Paper 3, Section II, A

Vector Calculus | Part IA, 2014

(a) State Stokes' Theorem for a surface SS with boundary S\partial S.

(b) Let SS be the surface in R3\mathbb{R}^{3} given by z2=1+x2+y2z^{2}=1+x^{2}+y^{2} where 2z5\sqrt{2} \leqslant z \leqslant \sqrt{5}. Sketch the surface SS and find the surface element dS\mathbf{d} \mathbf{S} with respect to the Cartesian coordinates xx and yy.

(c) Compute ×F\nabla \times \mathbf{F} for the vector field

F(x)=(yxxy(x+y))\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ x y(x+y) \end{array}\right)

and verify Stokes' Theorem for F\mathbf{F} on the surface SS.

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