Paper 3, Section II, C

Vector Calculus | Part IA, 2018

State the formula of Stokes's theorem, specifying any orientation where needed.

Let F=(y2z,xz+2xyz,0)\mathbf{F}=\left(y^{2} z, x z+2 x y z, 0\right). Calculate ×F\boldsymbol{\nabla} \times \mathbf{F} and verify that ×F=0\boldsymbol{\nabla} \cdot \boldsymbol{\nabla} \times \mathbf{F}=0.

Sketch the surface SS defined as the union of the surface z=1,1x2+y24z=-1,1 \leqslant x^{2}+y^{2} \leqslant 4 and the surface x2+y2+z=3,1x2+y24x^{2}+y^{2}+z=3,1 \leqslant x^{2}+y^{2} \leqslant 4.

Verify Stokes's theorem for F\mathbf{F} on SS.

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