Past exam of the mathematical methods II course of the University of Oxford / 2019 resit / 10 / a

For general coordinates $q=(a,b,c)$ expressed as functions of $u=(u,v,w)$, the chain rule gives $dq=Jdu$, whereThe Jacobian determinant is the oriented volume scaling of this linearized map. Thus the unsigned volume element transforms as $dadbdc=|\det J|dudvdw$. Where the coordinate change is invertible, the inverse Jacobian determinant is $1/\det J$, by the Jacobian chain rule.

For cylindrical coordinates, use $x=r\cos \varphi$, $y=r\sin \varphi$, and unchanged height $z$. Direct differentiation giveswhose determinant is $r$. ThereforeBoth directions are displayed because “from Cartesian to cylindrical” can refer either to the derivative of the new coordinates or to the factor used in transforming a Cartesian volume integral. They are reciprocals; the cylindrical axis is a coordinate degeneracy.

Take spherical polar coordinates with polar angle $\theta$ measured from positive $z$ and azimuthal angle $\varphi$:The derivative matrix isExpanding its determinant gives $r^2\sin \theta$. ThusThe inverse formula holds where $r>0$ and $0<\theta <\pi$; it excludes the origin and polar axis. These formulas use the general chain rule and determinant construction in solution.