Home | 18.013A | | 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.

Up Previous Next