calculating the Jacobian in polar coordinates geometrically

Question

The task sounds like this: “geometrically calculate the Jacobian of polar substitution.” In this case, we get the expression dxdy=rdrd(theta)+ 4s, where 4s are the remaining pieces when inserted. But at the same time, this is wrong or not entirely correct, I was told that I need to solve this somehow differently, and at the same time it can be solved very simply, as I was told.enter image description here

Answer 1

The transformation of the volume element in $(x,y)\in \mathbb R^2$ is the alternating wedge product

$$\begin{align}\mathrm dx \wedge \mathrm dy &= \mathrm d(r \cos \theta )\wedge \mathrm d(r \sin \theta) \\&=(\mathrm dr \cos \theta- \mathrm d\theta\,r \sin \theta) \wedge (\mathrm d r \sin \theta + \mathrm d\theta\,r \cos \theta)\\& = r\,\mathrm dr\wedge \mathrm d\theta\,(\cos^2 \theta + \sin^2 \theta) \\& = \mathrm dr \wedge r\,\mathrm d\theta \end{align}$$

The last expression is simply the rectangular area between to circles with radial distance $\mathrm dr$ and two radial rays with arc length $r\,\mathrm d\theta$ between them.

In orthogonal coordinates the volume element is always the product of the coordinate differentials multiplied by their scale factors