Ridiculous Question (Derivatives)

[tshafer]

Homework Statement

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.

 

[mathfeel]

Homework Statement

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.

partial derivative? Any time you want, I think.

 

[tshafer]

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

 

[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,

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

 

[Redbelly98]

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.

 

[Phrak]

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.

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

 

[Redbelly98]

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

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

 

[Phrak]

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

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.

 

[Redbelly98]

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.