Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 16

The logarithm is defined away from the $z$ axis, where $x^2+y^2>0$. For the vector potential $\psi {\hat{\mathbf{e}}}_z=(0,0,\psi )$, its curl is $({\psi }_y,-{\psi }_x,0)$. Sincewe obtainFor $a>0$ this is clockwise circulation about the axis, the velocity structure of a point vortex in each transverse plane.

On the same domain, differentiate the Cartesian components:ThusThis is also an instance of divergence of a curl is zero, since the potential is smooth off the axis. The classical field is undefined on the axis, so that identity must not be applied there without specifying an extension.

In cylindrical coordinates, $x^2+y^2=r^2$ and $r>0$, soThe potential vector has components $(A_r,A_{\theta },A_z)=(0,0,a\ln r)$. Expanding the prescribed cylindrical curl formula givesThereforeSince ${\hat{\mathbf{e}}}_{\theta }=(-\sin \theta ,\cos \theta ,0)$, this agrees with the Cartesian result and its clockwise orientation.

The divergence isFor the tetrahedral domain, convenient integration bounds areConsequently its volume integral isPutting $h=2(1-x-y)$, the inner integral is $(2y+6)h+h^2=(16-4x)(1-x-y)$. ThereforeThusThere is an independent centroid check. The vertices are $(0,0,0),(1,0,0),(0,1,0),(0,0,2)$, giving volume $1/3$ and centroid $(1/4,1/4,1/2)$. An affine function averages to its value at the centroid, so $I=(1/3)(6+2/4+2/2)=5/2$. This is the integral of an affine function over a simplex.