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:
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
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