Hello there I have trouble with the goal of following question about divergence and curl. More specifically I do know what these terms mean and how the maths related to them work, but the goal of the question is rather veague to me. The question goes as follows.
Let $\vec{K}(\vec{r})$ be a constant vector field and $g(\vec{r})$ a scalar field. Let $\vec{Z} = g(\vec{r})\vec{K}(\vec{r})$. What conditions must g meet in order for the divergence of $\vec{Z}$ to be zero. Secondly same question but now the divergence need not to be zero but the curl of $\vec{Z}$ needs to be zero.
Now I approached this problem using the product rules for divergence and curl and notices that the curl and divergence of the constant vector field is of course zero. This lead me to the conclusion that in both cases the gradient of the scalar field g must be zero, but i'm quite uncertain if this is the right anser and if there maybe exist a condition that's less strict.
Any help would be greatly appreciated!
Best Answer
Define an Operator $\vec{D}$ as divergence so for $\vec{D}(\vec{Z})$ we need $$0=\vec{D}(\vec{Z})=\vec{g}\vec{D}(\vec{K})+\vec{K}\vec{D}(\vec{g})$$ Equivalently, $$\vec{g}\vec{D}(\vec{K})=-\vec{K}\vec{D}(\vec{g})$$ Unless otherwise specified I am not taking case where operators applied or vectors might take zero values as those cases can be treated trivially!!!
Hence, whenever $\vec{g}\neq{0}$ and. $\vec{K}\neq{0}$ we can take, $$\boxed{\frac{\vec{D}(\vec{K}).\vec{K}}{||K||^{2}}=-\frac{\vec{g}.\vec{D}(\vec{g})}{||g||^2}}$$ Hope it helps!!! For curl do it similarly!!!
$\text{Remark:}$This is a general version to be zero , so whenever operator applied is zero you can think like it's a constant vector field!!!