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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2002

Paper 3

Use the <product rule> for the <divergence> of the <vector field> $F=\phi\nabla\psi-\psi\nabla\phi$:
$$
\begin{aligned}
\nabla\cdot F
&=\nabla\phi\cdot\nabla\psi+\phi\nabla^2\psi
-\nabla\psi\cdot\nabla\phi-\psi\nabla^2\phi\\
&=\phi\nabla^2\psi-\psi\nabla^2\phi.
\end{aligned}
$$
The two <dot products> cancel. Apply the <divergence theorem> with outward unit <normal vector> $n$ to obtain <Green's second identity>:
$$
\boxed{\int_V(\phi\nabla^2\psi-\psi\nabla^2\phi)\,dV
=\int_S(\phi\nabla\psi-\psi\nabla\phi)\cdot n\,dS.}
$$
Smoothness of the functions and boundary supplies the regularity needed for the <divergence theorem>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 3 / 12A / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 3 / 12A / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 3 / 12A / b / Solution

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