integral would not vanish.) If we set $u\,$ equal to this line-integral, supposing $P''\,$ to be variable and $P'\,$ to be constant in position, $u\,$ will be a scalar function of the position of the point $P''\,$, satisfying the condition $du=\omega \cdot d\rho \,$, or, by No. 51, $\nabla u=\omega \,$. There will evidently be an infinite number of functions satisfying this condition, which will differ from one another by constant quantities.
If the region for which $\nabla \times \omega =0\,$ is unlimited, these functions will be single-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, $\nabla \cdot \nabla \times \omega =0\,$. 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
* $\nabla \cdot \nabla u=\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)u.\,$
70. Def.—If $\omega \,$ is any vector function of position in space, we may define $\nabla \cdot \nabla \omega \,$ by the equation
* $\nabla \cdot \nabla \omega =\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)\omega ,\,$
# Schwere, Elektricität und Magnetismus:385

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By direct multiplication of equations (1) with each other, and of equations (2) with each other, we obtain six of the type
| $(i\cdot i')(i\cdot j')+(j\cdot i')(j\cdot j')+(k\cdot i')(k\cdot j')=0.\,$ | (4) |
By skew multiplication of equations (1) with each other, we obtain three of the type
* $k'=\lbrace (j\cdot i')(k\cdot j')-(k\cdot i')(j\cdot j')\rbrace i+\lbrace (k\cdot i')(i\cdot j')-(i\cdot i')(k\cdot j')\rbrace j\,$
* $+\lbrace (i\cdot i')(j\cdot j')-(j\cdot i')(i\cdot j')\rbrace k.\,$
Comparing these three equations with the original three, we obtain nine of the type
| $i\cdot k'=(j\cdot i')(k\cdot j')-(k\cdot i')(j\cdot j').\,$ | (5) |
Finally, if we equate the scalar product of the three right hand members of (1) with that of the three left hand members, we obtain
| $(i\cdot i')(j\cdot j)(k\cdot k')+(i\cdot j')(j\cdot k')(k\cdot i')+(i\cdot k')(j\cdot i')(k\cdot j')\,$ $-(k\cdot i')(j\cdot j')(i\cdot k')-(k\cdot j')(j\cdot k')(i\cdot i')-(k\cdot k')(j\cdot i')(i\cdot j')=1.\,$ | (6) |
Equations (1) and (2) (if the expressions in the parentheses are supposed replaced by numerical values) represent the linear relations which subsist between one vector of one system and the three vectors of the other system. If we desire to express the similar relations which subsist between two vectors of one system and two of the other, we may take the skew products of equations (1) with equations (2), after transposing all terms in the latter. This will afford nine equations of the type
| $(i\cdot j')k'-(i\cdot k')j'=(k\cdot i')j-(j\cdot i')k.\,$ | (7) |
# Schwere, Elektricität und Magnetismus:390

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This quotient must therefore be independent of the form of the surface. We may define $\nabla \cdot \omega \,$ as representing that quotient, and then obtain equation (1) of No. 54 by applying the general principle to the ease of the rectangular parallelopiped.
56. Skew surface-integrals.—The integral $\iint d\sigma \times \omega \,$ may be called the skew surface-integral of $\omega \,$. It is evidently a vector. For a closed surface bounding a space $dv\,$ infinitely small in all dimensions, this integral reduces to $\nabla \times \omega dv\,$, as is easily shown by reasoning like that of No. 55.
57. Integration.—If $dv\,$ represents an element of any space, and $d\sigma \,$ an element of the bounding surface,
* $\iiint \nabla \cdot \omega dv=\iint \omega \cdot d\sigma .\,$
For the first member of this equation represents the sum of the surface integrals of all the elements of the given space. We may regard this principle as affording a means of integration, since we may use it to reduce a triple integral (of a certain form) to a double integral.
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 $\int \omega \cdot d\rho \,$, 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-clock-wise.
59. Integration.—From No. 51 we obtain directly
* $\int \nabla \cdot ud\rho =u''-u',\,$
where the single and double accents distinguish the values relating to the beginning and end of the line.
In other words,—The line-integral of the derivative of any (continuous) 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 $d\rho \,$ an clement of the bounding line,
* $\iint \nabla \times \omega \cdot d\sigma =\int \omega \cdot d\rho .\,$
In other words,—The line-integral of any vector function of position in space for a closed line is equal to the surface-inte-
# Schwere, Elektricität und Magnetismus:394

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the expression $\nabla \cdot \nabla \,$ being regarded, for the present at least, as a single operator when applied to a vector. (It will be remembered that no meaning has been attributed to $\nabla \,$ before a vector.) It should be noticed that, if
* $\omega =iX+jY+kZ,\,$
* $\nabla \cdot \nabla \omega =i\nabla \cdot \nabla X+j\nabla \cdot \nabla Y+k\nabla \cdot \nabla Z,\,$
that is, the operator $\nabla \cdot \nabla \,$ applied to a vector affects separately its scalar components.
71. From the above definition with those of Nos. 52 and 54 we may easily obtain
* $\nabla \cdot \nabla \omega =\nabla \nabla \omega -\nabla \times \nabla \times \omega .\,$
The effect of the operator $\nabla \cdot \nabla \,$ is therefore independent of the directions of the axes used in its definition.
72. The expression $-{\frac {1}{6}}a^{2}\nabla \cdot \nabla u\,$ 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, $-{\frac {1}{6}}a^{2}\nabla \cdot \nabla u\,$.
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 $-\nabla \cdot \nabla u\,$ the concentration of $u\,$, whether $u\,$ is scalar or vector. We may call $\nabla \cdot \nabla u\,$ (or $\nabla \cdot \nabla \omega \,$), 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 $\omega \,$).
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 formulae (like those of Nos. 57, 59, and 60) are written for the case of continuous values of the quantities (scalar and vector) to which the signs $\nabla \cdot \nabla \cdot \,$, and $\nabla \times \,$ are applied. It is left to the student to complete the formulae for cases of discontinuity in these values. The manner in which this is to be done may in each case be inferred
# Schwere, Elektricität und Magnetismus:389

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we obtain by substitution the equations
* $\nabla \cdot \omega ={\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}},\,$
| and | $\nabla \times \omega =i\left({\frac {dZ}{dy}}-{\frac {dY}{dz}}\right)+j\left({\frac {dX}{dz}}-{\frac {dZ}{dx}}\right)+k\left({\frac {dY}{dx}}-{\frac {dX}{dy}}\right),\,$ |
which may also be regarded as defining $\nabla \cdot \omega \,$ and $\nabla \times \omega \,$.
55. Surface-integrals.—The integral $\iint \omega \cdot d\sigma \,$, 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 (i. e., 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
* $\nabla \cdot \omega dv.\,$
This follows immediately from the definition of $\nabla \omega \,$, when the space is a parallelopiped bounded by planes perpendicular to $(i,\,j,\,k)\,$. In other cases, we may imagine the space—or rather a space nearly coincident with the given space and of the same volume $dv\,$—to be divided up into such parallelopipeds. The surface-integral for the space made up of the parallelopipeds will be the sum of the surface-integrals of all the parallelopipeds, and will therefore be expressed by $\nabla \cdot \omega dv\,$. The surface-integral of the original space will have sensibly the same value, and will therefore be represented by the same formula. It follows that the value of $\nabla \cdot \omega \,$ does not depend upon the system of unit vectors employed in its definition.
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.
# Schwere, Elektricität und Magnetismus:392

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on which a positive motion in the circle appears counter-clockwise, gives the direction of $\nabla \times \omega \,$, and the quotient of the integral divided by the area of the circle gives the magnitude of $\nabla \times \omega \,$.
62. A constant scalar factor after $\nabla \,$, $\nabla \cdot \,$ or $\nabla \times \,$ may he placed before the symbol.
63. If $f(u)\,$ denotes any scalar function of $u\,$, and $f'(u)\,$ the derived function,
* $\nabla f(u)=f'(u)\nabla u.$
64. If $u\,$ or $\omega \,$ is a function of several scalar or vector variables, which are themselves functions of the position of a single point, the value of $\nabla u\,$ or $\nabla \cdot \omega \,$ or $\nabla \times \omega \,$ will be equal to the sum of the values obtained by making successively all but each one of these variables constant.
65. By the use of this principle, we easily derive the following identical equations:
| $\nabla (t+u)=\nabla t+\nabla u.$ | (1) |
| $\nabla \cdot (\tau +\omega )=\nabla \cdot \tau +\nabla \cdot \omega .\quad \nabla \times \lbrack \tau +\omega \rbrack =\nabla \times \tau +\nabla \times \omega .\,$ | (2) |
| $\nabla (tu)=u\nabla t+t\nabla u.\,$ | (3) |
| $\nabla \cdot (u\omega )=\omega \cdot \nabla u+u\cdot \nabla \omega .\,$ | (4) |
| $\nabla \times \lbrack u\omega \rbrack =u\nabla \times \omega -\omega \times \nabla u.\,$ | (5) |
| $\nabla \cdot \lbrack \tau \times \omega \rbrack =\omega \cdot \nabla \times \tau -\tau \cdot \nabla \times \omega .\,$ | (6) |
The student will observe an analogy between these equations and the formulae of multiplication. (In the last four equations the analogy appears most distinctly when we regard all the factors but one as constant.) Some of the more curious features of this analogy are due to the fact that the $\nabla \,$ contains implicitly the vectors $i,\,j\,$ and $k\,$, which are to be multiplied into the following quantities.
66. If $u\,$ is any scalar function of position in space,
* $\nabla \times \nabla u=0,\,$
as may be derived directly from the definitions of these operators.
67. Conversely, if $\omega \,$ is such a vector function of position in space that
* $\nabla \times \omega =0,\,$
# Topographia Alsatiae: Steg

Topographia Germaniae :
| Steg ||
| <<<Vorheriger Stauffen | Nächster>>> Steinbach |
| aus: Matthäus Merian (Herausgeber und Illustrator) und Martin Zeiller (Textautor): Topographia Alsatiae. Merian, Frankfurt am Main 1643/44, S. 51. ||
| [[| in Wikisource]] | |
| Steige (Bas-Rhin) in der Wikipedia ||
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Ein Dorff oberhalb Andlau dem Gebürge zu gelegen.
# Topographia Alsatiae: Steinberg

Topographia Germaniae :
| Steinberg (heute: Steinburg) ||
| <<<Vorheriger Steinbach | Nächster>>> Stephansfelden |
| aus: Matthäus Merian (Herausgeber und Illustrator) und Martin Zeiller (Textautor): Topographia Alsatiae. Merian, Frankfurt am Main 1643/44, S. 51. ||
| [[| in Wikisource]] | |
| Steinbourg in der Wikipedia ||
| Bearbeitungsstand ||
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Ein Dorff / und Schloß / so / nach Absterben Wilhelms von Wilsperg / deß letzten dieses Namens / und Stammens / als ein Lehen / an das Hohe Stifft Straßburg gefallen ist.