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Unformatted text preview: 1 Example 1. Let F = ( yze x + xyze x ) i + xze x j + xye x k . Show that the integral of F around an oriented simple curve C that is the boundary of a surface S is zero. Solution. By Stokes theorem, Z C F d s = ZZ S ( F ) n dS. We calculate F = i j k x y z yze x + xyze x xze x xye x = [ xe x- xe x ] i- [( y + xy ) e x- ( y + xy ) e x ] j +[( z + xz ) e x- ( z + xz ) e x ] k = 0 . Hence the integrals are zero. Alternatively, one can observe that F = ( xyze x ) and so its integral around any closed curve is zero. 2 Example 2. Find the integral of F ( x,y,z ) = x 2 i + y 2 j- z k around the triangle with vertices (0 , , 0) , (0 , 2 , 0) and (0 , , 3), using Stokes theorem. Solution. In the figure, C is the triangle in question and S is a surface it bounds. By Stokes theorem, Z C F d r = ZZ S ( F ) n dS. Now i j k x y z x 2 y 2- z = 0 i + 0 j + 0 k = 0 ....
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