# building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta$

$$\begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin \theta \cos \phi & \cos \theta \cos \phi & -\sin\phi\\ \sin \theta \sin \phi & \cos \theta \sin \phi & \cos\phi\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} A_r\\ A_\theta\\ A_\phi \end{bmatrix}$$

Also how show that $$\begin{bmatrix} \hat i\\ \hat j\\ \hat k \end{bmatrix} = \begin{bmatrix} \sin \theta \cos \phi & \cos \theta \cos \phi & -\sin\phi\\ \sin \theta \sin \phi & \cos \theta \sin \phi & \cos\phi\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \hat e_r\\ \hat e_\theta\\ \hat e_\phi \end{bmatrix}$$ How to change $(a,b,c)$ into spherical polar coordinates and $(r ,\theta, \phi)$ into cartesian coordinates using this matrix? Thank you!!

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Find a transformation from Cartesian to cylindrical coordinates. Find a transformation from cylindrical to spherical coordinates. Compose them. –  Potato Jun 24 '12 at 0:36