Writing Unit Vectors for a Cartesian Coordinate System in Terms of Unit Vectors for a Spherical Coordinate System

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 [2] 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 $\vec{e}_y$

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