### Objective of this Post

The objective of this post is to form (4) in this reference.

### Partial Derivatives Relating Cartesian Coordinates to Spherical Coordinates

Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. In the previous post, the following three equations were written.

$\frac{\partial F(x,y,z)}{\partial x} = \cos\phi \sin\theta \frac{\partial F}{\partial r} + \frac{\cos \phi \cos\theta}{r} \frac{\partial F}{\partial \theta} – \frac{\sin\phi}{r \sin\theta}\frac{\partial F}{\partial \phi}$.

$\frac{\partial F(x,y,z)}{\partial y} = \sin\phi \sin\theta \frac{\partial F}{\partial r} + \frac{\sin\phi \cos\theta}{r} \frac{\partial F}{\partial \theta} + \frac{\cos \phi}{r \sin\theta} \frac{\partial F}{\partial \phi}$.

$\frac{\partial F(x,y,z)}{\partial z} = \cos\theta \frac{\partial F}{\partial r} – \frac{\sin\theta}{r} \frac{\partial F}{\partial \theta}$.

These three equations correspond to (2) in this reference.

### Unit Vectors for a Spherical Coordinate System

Next, recall from this post about unit vectors that three unit vectors for a right-handed spherical coordinate system are

$\vec{e}_r = \sin \theta \cos\phi \vec{e}_x + \sin \theta \sin\phi \vec{e}_y + \cos \theta \vec{e}_z$

$\vec{e}_{\theta} = \cos\theta \cos\phi \vec{e}_x + \cos\theta \sin\phi \vec{e}_y -\sin\theta \vec{e}_z$

$\vec{e}_{\phi} = -\sin\phi \vec{e}_x + \cos\phi \vec{e}_{y}$.

These three equations correspond to (3) in this reference. In this coordinate system, a point is represented by $(r, \theta, \phi)$.

### Equations with Unit Vectors

Reference  suggests that the following three equations are valid.

$\vec{e}_x \stackrel{\text{?}}{=} \sin\theta \cos\phi \vec{e}_r + \cos\theta \cos\phi \vec{e}_\theta – \sin\phi \vec{e}_\phi$

$\vec{e}_y \stackrel{\text{?}}{=} \sin\theta\sin\phi \vec{e}_r + \cos\theta\sin\phi \vec{e}_\theta + \cos\phi \vec{e}_\phi$

$\vec{e}_z \stackrel{\text{?}}{=} \cos\theta \vec{e}_r – \sin\theta \vec{e}_\theta$

These three equations are now checked, using known information

#### Checking the Equation for $\vec{e}_x$ ####  Checking the Equation for $\vec{e}_z$ #### Conclusion

Therefore,

$\vec{e}_x = \sin\theta \cos\phi \vec{e}_r + \cos\theta \cos\phi \vec{e}_\theta – \sin\phi \vec{e}_\phi$

$\vec{e}_y = \sin\theta\sin\phi \vec{e}_r + \cos\theta\sin\phi \vec{e}_\theta + \cos\phi \vec{e}_\phi$

$\vec{e}_z = \cos\theta \vec{e}_r – \sin\theta \vec{e}_\theta$

These three equations correspond to (4) in Reference 2.

### References

 How a Multivariable Function Changes with Respect to the Cartesian Coordinates

 http://www.thphys.nuim.ie/Notes/MP469/Laplace.pdf