Component test for conservative fields

Question

I have a question concerning the component test for conservative fields. So the component test tells us that the vector field is conservative if the following three conditions are met. $$ P_y = N_z \\ M_z = P_x \\ N_x = M_y $$ I am having trouble memorizing, because I can't seem to find why these values are used, I created a little tree diagram and noticed that it looks as thought they are taking one of the partials and setting it equal to to one of the partials in the other respective categories. However it seems as though there a few other ways to do this, I am wondering if theses values they use are just the standard way of doing however you really can use one of the other ways you could find, or if there is some deeper reasoning why these values are used.

Answer 1

The reason behind it is that line integrals around closed curves do not contribute$$ \DeclareMathOperator{curl}{curl} 0 = \int\limits_{\partial A} F \cdot dx = \int\limits_A\curl F \cdot dA $$ or in other words (Stokes' theorem) the field is curl free.

$$ 0 = \curl F = \nabla \times F = (\partial_1, \partial_2, \partial_3) \times F = e_i \epsilon_{ijk} \partial_j F_k $$ e.g. \begin{align} 0 &= (\curl F)_1 = \epsilon_{123}\partial_2 F_3 + \epsilon_{132} \partial_3 F_2 = \partial_2 F_3 - \partial_3 F_2 \\ 0 &= (\curl F)_2 = \partial_3 F_1 - \partial_3 F_2 \\ 0 &= (\curl F)_3 = \partial_1 F_2 - \partial_2 F_1 \\ \end{align} where $\partial_i = \partial / \partial x_i$.

This seems to give your case if $F = (M, N, P)$ and $(x_1, x_2, x_3) = (x,y,z)$.

Answer 2

This might help you remember until you’ve learned about curl.

If the vector field is conservative, then $P=\frac{\partial\phi}{\partial x}$, $M=\frac{\partial\phi}{\partial y}$ and $N=\frac{\partial\phi}{\partial z}$ for some function $\phi$. The component test says that the mixed second partial derivatives of $\phi$ must be equal.