Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
This important result is known as the second form of Green's theorem, and has
many applications.
Green's theorem in the plane
Consider the two-dimensional vector ®eld A ˆ M…x 1 ; x 2 †^e 1 ‡ N…x 1 ; x 2 †^e 2 . From
Stokes' theorem
I Z
Z
@N
A dr ˆ rA da ˆ @M
dx
@x 1 @x 1 dx 2 ; …1:88†
2
S
which is often called Green's theorem in the plane.
Since H A dr ˆ H …Mdx 1 ‡ Ndx 2 †, Green's theorem in the plane can be written
as
I
Z
@N
Mdx 1 ‡ Ndx 2 ˆ @M
dx
@x 1 @x 1 dx 2 : …1:88a†
2
S
As an illustrative example, let us apply Green's theorem in the plane to show
that the area bounded by a simple closed curve is given by
I
1
x
2 1 dx 2 x 2 dx 1 :
Into Green's theorem in the plane, let us put M ˆx 2 ; N ˆ x 1 , giving
I
Z
@
x 1 dx 2 x 2 dx 1 ˆ x
S @x 1 @ Z
…x
1 @x 2 † dx 1 dx 2 ˆ 2 dx 1 dx 2 ˆ 2A;
2 S
H
where A is the required area. Thus A ˆ 1
2 x 1dx 2 x 2 dx 1 .
S
Helmholtz's theorem
The divergence and curl of a vector ®eld play very important roles in physics. We
learned in previous sections that a divergence-free ®eld is solenoidal and a curlfree
®eld is irrotational. We may classify vector ®elds in accordance with their
being solenoidal and/or irrotational. A vector ®eld V is:
(1) Solenoidal and irrotational if rV ˆ 0andrV ˆ 0. A static electric
®eld in a charge-free region is a good example.
(2) Solenoidal if rV ˆ 0 but rV 6ˆ 0. A steady magnetic ®eld in a currentcarrying
conductor meets these conditions.
(3) Irrotational if rV ˆ 0 but rV ˆ 0. A static electric ®eld in a charged
region is an irrotational ®eld.
The most general vector ®eld, such as an electric ®eld in a charged medium with
a time-varying magnetic ®eld, is neither solenoidal nor irrotational, but can be
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