# Ridiculous Question (Derivatives)

1. Aug 14, 2009

### tshafer

1. The problem statement, all variables and given/known data

Under what conditions may I change temporal and spatial derivatives? I cannot remember for the life of me.

EG:
$$\frac{\partial }{\partial t}\left( \nabla \cdot \mathbf{A} \right) = \nabla \cdot \left( \frac{\partial \mathbf{A}}{\partial t}\right)$$

Thanks.

2. Aug 14, 2009

### mathfeel

partial derivative? Any time you want, I think.

3. Aug 14, 2009

### tshafer

That's what I thought, too, but I was unsure. Trying to de-rust, here, heh.

4. Aug 14, 2009

### Phrak

Go back to the definition of the divergence, so that your equation contains terms like

$$\frac {\partial } {\partial t} \frac {\partial } {\partial x} \mathbf{A}_x \right) \ .$$

While x and t are independent variables,

$$\frac {\partial } {\partial t} \frac {\partial } {\partial x} = \frac {\partial } {\partial x} \frac {\partial } {\partial t} \ .$$

5. Aug 14, 2009

### Redbelly98

Staff Emeritus
This works for gradient and curl too, by the way. It's basically a matter of

∂²A/∂x∂t = ∂²A/∂t∂x

and similarly for y and z.

6. Aug 14, 2009

### Phrak

Could you use this to show that electric and magnetic fields don't move in space?

7. Aug 14, 2009

### Redbelly98

Staff Emeritus
Hmmm, not sure I understand your question ... E-M fields can and do move through space.

8. Aug 14, 2009

### Phrak

The value of the fields change over time for any given coordinate, but the fields are not considered moving objects. The field is attached to a coordinate. A propagating electromagnetic wave is like a sound wave. The wave moves, the molecules stay (nominally) in place. I may be confused about the math, though.

9. Aug 14, 2009

### Redbelly98

Staff Emeritus
Okay. Well, if you're going to say by definition that the fields don't move through space, not sure why you'd need a proof of that.

Or even if you're not saying this is by definition, I don't see how my earlier statement in post #5 could be used to prove it. That statement is a basic consequence of multivariable calculus.