Mathematical Methods for Physicists: A concise introduction - Site Map

SOME USEFUL INTEGRAL RELATIONS
considered as the sum of a solenoidal ®eld and an irrotational ®eld. This is made
clear by Helmholtz's theorem, which can be stated as (C. W. Wong: Introduction
to Mathematical Physics, Oxford University Press, Oxford 1991; p. 53):
A vector ®eld is uniquely determined by its divergence and curl in
a region of space, and its normal component over the boundary
of the region. In particular, if both divergence and curl are
speci®ed everywhere and if they both disappear at in®nity
suciently rapidly, then the vector ®eld can be written as a
unique sum of an irrotational part and a solenoidal part.
In other words, we may write
V…r† ˆr…r†‡rA…r†;
…1:89†
where r is the irrotational part and rA is the solenoidal part, and (r) and
A…r† are called the scalar and the vector potential, respectively, of V…r). If both A
and can be determined, the theorem is veri®ed. How, then, can we determine A
and ? If the vector ®eld V…r† is such that
rV…r† ˆ; and rV…r† ˆv;
then we have
rV…r† ˆ ˆr…r†‡r…rA†
or
r 2 ˆ;
which is known as Poisson's equation. Next, we have
rV…r† ˆv ˆ r ‰r ‡rA…r†Š
or
r 2 A ˆ v;
or in component, we have
r 2 A i ˆ v i ; i ˆ 1; 2; 3
where these are also Poisson's equations. Thus, both A and can be determined
by solving Poisson's equations.
Some useful integral relations
These relations are closely related to the general integral theorems that we have
proved in preceding sections.
(1) The line integral along a curve C between two points a and b is given by
Z b
a
… r†dl ˆ …b†…a†: …1:90†
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