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jonathanpun
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I am facing some problem about derivatives in spherical coordinates
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?
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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