 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

2016

Paper 2

Apply the <product rule> to 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 two mixed <dot products> cancel. The <divergence theorem>, using the outward <unit normal>, then 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 usual assumptions are a sufficiently regular boundary and fields with the derivatives and integrability needed for the theorem; singular sources can instead be treated by excising small balls or using <distributions>.

Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / 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 / 2016 / 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 / 2016 / Paper 2 / 3 / a / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)