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## Homework Statement

Express in unit vectors [tex]\hat{r}[/tex], [tex]\hat{\theta}[/tex], [tex]\hat{\phi}[/tex] in terms of [tex]\hat{x}[/tex], [tex]\hat{y}[/tex], [tex]\hat{z}[/tex] (that is derive the relevant equations). ... Also work out the inverse forulas giving [tex]\hat{x}[/tex], [tex]\hat{y}[/tex], [tex]\hat{z}[/tex] in terms of [tex]\hat{r}[/tex], [tex]\hat{\theta}[/tex], [tex]\hat{\phi}[/tex] (and [tex]\theta[/tex], [tex]\phi[/tex]).

## Homework Equations

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

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

[tex]\hat{\phi} = -sin \phi \hat{x} +cos \phi \hat{y}[/tex]

## The Attempt at a Solution

That looked like a linear system to me so I put the spherical coordinate unit vectors in a 3x1 matrix, the trig in a 3x3 matrix, and the Cartesian in a 3x1 matrix. Called the 3x3 [A] and took its inverse.

The method I used to arrive with the inverse is [A]

^{-1}=[tex]\frac{adj A}{det A}[/tex]=[tex]\frac{adj A}{|[A]|}[/tex]. The determinate of A is [tex]sin^{2}\theta sin^{2}\phi + cos^{2}\theta cos^{2}\phi + cos^{2}\theta sin^{2}\phi + sin^{2}\theta cos^{2}\phi = 1[/tex]

The solution I came up with is:

[tex]\hat{x} = sin \theta cos \phi \hat{r} -sin \theta sin \phi \hat{\theta} +cos \theta \hat{\phi}[/tex]

[tex]\hat{y} = -cos \theta cos \phi \hat{r} +cos \theta sin \phi \hat{\theta} +sin \theta \hat{\phi}[/tex]

[tex]\hat{z} = -sin \phi \hat{r} +cos \phi \hat{\theta}[/tex]

The problem I'm having is that [A][A]

^{-1}[tex]\neq[/tex]the identity matrix. Some of the symptoms are [tex]\frac{d}{d\theta}[/tex]|[A][A]

^{-1}|=0, [tex]\frac{d}{d\phi}[/tex]|[A][A]

^{-1}|[tex]\neq[/tex]0, |A

^{-1}|=1, [tex]\frac{d}{d\theta}[/tex]|[A]|=0, and [tex]\frac{d}{d\phi}[/tex]|[A]|[tex]\neq[/tex]0.

Either this operation is not allowed, not defined in this manner, or I screwed up. If you happen to know which and in what manner please say so. I have not done the first part yet figuring I would take care of what looked to be the easy part first. That may help in this second part. If you happen to have pointers on how to proceed in the first part, please say so. I'm going to go poking through my notes now to see what might work there.