General:

[3]

number 3 in list of references on pages 430–32

Ex. 12.4

fourth exercise at the end of Section 12

Eq. (3.5), Th. 7.2

similar references to numbered equations and theorems

indicates the end of a proof

⇒

implies

⇔

if and only if

x ∈ A

x is an element of the set A

A ⊂ B

the set A is included in the set B

log x

natural logarithm of x (base e)

exp x

e^x

f(b) – f(a)

f(x, y) evaluated at (x₀, y₀)

Specific:

a, b

vector with components a, b

v

boldface indicates a vector quantity

|v|

magnitude of the vector v

T_α

unit vector in the direction α

v · w

scalar product of vectors

partial derivatives

∇_α

directional derivative in the direction α

∇

gradient

f_c(t)

function f(x, y) restricted to curve C with parameter t

f_xx, f_xy, f_yy

second order derivatives

class of functions all of whose nth-order partial derivatives exist and are continuous

for transformations

for vector fields

the class of continuous functions

nth-order derivatives

higher order directional derivatives

F: (x, y) → (u, υ)

the mapping F takes (x, y) onto (u, υ)

F^-1

inverse of mapping F

F^-1(S)

the inverse image of the set S under the mapping F

F₂ F₁

composition of mappings

dF

differential of transformation F

J_F

Jacobian matrix of F

Jacobian of a mapping

Δf

Laplacian of function f

∫_cf dx, ∫_cf dy

line integrals along a curve C

∫_cf ds

line integral with respect to arc length

ν_T

tangential component of vector v

∪

union

double integral of the function f over the figure F

normal derivative of the function u

n(C; X, Y)

winding number of curve C about point (X, Y)

∂F

oriented boundary of a figure F

υ_N

normal component of vector v

div v

divergence of vector field v