Curl of a vector field:

Question

So I have a force given by

$$ F = (x^2 + y^2 + z^2)^n(xi+yj+zk)$$

I was wondering how we handle this for the curl...an explanation of the setup would be excellent.

Answer 1

In components, if $F$ is a vector, $$ (\operatorname{curl} F)_i = \varepsilon_{ijk} \partial_j F_k. $$ Using this, it is easy to show that if $g$ is a scalar-valued function, $$ (\operatorname{curl} (gF))_i = \varepsilon_{ijk}( \operatorname{grad} g )_j F_k + g \varepsilon_{ijk} \partial_j F_k = ((\operatorname{grad} g) \times F + g \operatorname{curl} F )_i, $$ so you just have to compute this for $g=(x^2+y^2+z^2)^n$, $F = (x,y,z)$. It is easy to see that $\operatorname{curl}F=0$, since each component doesn't depend on the others (do the calculation to see what I mean). And further, I showed here that $\operatorname{grad} g $ is parallel to $F$, so the cross product term is also zero. Hence the whole lot is zero.

Answer 2

I would suggest using this reference. It has a nice concise description of how to compute the curl of a v-field.

http://mathworld.wolfram.com/Curl.html

I specifically recommend equation (5). This matrix allows you to compute curl very easily.