]> Glossary of Notations
 Tools     Glossary     Index      Up      Previous      Next

# Glossary of Notations

 $i$ The square root of minus one. $f ( x )$ The value of the function $f$ at argument $x$ . $sin ⁡ x$ The value of the sine function at argument $x$ . $exp ⁡ x$ 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. $ln ⁡ x$ The inverse function to $exp ⁡ x$ . $a x$ Same as $a ^ x$ . $log ⁡ b a$ The power you must raise $b$ to in order to get $a$ ; $b log ⁡ b a = a$ . $cos ⁡ x$ The value of the cosine function (complement of the sine) at argument $x$ . $tan ⁡ x$ Works out to be $sin ⁡ x cos ⁡ x$ . $cot ⁡ x$ The value of the complement of the tangent function or $cos ⁡ x sin ⁡ x$ . $sec ⁡ x$ Value of the secant function, which turns out to be $1 cos ⁡ x$ . $csc ⁡ x$ Value of the complement of the secant, called the cosecant. It is $1 sin ⁡ x$ . $asin ⁡ x$ The value, $y$ , of the inverse function to the sine at argument $x$ . Means $x = sin ⁡ y$ . $acos ⁡ x$ The value, $y$ , of the inverse function to cosine at argument $x$ . Means $x = cos ⁡ y$ . $atan ⁡ x$ The value, $y$ , of the inverse function to tangent at argument $x$ . Means $x = tan ⁡ y$ . $acot ⁡ x$ The value, $y$ , of the inverse function to cotangent at argument $x$ . Means $x = cot ⁡ y$ . $asec ⁡ x$ The value, $y$ , of the inverse function to secant at argument $x$ . Means $x = sec ⁡ y$ . $acsc ⁡ x$ The value, $y$ , of the inverse function to cosecant at argument $x$ . Means $x = csc ⁡ y$ . $θ$ A standard symbol for angle. Measured in radians unless stated otherwise. Used especially for $a tan ⁡ x y$ when $x , y$ , and $z$ are variables used to describe point in three dimensional space. $i ^ , j ^ , k ^$ 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 $b$ . $( a ⟶ , b ⟶ )$ The dot product of vectors $a ⟶$ and $b ⟶$ . $a ⟶ · b ⟶$ The dot product of vectors $a ⟶$ and $b ⟶$ . $( a ⟶ · b ⟶ )$ The dot product of vectors $a ⟶$ and $b ⟶$ . $| v ⟶ |$ The magnitude of the vector $v ⟶$ . $| x |$ The absolute value of the number $x$ . $∑$ 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 $∑ j = 1 n j$ or $∑ n j$ . This signifies $1 + 2 + ⋯ + n$ . $M$ 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. $d x$ An "infinitesimal" or very small change in the variable $x$ ; also similarly $d y , d z , d r$ etc. $d s$ A small change in distance. $ρ$ The variable $( x 2 + y 2 + z 2 ) 1 / 2$ or distance to the origin in spherical coordinates. $r$ The variable $( x 2 + y 2 ) 1 / 2$ or distance to the z-axis in three dimensions or in polar coordinates. $| M |$ The determinant of a matrix $M$ (whose magnitude is the area or volume of the parallel sided region determined by its columns or rows). $‖ M ‖$ The magnitude of the determinant of the matrix $M$ , which is a volume or area or hypervolume. $det ⁡ M$ The determinant of $M$ . $M − 1$ The inverse of the matrix $M$ . $v ⟶ × w ⟶$ The vector product or cross product of two vectors, $v ⟶$ and $w ⟶$ . $θ v ⟶ w ⟶$ The angle made by vectors $v ⟶$ and $w ⟶$ . $A ⟶ · B ⟶ × C ⟶$ The scalar triple product, the determinant of the matrix formed by columns $A ⟶ , B ⟶ , C ⟶$ . $u ^ w ⟶$ A unit vector in the direction of the vector $w ⟶$ ; it means the same as $w ⟶ | w ⟶ |$ . $d f$ A very small change in the function $f$ , sufficiently small that the linear approximation to all relevant functions holds for such changes. $d f d x$ The derivative of $f$ with respect to $x$ , which is the slope of the linear approximation to $f$ . $f '$ The derivative of $f$ with respect to the relevant variable, usually $x$ . $∂ f ∂ x$ 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 $d f$ to $d q$ 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. $∂ f ∂ x | y , z$ The partial derivative of $f$ with respect to $x$ keeping $y$ and $z$ fixed. $grad ⟶ f$ The vector field whose components are the partial derivatives of the function $f$ with respect to $x , y$ and $z$ : $( ∂ f ∂ x , ∂ f ∂ y , ∂ f ∂ z )$ or $∂ f ∂ x i ^ + ∂ f ∂ y j ^ + ∂ f ∂ z k ^$ ; called the gradient of $f$ . $∇ ⟶$ The vector operator $∂ ∂ x i ^ + ∂ ∂ y j ^ + ∂ ∂ z k ^$ , called "del" . $∇ ⟶ f$ The gradient of $f$ ; its dot product with $u ^ w$ is the directional derivative of $f$ in the direction of $w ⟶$ . $∇ ⟶ · w ⟶$ The divergence of the vector field $w ⟶$ ; it is the dot product of the vector operator $∇ ⟶$ with the vector $w ⟶$ , or $∂ w x ∂ x + ∂ w y ∂ y + ∂ w z ∂ z$ . $curl ⟶ w ⟶$ The cross product of the vector operator $∇ ⟶$ with the vector $w ⟶$ . $∇ ⟶ × w ⟶$ The curl of $w ⟶$ , with components $( ∂ f z ∂ y − ∂ f y ∂ z , ∂ f x ∂ z − ∂ f z ∂ x , ∂ f y ∂ x − ∂ f x ∂ y )$ . $∇ ⟶ · ∇ ⟶$ The Laplacian, the differential operator: $∂ 2 ∂ x 2 + ∂ ∂ y 2 + ∂ ∂ z 2$ . $f " ( x )$ The second derivative of $f$ with respect to $x$ ; the derivative of $f ' ( x )$ . $d 2 f d x 2$ 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 k-th derivative of $f$ with respect to $x$ ; the derivative of $f ( k − 1 ) ( x )$ . $T ^$ Unit tangent vector along a curve; if curve is described by $r ⟶ ( t ) , T ^ = d r ⟶ / d t | d r ⟶ / d t |$ . $d s$ A differential of distance along a curve. $κ$ The curvature of a curve; the magnitude of the derivative of its unit tangent vector with respect to distance on the curve: $| d T ^ d s |$ . $N ^$ A unit vector in the direction of the projection of $d T ^ d s$ normal to $T ^$ . $B ^$ A unit vector normal to the plane of $T ^$ and $N ^$ , which is the plane of curvature. $τ$ The torsion of a curve; $| d B ^ d s |$ . $g$ The gravitational constant. $F ⟶$ The standard symbol for force in mechanics. $k$ The spring constant of a spring. $p ⟶ i$ The momentum of the i-th particle. $H$ The Hamiltonian of a physical system, which is its energy expressed in terms of ${ r ⟶ i }$ and ${ p ⟶ i }$ , position and momentum. ${ Q , H }$ The Poisson bracket of $Q$ and $H$ . $∫ x f ( u ) d u$ An antiderivative of $f ( x )$ expressed as a function of $x$ . $∫ a b f ( x ) d x$ The definite integral of $f$ from $a$ to $b$ . When $f$ is positive and $a < b$ holds, then this is the area between the x-axis 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.