Can a Vector Field in 3D and Time Have a Fourth Component in its Divergence?

[jem]

Homework Statement

I attempted to solve the problem. I would like to know if my work/thought process or even answer is correct, and if not, what I can do to fix it.
I am given:

Calculate the divergence of the vector field :
A=0.2R^(3)∅ sin^2(θ) (R hat+θ hat+ ∅ hat)

Homework Equations

[/B]
The equation I used was the divergence of a vector field in spherical coordinates:
The file is attached:

The Attempt at a Solution

The file is attached:
My final answer is 3.60.

 

[Ray Vickson]

Homework Statement

I attempted to solve the problem. I would like to know if my work/thought process or even answer is correct, and if not, what I can do to fix it.
I am given:

Calculate the divergence of the vector field :
A=0.2R^(3)∅ sin^2(θ) (R hat+θ hat+ ∅ hat)

Homework Equations

[/B]
The equation I used was the divergence of a vector field in spherical coordinates:
The file is attached:

The Attempt at a Solution

The file is attached:
My final answer is 3.60.

What does your notation mean? What is $\hat{R}$. What is the difference between $\theta$ and $\hat{\theta}$, and between $\phi$ and $\hat{\phi}$? I can guess, but why should I need to, and maybe my guess is wrong. Finally, which of the two common forms of spherical polar coordinates are you using? Some sources use $\theta$ as the polar angle (latitude) and $\phi$ as longitude, while others choose the exact opposite convention. Both appear in this Forum from time to time.

 

[quiet]

Hi. When a vector field is a function of the 3 Cartesian coordinates and time, for example in the case of an electromagnetic wave in a vacuum, can it happen that a divergence of 4 components appears, with the fourth component of the type
$$\dfrac{1}{C} \ \dfrac{\partial A}{\partial t}$$
(A symbolizes some field of the wave)