]> Glossary of Notations

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.