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## Main Question or Discussion Point

We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate ##r, r+dr, \theta, \theta + d\theta##).

Now, I tried to calculate it in a different way. We know,

## x = r \cos {\theta} ; y= r\sin{\theta}; ##

Hence,

## dx = - r \sin{\theta}~ d\theta + \cos{\theta}~ dr##

& ## dy = r \cos{\theta}~ d\theta + \sin{\theta} ~dr##

Now, we have, infinitesimal area element,

## dA = dx ~dy = - r^2 \cos {\theta} \sin {\theta} ~d\theta ^2 + (\cos^2 {\theta}-\sin^2{\theta} ) r ~dr~ d\theta + \sin{\theta}\cos{\theta}~ dr^2##

Where did I make the mistake?

Now, I tried to calculate it in a different way. We know,

## x = r \cos {\theta} ; y= r\sin{\theta}; ##

Hence,

## dx = - r \sin{\theta}~ d\theta + \cos{\theta}~ dr##

& ## dy = r \cos{\theta}~ d\theta + \sin{\theta} ~dr##

Now, we have, infinitesimal area element,

## dA = dx ~dy = - r^2 \cos {\theta} \sin {\theta} ~d\theta ^2 + (\cos^2 {\theta}-\sin^2{\theta} ) r ~dr~ d\theta + \sin{\theta}\cos{\theta}~ dr^2##

Where did I make the mistake?