*©
Copyright 19942000 by Evans M. Harrell
II and James V. Herod. All rights reserved.
This document collects some standard vector identities
and relationships among coordinate systems in three dimensions. It is assumed
that all vector fields are differentiable arbitrarily often; if the vector
field is not sufficiently smooth, some of these formulae are in doubt.
Basic formulae

The dot product
quantifies the correlation between the vectors a and b .

The cross product
is the area of the parallelogram spanned by the vectors a and b.
Notice that
Unlike the dot product, which works in all dimensions, the cross product is
special to three dimensions.

The triple product
has the value of the determinant of the matrix consisting of a,
b, and c as row vectors. It is unchanged by cyclic permutation:
Although the cross product is strictly threedimensional, the generalization
of the triple product as a determinant is useful in all dimensions.

Other multiple products.

The gradient is defined on a scalar field f and produces a vector field,
denoted
It quantifies the rate of change and points in the direction of greatest
change.

The divergence is defined on a vector field v and produces a scalar
field, denoted
It quantifies the tendency of neighboring vectors to point away from one
another (or towards one another, if negative)

The curl is defined on a vector field and produces another vector field,
except that the curl of a vector field is not affected by reflection in the
same way as the vector field is. It is denoted
Unlike the gradient and the divergence, which work in all dimensions, the curl
is special to three dimensions.

The Laplacian is defined as

Product rules:
or, equivalently, grad (f g) = f grad g + g grad f
or, equivalently, div(f v) = f div v + grad f . v
or, equivalently, curl(f v) = f curl v + grad f X v

Chain rules
or, equivalently, grad f(g(x)) = f'(g(x)) grad g(x)
or, equivalently, df(w(t))/dt = grad f(w(t)) w'(t)

Integral identities (
Green's,
Gauss's,
and
Stokes's
identities):
Green's identities:


Gauss's divergence theorem:

Stokes's theorem:


Relationships among the common threedimensional coordinate systems.

Cartesian in spherical

Cartesian in cylindrical
 spherical in Cartesian

spherical in cylindrical

cylindrical in Cartesian

cylindrical in spherical

Cylindrical vector calculus. Let e_{k} denote the unit vector in the
direction of increase of coordinate k. Then

Spherical vector calculus. Let e_{k} denote the unit vector in the
direction of increase of coordinate k. Then
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