Question

Consider a vector field given in cartesian coordinates (r, y,2) by uy (A) Calculate the curl of this vector field ▽ ˇ (B) Verify that Stokes theorem holds if the contour is the square with corners (d, d, 0), (-d, d, 0), ( d, d, 0), and (d,-d,0) aid the surface spanned by this (ont our is at 0.

Answer 1

Consider a vector field given in cartesian coordinates (x, y, z) by v^vector = y x^^(A) the curl of vector field nabla times v^vector since nabla = operator nabla^vector = partial differential/partial differential x x^^+ partial differential/partial differential y y^^+ partial differential/partial differential z z^^then nabla times nabla^vector = x^^partial differential/partial differential x y y^^partial differential/partial differential y 0 z^^partial differential/partial differential z 0 nabla^vector times v^vector = x^^(0) + y^^(0) + z^^(0 - partial differential (y)/partial differential y) nabla^vector times v^vector = -z^^equation (a) (B) given: - square points: (d, d, 0), (- d, d, 0), (-d, -d, 0) and (d, -d, 0) at z = 0 we know that stoke�s theorem contourintegral v^vector middot ds^vector = doubleintegral (nabla^vector times v^vector) middot ds^vector contourintegral v^vector middot ds^vector = doubleintegral_s (nabla^vector times v^vector) middot n^^ds equation (b) to take R.H.S. part of eqn (b) from eqn (a)