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teddd

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Very simple question:

Let [itex]x^0,x^1,...,x^n[/itex] be some fixed coordinate system, so that the infinitesimal volume element is [itex]dV=dx^0dx^1...dx^n[/itex].

Then any change to a new (primed) coordinate system [itex]x^{0'},x^{1'},...,x^{n'}[/itex] transforms the volume to [tex]dV=\frac{\partial (x^0,x^1,...,x^n)}{\partial (x^{0'},x^{1'},...,x^{n'})}dx^{0'}dx^{1'}...dx^{n'}[/tex] where [itex]\frac{\partial (x^0,x^1,...,x^n)}{\partial (x^{0'},x^{1'},...,x^{n'})}[/itex] is the determinat of the jacobian of the transformation.

So let's try to do this in a concrete example: the transformation from cartesian [itex]x,y[/itex] to polar [itex]r,\theta[/itex] coordinates.

The Jacobian is simply [itex]r[/itex] and so i get to [itex]dV=dxdy=rdrd\theta[/itex].

Doing the math i get [tex]dV^{pol.}=r(\cos\theta dx+\sin\theta dy)(-\frac{1}{r}\sin\theta dx+\frac{1}{r}\cos\theta dy)[/tex]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].

But then:[tex]dV^{pol.}=-\cos\theta\sin\theta dx^2+\cos^2\theta dxdy-\sin^2\theta dxdy + \cos\theta\sin\theta dy^2[/tex] and is clear that [itex]dV^{cart.}\neq dV^{pol.}[/itex] !

Where is my mistake?? Thanks for your help guys!

Let [itex]x^0,x^1,...,x^n[/itex] be some fixed coordinate system, so that the infinitesimal volume element is [itex]dV=dx^0dx^1...dx^n[/itex].

Then any change to a new (primed) coordinate system [itex]x^{0'},x^{1'},...,x^{n'}[/itex] transforms the volume to [tex]dV=\frac{\partial (x^0,x^1,...,x^n)}{\partial (x^{0'},x^{1'},...,x^{n'})}dx^{0'}dx^{1'}...dx^{n'}[/tex] where [itex]\frac{\partial (x^0,x^1,...,x^n)}{\partial (x^{0'},x^{1'},...,x^{n'})}[/itex] is the determinat of the jacobian of the transformation.

So let's try to do this in a concrete example: the transformation from cartesian [itex]x,y[/itex] to polar [itex]r,\theta[/itex] coordinates.

The Jacobian is simply [itex]r[/itex] and so i get to [itex]dV=dxdy=rdrd\theta[/itex].

Doing the math i get [tex]dV^{pol.}=r(\cos\theta dx+\sin\theta dy)(-\frac{1}{r}\sin\theta dx+\frac{1}{r}\cos\theta dy)[/tex]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].

But then:[tex]dV^{pol.}=-\cos\theta\sin\theta dx^2+\cos^2\theta dxdy-\sin^2\theta dxdy + \cos\theta\sin\theta dy^2[/tex] and is clear that [itex]dV^{cart.}\neq dV^{pol.}[/itex] !

Where is my mistake?? Thanks for your help guys!

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