hcju/proofwikips huggingface dataset:
Definition:Classical Algorithm/Primitive Multiplication/Base 10 Multiplication Table
The primitive multiplication operation for conventional base $10$ arithmetic of two $1$-digit integers can be presented as a pair of operation tables as follows: :$\begin{array}{c|cccccccccc} p & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 0 & 2 & 4 & 6 & 8 & 0 & 2 & 4 & 6 & 8 \\ 3 & 0 & 3 & 6 & 9 & 2 & 5 & 8 & 1 & 4 & 7 \\ 4 & 0 & 4 & 8 & 2 & 6 & 0 & 4 & 8 & 2 & 6 \\ 5 & 0 & 5 & 0 & 5 & 0 & 5 & 0 & 5 & 0 & 5  \\ 6 & 0 & 6 & 2 & 8 & 4 & 0 & 6 & 2 & 8 & 4  \\ 7 & 0 & 7 & 4 & 1 & 8 & 5 & 2 & 9 & 6 & 3  \\ 8 & 0 & 8 & 6 & 4 & 2 & 0 & 8 & 6 & 4 & 2  \\ 9 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1  \\ \end{array} \qquad \begin{array}{c|cccccccccc} c & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 \\ 4 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 2 & 3 & 3 \\ 5 & 0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 \\ 6 & 0 & 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 \\ 7 & 0 & 0 & 1 & 2 & 2 & 3 & 4 & 4 & 5 & 6 \\ 8 & 0 & 0 & 1 & 2 & 3 & 4 & 4 & 5 & 6 & 7 \\ 9 & 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \end{array}$
Definition:Classical Algorithm/Addition
Let $u = \sqbrk {u_{n - 1} u_{n - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $n$-digit integers. The '''classical addition algorithm''' forms their $n + 1$-digit sum $u + v$: :$w = \sqbrk {w_n w_{n - 1} \dotsm w_1 w_0}_b$ where $w_n$ is either $0$ or $1$. The steps are: :$(\text A 1): \quad$ Set $j = 0$, $k = 0$. ::::$j$ is used to run through all the digit positions ::::$k$ keeps track of the carry digit between each step. :$(\text A 2): \quad$ Calculate digit $j$: ::::Calculate $\begin {cases} s = \paren {u_j + v_j + k} \pmod b \\ c = \floor {\dfrac {u_j + v_j + k} b} \end {cases}$ using the primitive addition. ::::Set $w_j$ to $s$. ::::Set $k$ to $c$. :$(\text A 3): \quad$ Add $1$ to $j$, using conventional integer addition. ::::If $j < n$, return to $(\text A 2)$. ::::Otherwise, set $w_n$ equal to $k$ and exit.
Definition:Classical Algorithm/Subtraction
Let $u = \sqbrk {u_{n - 1} u_{n - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $n$-digit integers. The '''classical subtraction algorithm''' forms their $n$-digit difference $u - v$: :$w = \sqbrk {w_n w_{n - 1} \dotsm w_1 w_0}_b$ where $w_n$ is either $0$ or $1$. The steps are: :$(\text S 1): \quad$ Set $j = 0$, $k = 0$. ::::$j$ is used to run through all the digit positions ::::$k$ keeps track of the carry digit between each step. :$(\text S 2): \quad$ Calculate digit $j$: ::::Calculate $\begin {cases} d = \paren {u_j + v_j - k} \pmod b \\ c = \floor {\dfrac {u_j - v_j + k} b} \end {cases}$ using the primitive subtraction. ::::Set $w_j$ to $d$. ::::Set $k$ to $c$. :$(\text S 3): \quad$ Add $1$ to $j$, using conventional integer addition. ::::If $j < n$, return to $(\text S 2)$. ::::Otherwise exit.
Definition:Classical Algorithm/Multiplication
Let $u = \sqbrk {u_{m - 1} u_{m - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $m$-digit and $n$-digit integers respectively. The '''classical multiplication algorithm''' forms their $m + n$-digit product $u v$: :$w = \sqbrk {w_{m + n - 1} w_{m + n - 2} \dotsm w_1 w_0}_b$ The steps are: :$(\text M 1): \quad$ Initialise: ::::Set $w_{m - 1}, w_{m - 2}, \dotsc, w_1, w_0 := 0$ ::::Set $j := 0$ :$(\text M 2): \quad$ Is $v_j = 0$? ::::If so, go to Step $(\text M 6)$. :$(\text M 3): \quad$ Initialise loop: ::::Set $i := 0$ ::::Set $k := 0$ :$(\text M 4): \quad$ Multiply and Add: ::::Set $t := u_i \times v_j + w_{i + j} + k$ ::::Set $w_{i + j} := t \pmod b$ ::::Set $k := \floor {\dfrac t b}$ :$(\text M 5): \quad$ Loop on $i$: ::::Increase $i$ by $1$. ::::If $i < m$, go back to Step $(\text M 4)$. ::::Otherwise, set $w_{j + m} := k$ :$(\text M 6): \quad$ Loop on $j$: ::::Increase $j$ by $1$. ::::If $j < n$, go back to Step $(\text M 2)$. ::::Otherwise exit.
Definition:Classical Algorithm/Division
Let $u = \sqbrk {u_{m + n - 1} u_{m - 2} \dotsm u_1 u_0}_b$ and $v = \sqbrk {v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ be $m + n$-digit and $n$-digit integers respectively, where $v_{n - 1} \ne 0$ and $n > 1$. The '''classical division algorithm''' forms: :their $m + 1$-digit quotient $\floor {\dfrac u v} = q = \sqbrk {q_m q_{m - 1} \dotsm q_1 q_0}_b$ and: :their $n$-digit remainder $u \pmod v = r = \sqbrk {r_{n - 1} \dotsm r_1 r_0}_b$ The steps are: :$(\text D 1): \quad$ Normalize: ::::Set: {{begin-eqn}} {{eqn | l = d       | o = :=       | r = \floor {\dfrac b {v_{n - 1} - 1} } }} {{eqn | l = \sqbrk {u_{m + n} u_{m - 1} \dotsm u_1 u_0}_b       | o = :=       | r = \sqbrk {u_{m + n - 1} u_{m - 2} \dotsm u_1 u_0}_b \times d }} {{eqn | l = \sqbrk {v_{n - 1} \dotsm v_1 v_0}_b       | o = :=       | r = \sqbrk {v_{n - 1} \dotsm v_1 v_0}_b \times d }} {{end-eqn}} ::::Note that if $d = 1$, all that needs to be done is to set $u_{m + n} := 0$. :$(\text D 2): \quad$ Initialise $j$: ::::Set $j := m$ :$(\text D 3): \quad$ Calculate the quotient $\hat q$ and remainder $\hat r$: ::::Set $\hat q := \floor {\dfrac {u_{j + n} b + u_{j + n - 1} } {v_{n - 1} } }$ ::::Set $\hat r := \paren {u_{j + n} b + u_{j + n - 1} } \pmod {v_{n - 1} }$ ::::Test whether $\hat q = b$ or $\hat q v_{n - 1} > b \hat r + u_{j + n - 2}$. ::::If so, decrease $\hat q$ by $1$ and increase $\hat r$ by $v_{n - 1}$. ::::If $\hat r < b$ repeat this test. :$(\text D 4): \quad$ Multiply and Subtract: ::::Replace $\sqbrk {u_{j + n} u_{j + n - 1} \dotsm u_j}_b$ by $\sqbrk {u_{j + n} u_{j + n - 1} \dotsm u_j}_b - \hat q \sqbrk {0 v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ ::::If the result of this step is negative, add $b_{n + 1}$ to it. :$(\text D 5): \quad$ Test remainder: ::::Set $q_j := \hat q$. ::::If the result of '''Step $(\text D 4)$''' was negative, go to '''Step $(\text D 6)$''' ::::Otherwise go to '''Step $(\text D 7)$''' :$(\text D 6): \quad$ Add back: ::::Decrease $q_j$ by $1$ ::::Add $\sqbrk {0 v_{n - 1} v_{n - 2} \dotsm v_1 v_0}_b$ to $\sqbrk {u_{j + n} u_{j + n - 1} \dotsm u_j}_b$ (ignoring the carry) :$(\text D 7): \quad$ Loop on $j$: ::::Decrease $j$ by $1$ ::::If $j \ge 0$ go back to '''Step $(\text D 3)$''' :$(\text D 8): \quad$ Unnormalise: ::::The required quotient is $\sqbrk {q_m q_{m - 1} \dotsm q_1 q_0}_b$ ::::The required remainder is obtained by dividing $\sqbrk {u_{n - 1} u_{m - 2} \dotsm u_1 u_0}_b$ by $d$.
Definition:Unit Disk
A '''unit disk''' is a disk whose radius is equal to $1$.
Definition:Disk/Radius
Let $\map B {a, r}$ be a disk. The '''radius''' of $\map B {a, r}$ is the parameter $r$.
Definition:Disk/Center
Let $\map B {a, r}$ be a disk. The '''center''' of $\map B {a, r}$ is the parameter $a$.
Definition:Transcendental Function
A '''transcendental function''' is an analytic function which is not an algebraic function. That is, it cannot be expressed as a polynomial equation. === Defined by Integral === {{:Definition:Transcendental Function/Defined by Integral}} === Defined by Differential Equation === {{:Definition:Transcendental Function/Defined by Differential Equation}}
Definition:Transcendental Function/Also defined as
Some sources define a '''transcendental function''' as a real function or complex function which is not an elementary function. However, the distinction between what is and is not an elementary function is more or less arbitrary, consisting of both algebraic functions and those derived from the exponential function, which itself is not algebraic. The current school of thought appears to be that this definition: "not an elementary function" is actually considered to be erroneous. However, the distinction is not considered particularly important nowadays. As long as it is made clear which definition is being used at the time, that would be adequate.
Definition:Algebraic Function
=== Real Algebraic Function === {{:Definition:Algebraic Function/Real}} === Complex Algebraic Function === {{:Definition:Algebraic Function/Complex}}
Definition:Algebraic Function/Real
Let $y$ be a solution to the polynomial equation: :$\map {p_0} x + \map {p_1} x y + \dotsb + \map {p_{n - 1} } x y^{n - 1} + \map {p_n} x y^n = 0$ where $\map {p_0} x \ne 0, \map {p_1} x, \dotsc, \map {p_n} x$ are real polynomial functions in $x$. Then $y = \map f x$ is a '''(real) algebraic function''':
Definition:Algebraic Function/Complex
Let $w$ be a solution to the polynomial equation: :$\map {p_0} z + \map {p_1} z w + \dotsb + \map {p_{n - 1} } z w^{n - 1} + \map {p_n} z w^n = 0$ where $\map {p_0} z \ne 0, \map {p_1} z, \dotsc, \map {p_n} z$ are complex polynomial functions in $z$. Then $w = \map f z$ is a '''(complex) algebraic function''':
Definition:Perpendicular Projection
Let $\PP$ denote the plane. Let $L$ denote a straight line in $\PP$. For all $p \in \PP$, let $K_p$ denote the straight line through $P$ perpendicular to $L$. Let $p_L$ denote the point on $L$ where $K_p$ intersects $L$. Let $\pi_L: \PP \to L$ denote the mapping defined as: :$\forall p \in \PP: \map {\pi_L} p = p_L$ That is, $\pi_L$ sends every point $p$ in $\PP$ to the foot of the perpendicular from $p$ to $L$. $\pi_L$ is called the '''perpendicular projection of the plane onto $L$'''.
Definition:Ordering on Integers/Definition 1
The integers are ordered on the relation $\le$ as follows: :$\forall x, y \in \Z: x \le y$ : :$\exists c \in P: x + c = y$ where $P$ is the set of positive integers. That is, $x$ is '''less than or equal''' to $y$  $y - x$ is non-negative.
Definition:Strict Ordering on Integers/Definition 2
The integers are strictly ordered on the relation $<$ as follows: Let $x$ and $y$ be defined as from the formal definition of integers: :$x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$. Then: :$x < y \iff x_1 + y_2 < x_2 + y_1$ where: :$+$ denotes natural number addition :$a < b$ denotes natural number ordering $a \le b$ such that $a \ne b$.
Definition:Ordering on Integers/Definition 2
The integers are ordered on the relation $\le$ as follows: Let $x$ and $y$ be defined as from the formal definition of integers: :$x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$. Then: :$x < y \iff x_1 + y_2 \le x_2 + y_1$ where: :$+$ denotes natural number addition :$\le$ denotes natural number ordering.
Definition:Imperial/Area/Rood
One '''rood''' is equal to an oblong measuring $1$ chain by $2 \frac 1 2$ chains. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''rood''' }} {{eqn | r = 2 \tfrac 1 2       | c = square chains }} {{eqn | r = 40       | c = square rods, poles or perches }} {{eqn | r = 1210       | c = square yards }} {{eqn | r = 10 \, 890       | c = square feet }} {{end-eqn}} :400px
Definition:Parabola/Latus Rectum
The '''latus rectum''' of $P$ is the chord of $P$ passing through the focus of $P$ parallel to the directrix $D$.
Definition:Work
Let $P$ be a particle whose position vector at time $t$ is $\mathbf r$. Let a force applied to $P$ be represented by the vector $\mathbf F$. Suppose that, during the time interval $\delta t$, $P$ moves from $\mathbf r$ to $\mathbf r + \delta \mathbf r$. The '''work done''' by $\mathbf F$ during $\delta t$ is defined to be: :$\delta W = \mathbf F \cdot \delta \mathbf r$ where $\cdot$ denotes the dot product.
Definition:Cartesian Product/Cartesian Space/Three Dimensions
The '''cartesian $3$rd power of $S$''' is: :$S^3 = S \times S \times S = \set {\tuple {x_1, x_2, x_3}: x_1, x_2, x_3 \in S}$ The set $S^3$ called a '''cartesian space of $3$ dimensions'''.
Definition:Cartesian 3-Space
The '''Cartesian $3$-space''' is a Cartesian coordinate system of $3$ dimensions. === Definition by Axes === {{:Definition:Cartesian 3-Space/Definition by Axes}} === Definition by Planes === {{:Definition:Cartesian 3-Space/Definition by Planes}}
Definition:Cartesian Coordinate System/Z Coordinate
Let $z$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $z$-axis. Then $z$ is known as the '''$z$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $z$-axis, then $z$ is positive. If $Q$ is in the negative direction along the real number line that is the $z$-axis, then $z$ is negative.
Definition:Rectangular Coordinate System
A '''rectangular coordinate system''' is a '''Cartesian coordinate system''' in which each of the coordinate axes are perpendicular to each other.
Definition:Oblique Coordinate System
An '''oblique coordinate system''' is a coordinate system in which the position of a point is determined by its relation to a set of straight coordinate axes which are oblique. :400px
Definition:X-Y Plane
The '''$x$-$y$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $x$-axis and the $y$-axis. It consists of all the points in $S$ such that $z = 0$.
Definition:Y-Z Plane
The '''$y$-$z$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $y$-axis and the $z$-axis. It consists of all the points in $S$ such that $x = 0$.
Definition:X-Z Plane
The '''$x$-$z$ plane''' is the Cartesian plane embedded in Cartesian $3$-space which contains the $x$-axis and the $z$-axis. It consists of all the points in $S$ such that $x = 0$.
Definition:Cartesian 3-Space/Definition by Axes
600pxrightthumbA general point in Cartesian $3$-Space Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates  $\tuple {x, y, z}$, as follows: Construct a Cartesian plane, with origin $O$ and axes identified as the $x$-axis and $y$-axis. Recall the identification of the point $P$ with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane. Construct an infinite straight line through $O$ perpendicular to both the $x$-axis and the$y$-axis and call it the $z$-axis. Identify the point $P''$ on the $z$-axis such that $OP'' = OP$. Identify the $z$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P$ The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$. It is conventional to locate $P''$ as follows. Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.  Then $P''$ is then one unit ''above'' the $x$-$y$ plane. Let the $x$-$y$ plane be identified with the plane of the page or screen. The orientation of the $z$-axis is then: :coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive :going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.
Definition:Decimal Expansion/Size Less than 1
A number $x$ such that $\size x < 0$ has a units digit which is zero. Such a number may be expressed either with or without the zero, for example: :$0 \cdotp 568$ or: :$\cdotp 568$ While both are commonplace, the form with the zero is less prone to the mistake where decimal point is missed when reading it.
Definition:Cartesian 3-Space/Definition by Planes
600pxrightthumb Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates  $\tuple {x, y, z}$, as follows: Identify one distinct point in space as the origin $O$. Let $3$ distinct planes be constructed through $O$ such that all are perpendicular. Each pair of these $3$ planes intersect in a straight line that passes through $O$. Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection. Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively. Select a point $P$ on the $x$-axis different from $O$. Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane. Identify the point $P'$ on the $y$-axis such that $OP' = OP$. Identify the point $P''$ on the $z$-axis such that $OP'' = OP$. The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$. It is conventional to locate $P''$ as follows. Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.  Then $P''$ is then one unit ''above'' the $x$-$y$ plane. Let the $x$-$y$ plane be identified with the plane of the page or screen. The orientation of the $z$-axis is then: :coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive :going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.
Definition:Wave Profile
Let $\phi$ be a disturbance which is propagated along the $x$-axis with velocity $c$. At $t = 0$, let $\phi = \map f x$. Then $\map f x$ is the '''wave profile''' of $\phi$.
Definition:Switching Circuit
A '''switching circuit''' is a mechanical, electronic or electromechanical system for controlling the route of information or resources. The study of '''switching circuits''' can be considered a field of applied logic.
Definition:Electric Potential/Dimension
The dimension of measurement of '''electric potential''' is $\mathsf M \mathsf L^2 \mathsf T^{−3} \mathsf I^{−1}$.
Definition:Electric Potential/Units
The SI unit of '''electric potential''' is the volt $\mathrm V$.
Definition:Electric Potential
An '''electric potential''' is the amount of work needed to move a unit of electric charge from a given reference point to a specific point in an electric field without producing an acceleration. The reference point is usually either Earth or a point at infinity, although any point can be used.
Definition:Scalar Algebra
'''Scalar algebra''' is the application of the rules of conventional '''algebra''' in order to manipulate '''scalar quantities'''.
Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 6
{{begin-axiom}} {{axiom | n = \text B 6         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \cup \set x} \setminus \set y \in \mathscr B }} {{end-axiom}}
Definition:Cartesian 3-Space/Coordinate Planes
Consider the '''Cartesian $3$-space''' defined by $3$ distinct perpendicular planes through the origin $O$. These $3$ planes are known as the '''coordinate planes''' of the Cartesian $3$-space.
Definition:Electromagnetic Induction
'''Electromagnetic induction''' is the phenomenon whereby an electromotive force is generated across an electrical conductor in a changing magnetic field.
Definition:Electric Field
An '''electric field''' is a vector field that associates to each point in space the (electrostatic or Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point.
Definition:Magnetic Flux Density
'''Magnetic flux density''' is  the force exerted on a charged particle perpendicular to its velocity through a magnetic field. Hence it is a measure of the intensity of that magnetic field.
Definition:Magnetic Flux Density/Units
The SI unit for '''magnetic flux density''' is the tesla $\mathrm T$: :$1 \ \mathrm T = 1 \ \mathrm {kg} \, \mathrm s^{-2} \mathrm A^{-1}$ that is: :$1 \ \mathrm T = 1 \ \mathrm N \, \mathrm m^{-1} \mathrm A^{-1}$
Definition:Magnetic Flux Density/Dimension
'''Magnetic flux density''' has the dimension $\mathsf {M T}^{-2} \mathsf I^{-1}$.
Definition:Rate/Dimension
A '''rate''' has dimension $\mathsf T^{-1}$.
Definition:Charged Particle
A '''charged particle''' is an ideal particle which bears an electric charge.
Definition:Elementary Charge
The '''elementary charge''' is the value of the electric charge on a proton. It is defined as ''exactly'' $1.60217 \, 6634 \times 10^{−19}$ coulombs. === Symbol === {{:Definition:Elementary Charge/Symbol}}
Definition:Tesla (Unit)/Base Units
The SI base units of the '''tesla''' are: :$\mathrm T := \mathrm {kg} \, \mathrm s^{-2} \mathrm A^{-1}$ where: :$\mathrm {kg}$ denotes kilogram :$\mathrm A$ denotes amperes :$\mathrm s$ denotes seconds (of time).
Definition:Volume Charge Density
The '''volume charge density''' quantity of electric charge per unit volume, at any given point in that volume.
Definition:Volume Charge Density/Dimension
The dimension of '''volume charge density''' is $\mathsf {I T L}^{-3}$: electric charge per unit volume.
Definition:Volume Charge Density/Units
The SI units of '''volume charge density''' are $\mathrm C \, \mathrm m^{-3}$ (coulombs per cubic metre).
Definition:Electric Displacement Field
The '''electric displacement field''' is a vector quantity defined as the electric field strength multiplied by the permittivity of the medium through which it passes: :$\mathbf D = \varepsilon \mathbf E$ where: :$\varepsilon$ is the permittivity of the medium :$\mathbf E$ is the electric field strength
Definition:Magnetic Field
A '''magnetic field''' is a vector field that gives rise to a force on a moving electric charge perpendicular to both the velocity of the charge and the direction of the '''magnetic field'''.
Definition:Magnetic Field Strength
'''Magnetic field strength''' is the measure of the intensity of a magnetic field. It can be expressed as: :$\mathbf H = \dfrac {\mathbf B} \mu$ where: :$\mathbf B$ is the magnetic flux density :$\mu$ is the permeability of the medium.
Definition:Magnetic Field Strength/Units
The SI unit for '''magnetic field strength''' is given as amperes per metre: :$\mathrm A \, \mathrm m^{-1}$
Definition:Electrostatic Force
An '''electrostatic force''' is the force exerted on an electric charge caused by an electric field.
Definition:Schrödinger Wave Equation
=== Time Independent Version === {{:Definition:Schrödinger Wave Equation/Time Independent}} {{NamedforDef|Erwin Rudolf Josef Alexander Schrödinger|cat = Schrödinger}}
Definition:Schrödinger Wave Equation/Time Independent
:$-\dfrac {\hbar^2} {2 m} \nabla^2 \psi + V \psi = E \psi$ where: :$\psi$ is the wave function whose nature is to be determined :$V$ denotes the potential energy (usually positional) :$E$ denotes the total energy of the system :$m$ denotes the mass of the particle whose motion is described by $\psi$ :$\hbar$ is Planck's constant divided by $2 \pi$:  ::$\hbar = \dfrac h {2 \pi}$
Definition:Angular Momentum
The '''angular momentum''' of a body about a point $P$ is its moment of inertia about $P$ multiplied by its angular velocity about $P$. Angular momentum is a vector quantity.
Definition:Angular Momentum/Dimension
The dimension of measurement of '''angular momentum''' is $\mathsf {M L}^2 \mathsf T^{-1}$.
Definition:Vector Quantity/Arrow Representation
A '''vector quantity''' $\mathbf v$ is often represented diagramatically in the form of an '''arrow''' such that: :its length is proportional to the magnitude of $\mathbf v$ :its direction corresponds to the direction of $\mathbf v$. The head of the arrow then indicates the positive sense of the direction of $\mathbf v$. It can be rendered on the page like so: :300px It is important to note that a vector quantity, when represented in this form, is not in general fixed in space. All that is being indicated using such a notation is its magnitude and direction, and not, in general, a point at which it acts.
Definition:Vector Sum/Triangle Law
Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that: :$\mathbf u$ is represented by $\vec {AB}$ :$\mathbf v$ is represented by $\vec {BC}$ that is, so that the initial point of $\mathbf v$ is identified with the terminal point of $\mathbf u$. :400px Then their '''(vector) sum''' $\mathbf u + \mathbf v$ is represented by the arrow $\vec {AC}$.
Definition:Terminal Point of Vector
Let $\mathbf u$ be a vector quantity represented by an arrow $\vec {AB}$ embedded in the plane: :300px The point $B$ is the '''terminal point''' of $\mathbf u$.
Definition:Initial Point of Vector
Let $\mathbf u$ be a vector quantity represented by an arrow $\vec {AB}$ embedded in the plane: :300px The point $A$ is the '''initial point''' of $\mathbf u$.