Paper 3, Section II, A

Vector Calculus | Part IA, 2015

State Stokes' theorem.

Let SS be the surface in R3\mathbb{R}^{3} given by z2=x2+y2+1λz^{2}=x^{2}+y^{2}+1-\lambda, where 0z10 \leqslant z \leqslant 1 and λ\lambda is a positive constant. Sketch the surface SS for representative values of λ\lambda and find the surface element dS\mathbf{d} \mathbf{S} with respect to the Cartesian coordinates xx and yy.

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

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

and verify Stokes' theorem for F\mathbf{F} on the surface SS for every value of λ\lambda.

Now compute ×G\nabla \times \mathbf{G} for the vector field

G(x)=1x2+y2(yx0)\mathbf{G}(\mathbf{x})=\frac{1}{x^{2}+y^{2}}\left(\begin{array}{c} -y \\ x \\ 0 \end{array}\right)

and find the line integral SGdx\int_{\partial S} \mathbf{G} \cdot \mathbf{d x} for the boundary S\partial S of the surface SS. Is it possible to obtain this result using Stokes' theorem? Justify your answer.

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