Problems: Surface Independence
Suppose that F = V × G, where the components of G have continuous second partial
derivatives. Suppose also that S is a closed, positively-oriented surface divided into two
F · n dS = 0.
parts by a closed curve C. Apply Stokes’ theorem to show that
S
Answer: We start by drawing a picture.
G · dr =
By Stokes’ theorem,
F · n dS. Similarly,
curlG dA =
C
S1
S1
G · dr =
−C
F · n dS.
S2
We sum the two results to get:
F · n dS =
S
F · n dS +
S1
F · n dS =
S2
G · dr +
C
G · dr = 0.
−C
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18.02SC Multivariable Calculus
Fall 2010
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