How can I convert unit vectors from cartesian to spherical polar coordinates?

[Beer-monster]

Does anyone know where I can find what the unit vector in the y-direction would be expressed in spherical polar co-ordinates (assuming that the polar axis is along z axis)?

I can find polar unit vectors expressed in cartesians and but not the other way round?

Anyone have a clue?

 

[Dr Transport]

try inverting the matrix. you have the unit vectors in spherical coordinates expressed in terms of cartesian, invert the solution and you'll have it.

 

[Physics Monkey]

I'm sure its on the internet somewhere. You can also do it yourself, here's how:

The change of basis from $$\hat{x},\, \hat{y},\, \hat{z}$$ to $$\hat{r},\, \hat{\theta},\, \hat{\phi}$$ is orthogonal. If you know the matrix that relates one to the other, you would normally invert that matrix to find the inverse relationship. Since the matrix in your case is orthogonal, the transpose is the inverse. So just look up what $$\hat{r},\, \hat{\theta},\, \hat{\phi}$$ are in terms of $$\hat{x},\, \hat{y},\, \hat{z}$$, and then invert the matrix and the second row tells you $$\hat{y}$$ in terms of $$\hat{r},\, \hat{\theta},\, \hat{\phi}$$.