
$i$ 
The square root of minus one. 
$f(x)$ 
The value of the function $f$ at argument $x$ . 
$\mathrm{sin}x$ 
The value of the sine function at argument $x$ . 
$\mathrm{exp}x$ 
The value of the exponential function at argument $x$ . This is often written as ${e}^{x}$ . 
$a^x$ 
The number a raised to the power $x$ ; for rational $x$ is defined by inverse functions. 
$\mathrm{ln}x$ 
The inverse function to $\mathrm{exp}x$ . 
${a}^{x}$ 
Same as $a^x$ . 
${\mathrm{log}}_{b}a$ 
The power you must raise $b$ to in order to get $a$ ; ${b}^{{\mathrm{log}}_{b}a}=a$ . 
$\mathrm{cos}x$ 
The value of the cosine function (complement of the sine) at argument $x$ . 
$\mathrm{tan}x$ 
Works out to be $\frac{\mathrm{sin}x}{\mathrm{cos}x}$ . 
$\mathrm{cot}x$ 
The value of the complement of the tangent function or $\frac{\mathrm{cos}x}{\mathrm{sin}x}$ . 
$\mathrm{sec}x$ 
Value of the secant function, which turns out to be $\frac{1}{\mathrm{cos}x}$ . 
$\mathrm{csc}x$ 
Value of the complement of the secant, called the cosecant. It is $\frac{1}{\mathrm{sin}x}$ . 
$\mathrm{asin}x$ 
The value, $y$ , of the inverse function to the sine at argument $x$ . Means $x=\mathrm{sin}y$ . 
$\mathrm{acos}x$ 
The value, $y$ , of the inverse function to cosine at argument $x$ . Means $x=\mathrm{cos}y$ . 
$\mathrm{atan}x$ 
The value, $y$ , of the inverse function to tangent at argument $x$ . Means $x=\mathrm{tan}y$ . 
$\mathrm{acot}x$ 
The value, $y$ , of the inverse function to cotangent at argument $x$ . Means $x=\mathrm{cot}y$ . 
$\mathrm{asec}x$ 
The value, $y$ , of the inverse function to secant at argument $x$ . Means $x=\mathrm{sec}y$ . 
$\mathrm{acsc}x$ 
The value, $y$ , of the inverse function to cosecant at argument $x$ . Means $x=\mathrm{csc}y$ . 
$\theta $ 
A standard symbol for angle. Measured in radians unless stated otherwise. Used especially for $a\mathrm{tan}\frac{x}{y}$ when $x,y$ , and $z$ are variables used to describe point in three dimensional space. 
$\widehat{i},\widehat{j},\widehat{k}$ 
Unit vectors in the $x,y$ and $z$ directions respectively. 
$(a,b,c)$ 
A vector with $x$ component $a$ , $y$ component $b$ and $z$ component $c$ . 
$(a,b)$ 
A vector with $x$ component $a$ , $y$ component $b$ . 
$\left(\stackrel{\u27f6}{a},\stackrel{\u27f6}{b}\right)$ 
The dot product of vectors $\stackrel{\u27f6}{a}$ and $\stackrel{\u27f6}{b}$ . 
$\stackrel{\u27f6}{a}\xb7\stackrel{\u27f6}{b}$ 
The dot product of vectors $\stackrel{\u27f6}{a}$ and $\stackrel{\u27f6}{b}$ . 
$\left(\stackrel{\u27f6}{a}\xb7\stackrel{\u27f6}{b}\right)$ 
The dot product of vectors $\stackrel{\u27f6}{a}$ and $\stackrel{\u27f6}{b}$ . 
$\left\stackrel{\u27f6}{v}\right$ 
The magnitude of the vector $\stackrel{\u27f6}{v}$ . 
$\leftx\right$ 
The absolute value of the number $x$ . 
$\sum$ 
Used to denote a summation, usually the index and often their end values are written under it with upper end value above it. For example the sum of $j$ for $j=1$ to $n$ is written as $\sum _{j=1}^{n}}j$ or $\sum _{}^{n}}j$ . This signifies $1+2+\cdots +n$ . 
$M$ 
Used to represent a matrix or array of numbers or other entities. 
$v$ 
A column vector, that is one whose components are written as a column and treated as a $k$ by 1 matrix. 
$v$ 
A vector written as a row, or 1 by $k$ matrix. 
$dx$ 
An "infinitesimal" or very small change in the variable $x$ ; also similarly $dy,dz,dr$ etc. 
$ds$ 
A small change in distance. 
$\rho $ 
The variable ${\left({x}^{2}+{y}^{2}+{z}^{2}\right)}^{1/2}$ or distance to the origin in spherical coordinates. 
$r$ 
The variable ${\left({x}^{2}+{y}^{2}\right)}^{1/2}$ or distance to the zaxis in three dimensions or in polar coordinates. 
$\leftM\right$ 
The determinant of a matrix $M$ (whose magnitude is the area or volume of the parallel sided region determined by its columns or rows). 
$\Vert M\Vert $ 
The magnitude of the determinant of the matrix $M$ , which is a volume or area or hypervolume. 
$\mathrm{det}M$ 
The determinant of $M$ . 
${M}^{1}$ 
The inverse of the matrix $M$ . 
$\stackrel{\u27f6}{v}\times \stackrel{\u27f6}{w}$ 
The vector product or cross product of two vectors, $\stackrel{\u27f6}{v}$ and $\stackrel{\u27f6}{w}$ . 
${\theta}_{\stackrel{\u27f6}{v}\stackrel{\u27f6}{w}}$ 
The angle made by vectors $\stackrel{\u27f6}{v}$ and $\stackrel{\u27f6}{w}$ . 
$\stackrel{\u27f6}{A}\xb7\stackrel{\u27f6}{B}\times \stackrel{\u27f6}{C}$ 
The scalar triple product, the determinant of the matrix formed by columns $\stackrel{\u27f6}{A},\stackrel{\u27f6}{B},\stackrel{\u27f6}{C}$ . 
${\widehat{u}}_{\stackrel{\u27f6}{w}}$ 
A unit vector in the direction of the vector $\stackrel{\u27f6}{w}$ ; it means the same as $\frac{\stackrel{\u27f6}{w}}{\left\stackrel{\u27f6}{w}\right}$ . 
$df$ 
A very small change in the function $f$ , sufficiently small that the linear approximation to all relevant functions holds for such changes. 
$\frac{df}{dx}$ 
The derivative of $f$ with respect to $x$ , which is the slope of the linear approximation to $f$ . 
$f\text{'}$ 
The derivative of $f$ with respect to the relevant variable, usually $x$ . 
$\frac{\partial f}{\partial x}$ 
The partial derivative of $f$ with respect to $x$ , keeping $y$ , and $z$ fixed. In general a partial derivative of $f$ with respect to a variable $q$ is the ratio of $df$ to $dq$ when certain other variables are held fixed. Where there is possible misunderstanding over which variables are to be fixed that information should be made explicit. 
${\frac{\partial f}{\partial x}}_{y,z}$ 
The partial derivative of $f$ with respect to $x$ keeping $y$ and $z$ fixed. 
$\stackrel{\u27f6}{\text{grad}}f$ 
The vector field whose components are the partial derivatives of the function $f$ with respect to $x,y$ and $z$ : $\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$ or $\frac{\partial f}{\partial x}\widehat{i}+\frac{\partial f}{\partial y}\widehat{j}+\frac{\partial f}{\partial z}\widehat{k}$ ; called the gradient of $f$ . 
$\stackrel{\u27f6}{\nabla}$ 
The vector operator $\frac{\partial}{\partial x}\widehat{i}+\frac{\partial}{\partial y}\widehat{j}+\frac{\partial}{\partial z}\widehat{k}$ , called "del" . 
$\stackrel{\u27f6}{\nabla}f$ 
The gradient of $f$ ; its dot product with ${\widehat{u}}_{w}$ is the directional derivative of $f$ in the direction of $\stackrel{\u27f6}{w}$ . 
$\stackrel{\u27f6}{\nabla}\xb7\stackrel{\u27f6}{w}$ 
The divergence of the vector field $\stackrel{\u27f6}{w}$ ; it is the dot product of the vector operator $\stackrel{\u27f6}{\nabla}$ with the vector $\stackrel{\u27f6}{w}$ , or $\frac{\partial {w}_{x}}{\partial x}+\frac{\partial {w}_{y}}{\partial y}+\frac{\partial {w}_{z}}{\partial z}$ . 
$\stackrel{\u27f6}{\text{curl}}\stackrel{\u27f6}{w}$ 
The cross product of the vector operator $\stackrel{\u27f6}{\nabla}$ with the vector $\stackrel{\u27f6}{w}$ . 
$\stackrel{\u27f6}{\nabla}\times \stackrel{\u27f6}{w}$ 
The curl of $\stackrel{\u27f6}{w}$ , with components $\left(\frac{\partial {f}_{z}}{\partial y}\frac{\partial {f}_{y}}{\partial z},\frac{\partial fx}{\partial z}\frac{\partial {f}_{z}}{\partial x},\frac{\partial {f}_{y}}{\partial x}\frac{\partial {f}_{x}}{\partial y}\right)$ . 
$\stackrel{\u27f6}{\nabla}\xb7{\stackrel{\u27f6}{\nabla}}_{}$ 
The Laplacian, the differential operator: $\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{\partial}{\partial {y}^{2}}+\frac{\partial}{\partial {z}^{2}}$ . 
$f"(x)$ 
The second derivative of $f$ with respect to $x$ ; the derivative of $f\text{'}(x)$ . 
$\frac{{d}^{2}f}{d{x}^{2}}$ 
The second derivative of $f$ with respect to $x$ . 
${f}^{(2)}(x)$ 
Still another form for the second derivative of $f$ with respect to $x$ . 
${f}^{(k)}(x)$ 
The kth derivative of $f$ with respect to $x$ ; the derivative of ${f}^{(k1)}(x)$ . 
$\widehat{T}$ 
Unit tangent vector along a curve; if curve is described by $\stackrel{\u27f6}{r}(t),\widehat{T}=\frac{d\stackrel{\u27f6}{r}/dt}{\leftd\stackrel{\u27f6}{r}/dt\right}$ . 
$ds$ 
A differential of distance along a curve. 
$\kappa $ 
The curvature of a curve; the magnitude of the derivative of its unit tangent vector with respect to distance on the curve: $\left\frac{d\widehat{T}}{ds}\right$ . 
$\widehat{N}$ 
A unit vector in the direction of the projection of $\frac{d\widehat{T}}{ds}$ normal to $\widehat{T}$ . 
$\widehat{B}$ 
A unit vector normal to the plane of $\widehat{T}$ and $\widehat{N}$ , which is the plane of curvature. 
$\tau $ 
The torsion of a curve; $\left\frac{d\widehat{B}}{ds}\right$ . 
$g$ 
The gravitational constant. 
$\stackrel{\u27f6}{F}$ 
The standard symbol for force in mechanics. 
$k$ 
The spring constant of a spring. 
${\stackrel{\u27f6}{p}}_{i}$ 
The momentum of the ith particle. 
$H$ 
The Hamiltonian of a physical system, which is its energy expressed in terms of $\left\{{\stackrel{\u27f6}{r}}_{i}\right\}$ and $\left\{{\stackrel{\u27f6}{p}}_{i}\right\}$ , position and momentum. 
$\{Q,H\}$ 
The Poisson bracket of $Q$ and $H$ . 
${\int}_{}^{x}f(u)du$ 
An antiderivative of $f(x)$ expressed as a function of $x$ . 
${\int}_{a}^{b}f(x)dx$ 
The definite integral of $f$ from $a$ to $b$ . When $f$ is positive and $ab$ holds, then this is the area between the xaxis the lines $y=a,y=b$ and the curve that represents the function $f$ between these lines. 
$L(d)$ 
A Reimann sum with uniform interval size $d$ and $f$ evaluated at the left end of each subinterval. 
$R(d)$ 
A Reimann sum with uniform interval size $d$ and $f$ evaluated at the right end of each subinterval. 
$M(d)$ 
A Reimann sum with uniform interval size $d$ and $f$ evaluated at the maximum point of $f$ in each subinterval. 
$m(d)$ 
A Reimann sum with uniform interval size $d$ and $f$ evaluated at the minimum point of $f$ in each subinterval. 
