- #1

teddd

- 62

- 0

So here I go again..

I've a doubt on the way the infinitesimal volume element transfoms when performing a coordinate transformation from [itex]x^j[/itex] to [itex]x^{j'}[/itex]

It should change according to [tex]dx^1dx^2...dx^n=\frac{\partial (x^1,x^2...x^n)}{\partial (x^{1'}x^{2'}...x^{n'})}dx^{1'}dx^{2'}...dx^{n'}[/tex]where [itex]\frac{\partial (x^1,x^2...x^n)}{\partial (x^{1'}x^{2'}...x^{n'})}[/itex] is the Jacobian of the transformation.So i tried to do this in a concrete example: the transformation between cartesian [itex]x,y[/itex] to polar [itex]r,\theta[/itex] coordinates.

The jacobian of this transformation is [itex]r[/itex] and so, according to what I've written above[tex]dxdy=rdrd\theta[/tex]but since [itex]dr=cos\theta dx+sin\theta dy[/itex] and [itex]d\theta=-\frac{1}{r}sin\theta dx+\frac{1}{r}cos\theta dy[/itex] i get to [tex]dV=r(cos\theta dx+sin\theta dy)(-\frac{1}{r}sin\theta dx+\frac{1}{r}cos\theta dy)=(-sin\theta cos\theta dx^2+sin\theta cos\theta dy^2+cos^2\theta dxdy-sin^2\theta dxdy)[/tex]and this is not equal to [itex]dxdy[/itex], the volume element in cartesian coordinate, as it should be!

Where am I mistaking?

Thanks!