0
$\begingroup$

Can anyone help me to understand what is the difference between a sphere and a symmetrical sphere? I have to deal with spherical coordinates and I can find two different formulations, one without including the angle and the one where the angle is included. See example below:

$$\nabla\cdot A = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial A}{\partial r}\right) + \frac{1}{r^2 sin^2\phi}\frac{\partial}{\partial \phi }\left(\frac{\partial A}{\partial \phi}\right)+\frac{1}{r^2 sin \theta}\frac{\partial}{\partial \theta }\left(sin\theta \frac{\partial A}{\partial \theta}\right) \tag{1}$$ and: $$\nabla\cdot A = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial A}{\partial r}\right)\tag{2}$$

It appears that the angles are 0, 180 or 360 in the second formulation. Does that make it symmetrical? Isn't a sphere already symmetrical? Can anyone help me to understand this?

1 Answers 1

1

While a sphere has spherical symmetry, your vector field $A$ may not. Something has spherical symmetry if, when rotated in any direction by any amount, it always looks the same. For instance, imagine a vector field where the vectors always points radially outward and their magnitude decays as $1/r^2$ or, more generally, only depends on the distance from the origin, not on the direction. As an example of a vector field that isn't spherically symmetrical imagine almost anything else. Something where the vectors' magnitudes change with $\theta$ and $\phi$ or where they deviate from pointing radially as a function of $\theta$ and $\phi.$

Your second formula applies only to vector fields that have spherical symmetry.

Also, your formulas are written down wrong. You forgot to include the components of $A$. The $A$ in the first term needs to have a subscript $A_r,$ in the second term $A_\theta,$ etc.