2. Verify Stokes' theorem, ∬S​(∇×F)⋅dS=∮C​F⋅dr, for the vector field F=3yzi+x(2+z)j​+2xyk on the upside down paraboloida

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2 Verify Stokes Theorem S F Ds C F Dr For The Vector Field F 3yzi X 2 Z J 2xyk On The Upside Down Paraboloida 1 (196.68 KiB) Viewed 50 times

2. Verify Stokes' theorem, ∬S​(∇×F)⋅dS=∮C​F⋅dr, for the vector field F=3yzi+x(2+z)j​+2xyk on the upside down paraboloidal surface z=1−x2−y2,x2+y2≤1 (a) Calculate the line integral on the right side of Stokes' theorem for the appropriate, counterclockwise, boundary curve C in the z=0 plane. (b) Parameterize the surface, determine the vector element of surface dS, and calculate the surface integral on the left side of Stokes' theorem. Be sure to take advantage of the cylindrical symmetry of the surface.