16.13: Coordinate Systems

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Cartesian coordinates

Screen Shot 2019-11-01 at 12.39.33 PM.png

2 dimensions: area element: \(dA=dx.dy\).
3 dimensions: volume element: \(dV=dx.dy.dz\).

Polar coordinates

Screen Shot 2019-11-01 at 12.41.39 PM.png

\(x=r\cos\theta\)
\(y=r\sin\theta\)
\(r^2=x^2+y^2\)
\(tan \theta=y/x\)
\(dA=r.dr.d\theta\)

Spherical coordinates

Screen Shot 2019-11-01 at 12.42.10 PM.png

\(x=r\sin\theta\cos\phi\)
\(y=r\sin\theta\sin\phi\)
\(z=r\cos\theta\)
\(r^2=x^2+y^2+z^2\)
\(\theta=\cos^{-1}\left[\frac{z}{\sqrt{x^2+y^2+z^2}} \right]\)
\(\phi=\tan^{-1}\left(\frac{y}{x}\right)\)
\(dV=r^2\sin{\theta}dr.d\phi.d\theta\)

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