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 | 1 | initial version | posted 2014-07-31 12:43:04 -0600  |
I would just like to add that the intuitive reasoning for finding $\frac{\partial P}{\partial y}$ and $\frac{\partial Q}{\partial x}$ checking if they are equal is to show that $f$ is "sufficiently nice," as the book says. The reason that this condition shows whether $f$ exists or not is because of Clairaut’s Theorem. It is Theorem 14.6.2 in the book and it states:
If the mixed partial derivatives are continuous, they are equal.
In other words, if the mixed partials derivatives are not equal, then $f$ does not exist. Note that since $\vec F = \nabla f$, $\langle P, Q \rangle $ = $ \langle f_x, f_y \rangle $. This means that $\frac{\partial P}{\partial y}$ is $f_{xy}$ and $\frac{\partial Q}{\partial x}$ is $f_{yx}$ (i.e. the mixed partials of $f$).
