Vector Calculus | Part IA, 2003

Let CC be the closed curve that is the boundary of the triangle TT with vertices at the points (1,0,0),(0,1,0)(1,0,0),(0,1,0) and (0,0,1)(0,0,1).

Specify a direction along CC and consider the integral

CAdx\oint_{C} \mathbf{A} \cdot d \mathbf{x}

where A=(z2y2,x2z2,y2x2)\mathbf{A}=\left(z^{2}-y^{2}, x^{2}-z^{2}, y^{2}-x^{2}\right). Explain why the contribution to the integral is the same from each edge of CC, and evaluate the integral.

State Stokes's theorem and use it to evaluate the surface integral

T(×A)dS,\int_{T}(\boldsymbol{\nabla} \times \mathbf{A}) \cdot d \mathbf{S},

the components of the normal to TT being positive.

Show that dSd \mathbf{S} in the above surface integral can be written in the form (1,1,1)dydz(1,1,1) d y d z.

Use this to verify your result by a direct calculation of the surface integral.

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