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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 MIT OpenCourseWare http://ocw.mit.edu 18.02SC Multivariable Calculus Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.