[Math] Determining whether a vector field is conservative

Question

For a vector field $\vec{F}(x,y,z) = \langle F_1(x,y,z), F_2(x,y,z), F_3(x,y,z) \rangle$ in $\mathbb{R}^3$, how can I use mixed second-order partial derivatives of each of the components to determine whether it is conservative? Which partial derivatives should I compare?

Answer 1

You should check that

$$ \frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}, \\ \frac{\partial F_1}{\partial z} = \frac{\partial F_3}{\partial x}, \\ \frac{\partial F_2}{\partial z} = \frac{\partial F_3}{\partial y}. $$

An easy way to remember this is to call your variables $x^1,x^2,x^3$ (instead of $x,y,z$). Then you need to check that

$$ \frac{\partial F_i}{\partial x^j} = \frac{\partial F_j}{\partial x^i} $$

for all $i \neq j$. This works in all dimensions.