- #1
batboio
- 19
- 0
Hello! My problem is that I want to find [tex](\frac{\partial}{{\partial}x}, \frac{\partial}{{\partial}y}, \frac{\partial}{{\partial}z})[/tex] in spherical coordinates. The way I am thinking to do this is:
[tex]\frac{\partial}{{\partial}x}{\psi}(r(x,y,z),{\theta}(x,y,z),{\phi}(x,y,z))=\frac{{\partial}{\psi}}{{\partial}r}\frac{{\partial}r}{{\partial}x}+\frac{{\partial}{\psi}}{{\partial}{\theta}}\frac{{\partial}{\theta}}{{\partial}x}+\frac{{\partial}{\psi}}{{\partial}{\phi}}\frac{{\partial}{\phi}}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}\frac{{\partial}{\psi}}{{\partial}r}+\frac{1}{\frac{{\partial}x}{{\partial}{\theta}}}\frac{{\partial}{\psi}}{{\partial}{\theta}}+\frac{1}{\frac{{\partial}x}{{\partial}{\phi}}}\frac{{\partial}{\psi}}{{\partial}{\phi}}[/tex],
where [tex]\psi[/tex] is an arbitrary function. And so:
[tex]\frac{\partial}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}\frac{{\partial}}{{\partial}r}+\frac{1}{\frac{{\partial}x}{{\partial}{\theta}}}\frac{{\partial}}{{\partial}{\theta}}+\frac{1}{\frac{{\partial}x}{{\partial}{\phi}}}\frac{{\partial}}{{\partial}{\phi}}[/tex]
The problem is that I think I've seen somewhere that [tex]\frac{{\partial}r}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}[/tex] BUT I am really far from being sure that this is true.
And if this is true so far then what happens with [tex]\frac{{\partial}{\phi}}{{\partial}z}[/tex] when [tex]\frac{{\partial}z}{{\partial}{\phi}}=0[/tex]
[tex]\frac{\partial}{{\partial}x}{\psi}(r(x,y,z),{\theta}(x,y,z),{\phi}(x,y,z))=\frac{{\partial}{\psi}}{{\partial}r}\frac{{\partial}r}{{\partial}x}+\frac{{\partial}{\psi}}{{\partial}{\theta}}\frac{{\partial}{\theta}}{{\partial}x}+\frac{{\partial}{\psi}}{{\partial}{\phi}}\frac{{\partial}{\phi}}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}\frac{{\partial}{\psi}}{{\partial}r}+\frac{1}{\frac{{\partial}x}{{\partial}{\theta}}}\frac{{\partial}{\psi}}{{\partial}{\theta}}+\frac{1}{\frac{{\partial}x}{{\partial}{\phi}}}\frac{{\partial}{\psi}}{{\partial}{\phi}}[/tex],
where [tex]\psi[/tex] is an arbitrary function. And so:
[tex]\frac{\partial}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}\frac{{\partial}}{{\partial}r}+\frac{1}{\frac{{\partial}x}{{\partial}{\theta}}}\frac{{\partial}}{{\partial}{\theta}}+\frac{1}{\frac{{\partial}x}{{\partial}{\phi}}}\frac{{\partial}}{{\partial}{\phi}}[/tex]
The problem is that I think I've seen somewhere that [tex]\frac{{\partial}r}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}[/tex] BUT I am really far from being sure that this is true.
And if this is true so far then what happens with [tex]\frac{{\partial}{\phi}}{{\partial}z}[/tex] when [tex]\frac{{\partial}z}{{\partial}{\phi}}=0[/tex]