 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

2015

Paper 2

Use the <product rule> for the <vector field> $F=\phi\nabla\psi-\psi\nabla\phi$:
$$
\nabla\cdot F=\nabla\phi\cdot\nabla\psi+\phi\nabla^2\psi-\nabla\psi\cdot\nabla\phi-\psi\nabla^2\phi=\phi\nabla^2\psi-\psi\nabla^2\phi.
$$
The mixed <dot products> cancel. Applying the <divergence theorem> with the outward <unit normal> gives <Green's second identity>,
$$
\boxed{\int_V(\phi\nabla^2\psi-\psi\nabla^2\phi)dV=\int_S(\phi\,n\cdot\nabla\psi-\psi\,n\cdot\nabla\phi)dS.}
$$
The fields and boundary are assumed sufficiently regular; a Green-function singularity is handled by excising a small ball or by using <distributions>.

Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 3 / a / Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 3 / a / 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 / 2015 / Paper 2 / 3 / a / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)