Paper 3, Section II, C

Vector Calculus | Part IA, 2013

Consider the bounded surface SS that is the union of x2+y2=4x^{2}+y^{2}=4 for 2z2-2 \leqslant z \leqslant 2 and (4z)2=x2+y2(4-z)^{2}=x^{2}+y^{2} for 2z42 \leqslant z \leqslant 4. Sketch the surface.

Using suitable parametrisations for the two parts of SS, calculate the integral

S(×F)dS\int_{S}(\nabla \times \mathbf{F}) \cdot d \mathbf{S}

for F=yz2i\mathbf{F}=y z^{2} \mathbf{i}.

Check your result using Stokes's Theorem.

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