Mathematics of Physics and Engineering

150 The Three Theorems

3.2 The Three Integral Theorems of Vector Analysis

The theorems of Green, Gauss, and Stokes are usually the high point of
a multi-variable calculus class. We will use these theorems to derive the
equations of electromagnetic theory.

3.2.1 Green's Theorem

Theorem 3.2.1 In the space R^2 with cartesian coordinates, consider a

domain G whose boundary dG consists of finitely many simple closed piece-

wise smooth curves. Let P = P(x, y), Q = Q(x, y) be two functions having

continuous first-order partial derivatives in an open domain containing G.

Assume that the orientation of dG is positive (G stays to the left as you

walk around dG). Then

G

If the boundary of G consists of several pieces, then the integral on the left

is the sum of the corresponding integrals over each piece.

This result is known as Green's theorem, after the English scientist

GEORGE GREEN (1793-1841), who published it in 1828 as a part of an

essay on electricity and magnetism. The essay was self-published for pri-

vate distribution and did not receive much attention at the time. Green

did all his major scientific work while managing the family mill and bak-

ing business, with about two years of elementary school as his only formal

education. Later, he quit the family business and, in 1837, got an under-

graduate degree from Cambridge University.

EXERCISE 3.2.1. B Prove Green's theorem. Hint: Consider the vector field

F = Pi + Qj + Ok and verify that V x F = (Qx — Pv)it. Then break G into

small pieces Gk and note that, by (3.1.26), (Qx — Py)m(Gk) ~ fgG Pdx + Qdy,

where m(Gfc) is the area of Gk, and dGk is the boundary of Gk with the suitable

orientation. Now sum over all the small pieces and note that the line integrals

over the interior pieces cancel one another. Finally, pass to the limit m(Gfc) —• 0.

An interesting application of (3.2.1) is computing areas by line integrals:

m(G) = I dA= <p xdy = — d> ydx = -i xdy — ydx. (3.2.2)

J J JdG JdG^2 JdG