I don't know where is a contradiction to my understanding with the following argument. Can anyone help me to point out my mistake.

We know that the co-area states as follows (everything in $\mathbb{R}^{2}$):

$$ \intop_{\left\{ x^{2}+y^{2}<1\right\} }f\left(x,y\right)dxdy=\intop_{0}^{1}dt\intop_{\sqrt{x^{2}+y^{2}}=t}f\left(x,y\right)dS\left(x,y\right). $$ Now look at the right-hand side, we can write $$ \intop_{0}^{1}dt\intop_{\sqrt{x^{2}+y^{2}}=t}f\left(x,y\right)dS\left(x,y\right)=\intop_{0}^{1}dt\intop_{0}^{2\pi}f\left(t\cos\theta,t\sin\theta\right)d\theta. $$ However, using polar coordinate formula, the left-hand side is $$ \intop_{\left\{ x^{2}+y^{2}<1\right\} }f\left(x,y\right)dxdy=\intop_{0}^{1}\intop_{0}^{2\pi}f\left(t\cos\theta,t\sin\theta\right)\times\underbrace{t}_{Jacobian}\times d\theta dt. $$

The contradiction follows by the fact that the last two terms are not equal. Where is my mistake?



The arclength element $dS$ (in your notation) is $t\cdot d\theta$, not simply $d\theta$.

(Remember that the length of an arc subtended by angle $\theta$ in a circle with radius $r$ is $r\theta$.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.