 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2015

Paper 3

Apply the <divergence theorem> to $\mathbf F=\psi\nabla\phi$, for functions smooth enough on the closure of the bounded piecewise smooth volume. The <product rule> gives
$$
\nabla\cdot(\psi\nabla\phi)=\nabla\psi\cdot\nabla\phi+\psi\nabla^2\phi,
\qquad (\psi\nabla\phi)\cdot\mathbf n=\psi\frac{\partial\phi}{\partial n}.
$$
Thus <Green's first theorem>, also called <Green's first identity>, is
$$
\boxed{\int_V(\psi\nabla^2\phi+\nabla\psi\cdot\nabla\phi)\,dV
=\int_{\partial V}\psi\frac{\partial\phi}{\partial n}\,dS.}
$$
The normal is outward. This identifies the boundary term and its sign directly from the <divergence theorem>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 11A / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 11A / i / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 11A / i / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)