Question

Calculate the left and right hand side of Stokes Theorem for this problem. Explain why you obtained different values, and why it is not a contradiction

2. [5 Stokes' Theorem: Let S be a piecewise smooth oriented surface having a piecewise smooth boundary curve C. Let F = Mi+Nj+Pk be a vector field whose components have continuous first partial derivatives on an open region containing S. The the circulation of F around C in the direction counterclockwise with respect to the surface's unit normal vector n equals the integral of the curl vector field V x F over S: F dr = / /VxF n da counterclockwise Now consider the vector field F=- i + 2+y2 which is defined on the punctured disk D defined by those points in the ay-plane satisfying 0 < x2+y2 a2 for some positive real number, a.

Answer 1

"F vector dr vector = (- y/x^2 + y^2 i^^+ x/x^2 + y^2 j^^) middot (dx i^^+ dy j^^) = -y/x^2 + y^2 dx + x/x^2 + y^2 dy Using the parametric form of curve x^2 + y^2 = a^2, a > 0 x = a cos theta, y = a sin theta, theta elementof (0, 2 pi) So, dx = -a sin theta d theta, dy = a cos theta d theta F vector middot dr vector = -a sin theta/a^2 (-a sin theta d theta) + a cos theta/a^2 a cos theta d theta = sin^2 theta d theta + cos^2 theta d theta = d theta Thus, integral_c F vector middot dr vector = 2 pi integral theta = 0 d theta = 2 pi (1) Consider, nabla vector times F vector =