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

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

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

[1] How a Multivariable Function Changes with Respect to the Cartesian Coordinates

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

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