I am facing some problem about derivatives in spherical coordinates(adsbygoogle = window.adsbygoogle || []).push({});

in spherical coordinates:

x=r sinθ cos[itex]\phi[/itex]

y=r sinθ sin[itex]\phi[/itex]

z=r cosθ

and

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

θ=tan[itex]^{-1}[/itex][itex]\frac{\sqrt{x^{2}+y{2}}}{z}[/itex]

[itex]\phi[/itex]=tan[itex]^{-1}[/itex][itex]\frac{y}{x}[/itex]

[itex]\frac{\partial x}{\partial r}[/itex]=sinθ cos[itex]\phi[/itex]

then [itex]\frac{\partial r}{\partial x}[/itex]=[itex]\frac{1}{sinθ cos \phi }[/itex]

but if i calculate directly from r:

[itex]\frac{\partial r}{\partial x} = \frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}[/itex]

substitute:

=[itex]\frac{r sinθ cos \phi }{r}[/itex]

= sinθ cos[itex]\phi[/itex]

Why do the results are different? what i did wrong?

From https://www.physicsforums.com/showthread.php?t=63886

not this case is the second case? but why the inverse still not true?

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# Partial derivative in spherical coordinates

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