Paper 3, Section II, C

Vector Calculus | Part IA, 2012

State Stokes' Theorem for a vector field B(x)\mathbf{B}(\mathbf{x}) on R3\mathbb{R}^{3}.

Consider the surface SS defined by

z=x2+y2,19z1.z=x^{2}+y^{2}, \quad \frac{1}{9} \leqslant z \leqslant 1 .

Sketch the surface and calculate the area element dSd \mathbf{S} in terms of suitable coordinates or parameters. For the vector field

B=(y3,x3,z3)\mathbf{B}=\left(-y^{3}, x^{3}, z^{3}\right)

compute ×B\nabla \times \mathbf{B} and calculate I=S(×B)dSI=\int_{S}(\nabla \times \mathbf{B}) \cdot d \mathbf{S}.

Use Stokes' Theorem to express II as an integral over S\partial S and verify that this gives the same result.

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