 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Natural sciences course of the University of Cambridge

Year of the natural sciences course of the University of Cambridge

Year 2 of the natural sciences course of the University of Cambridge

Mathematics course of the year 2 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge

2014

Paper 2

For the <vector field> $F=\phi\nabla\psi-\psi\nabla\phi$, the <product rule> gives
$$
\nabla\cdot F=\nabla\phi\cdot\nabla\psi+\phi\nabla^2\psi-\nabla\psi\cdot\nabla\phi-\psi\nabla^2\phi=\phi\nabla^2\psi-\psi\nabla^2\phi.
$$
The mixed <dot products> cancel. The <divergence theorem> therefore yields <Green's second identity>,
$$
\boxed{\int_V(\phi\nabla^2\psi-\psi\nabla^2\phi)dV=\int_S(\phi\,\hat n\cdot\nabla\psi-\psi\,\hat n\cdot\nabla\phi)dS.}
$$
The fields and boundary are assumed sufficiently regular. A singular <Green function> can be treated by excising a small ball around its source and taking the radius to zero, or using <distributions>.

Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 3 / i / Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 3 / i / Solution

 Ancestors (13)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 3 / i / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)