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Past papers, worked solutions and errata: www.hep.phy.cam.ac.uk/~lester/teaching/partIIIparticles/exams/index.html

The archive lists papers for 2000–2025. Worked solutions are public for 2000–2008, 2012–2018 and 2020–2025; skip those years for new solutions. The archive does not provide solutions for 2009–2011 and marks the 2019 solutions as not released publicly.

2025 paper: www.hep.phy.cam.ac.uk/~lester/teaching/partIIIparticles/exams/PartIIIParticles_202425.pdf

2025 worked solutions: www.hep.phy.cam.ac.uk/~lester/teaching/partIIIparticles/exams/PartIIIParticles_202425_with_answers.pdf

The displayed three-dimensional parabolic coordinates use $u,v\ge 0$ and an azimuthal angle $\varphi$. Their coordinate tangent vectors have pairwise zero dot products, with scale factors of orthogonal coordinates $h_u=h_v=\sqrt{u^2+v^2}$ and $h_{\varphi }=uv$. The Jacobian determinant is $uv(u^2+v^2)$. Constant $u$ or $v$ gives a family of confocal circular elliptic paraboloids; the coordinate system degenerates on the axis where $uv=0$. For example, $0\le u\le v\le 1$ fills the cap $(x^2+y^2-1)/2\le z\le 0$, whose rim is the unit circle and whose vertex is $(0,0,-1/2)$.

For $a>0$, $u\ge 0$, $0\le v<2\pi$ and real $z$, these orthogonal coordinates have coordinate tangents $r_u=(a\sinh u\cos v,a\cosh u\sin v,0)$, $r_v=(-a\cosh u\sin v,a\sinh u\cos v,0)$ and $r_z=(0,0,1)$. Their scale factors of orthogonal coordinates are $h_u=h_v=a\sqrt{{\sinh }^{2}u+{\sin }^{2}v}$ and $h_z=1$; the ordered Jacobian determinant is positive away from the two focal lines. Positive constant $u$ gives confocal elliptical cylinders, while generic constant $v$ gives a hyperbolic-cylinder half-branch selected by the signs of $x$ and $y$. The limiting $u=0$ surface is the strip $y=0$, $|x|\le a$, with arbitrary $z$. Constant $z$ gives horizontal planes.

For a length scale $c>0$, planar bipolar coordinates may be defined byThe common scale factor is $h=c/(\cosh \tau -\cos \sigma )$, so Laplace's equation becomes ${\Phi }_{\sigma \sigma }+{\Phi }_{\tau \tau }=0$ away from coordinate singularities. For $\tau >0$, constant-$\tau$ curves are circlesTheir radii decrease as $\tau$ increases; for $0<a<b$ the $\tau =b$ circle lies inside the $\tau =a$ circle. The region $a<\tau <b$ is therefore an eccentric circular annulus. This ordering is useful when translating coordinate boundary data into geometric language.

For a ring torus of major radius $A>0$, circular torus coordinates use distance $r$ and angle $\alpha$ in the circular cross-section, and azimuth $\beta$ around the central axis. The coordinate tangent vectors are the three partial derivatives of the displayed position. They are mutually orthogonal, with scale factors of orthogonal coordinates $1,r,A+r\sin \alpha$. For $0<r<A$ their ordered Jacobian determinant is $r(A+r\sin \alpha )>0$, so $dV=r(A+r\sin \alpha )drd\alpha d\beta$. At $r=0$ the cross-section angle degenerates. These are tubular coordinates for a circular torus, rather than the confocal coordinate convention also called toroidal coordinates.