derive-it.com

Objective

The objective of this post is to investigate the validity of (6) in Reference 2. In this reference, (6) is del is written in terms of spherical coordinates and spherical unit vectors.

Partial Derivatives Relating Cartesian Coordinates to Spherical Coordinates

Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. In the this 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 Cartesian Coordinate System

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.

Step 1 – Definition of Del

The first step is to recall the Definition of Del in terms of Cartesian Coordinates. This definition is repeated here, for reference.

$\vec{\nabla} := \vec{e}_x\frac{\partial}{\partial x} + \vec{e}_y\frac{\partial}{dy} + \vec{e}_z\frac{d}{dz}$

Step 2 – Multiplication by $F(x,y,z)$

Recall that $F(x,y,z) \equiv f(x) g(y) h(z)$. Multiply both sides of the definition of Del by $F(x,y,z)$ from the right:

$\vec{\nabla}F(x,y,z) = \vec{e}_x\frac{\partial F(x,y,z)}{\partial x} + \vec{e}_y \frac{\partial F(x,y,z) }{\partial y}+ \vec{e}_z\frac{\partial F(x,y,z) }{\partial z}$

Step 3 – Substitution of Known Quantities

Now for some math! Substitute known quantities into the previous equation, and see what appears [2].

More substitution…
Group terms:

Simplify:

Conclusion

Therefore, if $F(x,y,z) \equiv f(x) g(y) h(z)$:

$\boxed{ \vec{\nabla}F(x,y,z) = \vec{e}_r \frac{\partial F(x,y,z) }{\partial r} + \vec{e}_\theta \frac{1}{r} \frac{\partial F(x,y,z) }{\partial \theta} + \vec{e}_\phi \frac{1}{r\sin\theta} \frac{\partial F(x,y,z) }{\partial \phi} }$.

This confirms (6) in Reference

References

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

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

[3] Defining Del in Terms of Cartesian Coordinates