Here is a small screenshot of something I'm reading:(adsbygoogle = window.adsbygoogle || []).push({});

http://img262.imageshack.us/img262/3585/sphericalcoords.png [Broken]

The first six equations are ok. (I don't think anyone actually needs the figure right? It's just general spherical coordinates). φ is the angle in the x-y plane.

I get how the seventh equation is found, because:

[tex]r=\sqrt{x^{2}+y^{2}+z^{2}}[/tex]

so they simply take the derivative of that and then they have the derivative in terms of x y and z and then they simply change it to spherical coordinates using one of the first three equations in the image.

But I cannot figure out how they get the eighth and ninth equations. (last 2).

Because:

[tex]\theta=cos^{-1}\left(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)[/tex]

and the derivative of that with respect to x (partial derivative) is:

[tex]\frac{\partial\theta}{\partial x}=\left(\frac{xz}{\left(x^{2}+y^{2}+z^{2}\right)^{3/2}\sqrt{1-\frac{z^{2}}{x^{2}+y^{2}+z^{2}}}}\right)[/tex]

and similarly for y and z.

OR:

[tex]\frac{\partial\theta}{\partial x}=\left(\frac{xz}{\sqrt{\left(x^{2}+y^{2}+z^{2}\right)^{3}-z^{2}\left(x^{2}+y^{2}+z^{2}\right)^{2}}}\right)[/tex]

I don't think I'm doing this right, then I'd still have to solve for [itex]\frac{\partial\phi}{\partial x}[/itex] and with y and z too.

The manner in which that document I'm reading goes over this topic leads me to believe that the method of doing this should be much simpler.

Can anyone offer some advice please?

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# Little help with spherical coordinates

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