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## Main Question or Discussion Point

This is driving me crazy, I just can't see how to do it. I want to express the cartesian unit vectors [tex]\hat{x}[/tex], [tex]\hat{y}[/tex] and [tex]\hat{z}[/tex] in terms of the spherical unit vectors [tex]\hat{r}[/tex], [tex]\hat{\theta}[/tex] and [tex]\hat{\phi}[/tex]. I have tried to do something similar in polar coordinates (just to make it a bit simpler for myself) but that didn't really help alot. I have figured out how to express the polar unit vectors in terms of cartesian ones:

[tex]\hat{\theta} = -sin(\theta)\hat{x} + cos(\theta) \hat{y} [/tex]

[tex]\hat{r} = cos(\theta) \hat{x} + sin(\theta) \hat{y} [/tex]

... and I think I can do that for spherical coordinates too. But I can't see how to do it the other way around (polar to cartesian). If you could just help me do it for polar coords I think I will be able to adapt it to spherical.

Thanks!

[tex]\hat{\theta} = -sin(\theta)\hat{x} + cos(\theta) \hat{y} [/tex]

[tex]\hat{r} = cos(\theta) \hat{x} + sin(\theta) \hat{y} [/tex]

... and I think I can do that for spherical coordinates too. But I can't see how to do it the other way around (polar to cartesian). If you could just help me do it for polar coords I think I will be able to adapt it to spherical.

Thanks!