Show that ∇· (∇ x F) = 0 for any vector field

Question

To solve this question, how do I define any vector field $F$, in order to solve it? I called $F = (ax,by,cz)$, in which case already $\nabla\times F = 0$. How would i go about proving this? Many thanks!

Answer 1

It's better if you define $F$ in terms of smooth functions in each coordinate. For instance I would write $F = (F_x, F_y, F_z) = F_x\hat{i} + F_y \hat{j} + F_z \hat{k}$ and compute each quantity one at a time. First you'll compute the curl:

$$ \nabla \times F \;\; =\;\; \left | \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ F_x & F_y & F_z \\ \end{array} \right | \;\; =\;\; G_x\hat{i} + G_y \hat{j} + G_z\hat{k} $$

where the functions $G_x, G_y, G_z$ are obtained by computing the determinant. Then you will want to compute

$$ \nabla\cdot (\nabla\times F) \;\; =\;\; \frac{\partial G_x}{\partial x} + \frac{\partial G_y}{\partial y} + \frac{\partial G_z}{\partial z}. $$

You should find that the last equation yields zero.