Question about Cross Product and Derivative

[bwest121]

Hi everyone,

Given a vector-valued function $\vec{A}$, how do I show that:

$$\vec{\nabla} \times \left(\frac{\partial \vec{A}}{\partial x}\right) = \frac{\partial}{\partial x}(\vec{\nabla} \times \vec{A})$$

In other words, are the cross product and derivative commutative w/ each other? I have an intuition that this is true, but I would like to know a good way to show this.

Thank you very much.

 

[pasmith]

If in doubt, write it out component by component and check.

Assuming the components of $\mathbf{A}$ are sufficiently smooth, partial derivative operators will commute.