http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Thu, 31 Jul 2014 12:43:04 -0500http://calc3.askbot.com/question/211/partial-derivatives/When showing that $\vec F$ is conservative, we take the partial derivatives, just wondering if that is the same as $\nabla f $? I know from there we find the function $f$ to make them equal? But I thought that the gradient was the partial derivatives?Thu, 31 Jul 2014 11:05:02 -0500http://calc3.askbot.com/question/211/partial-derivatives/http://calc3.askbot.com/question/211/partial-derivatives/?answer=214#post-id-214OHHHH! Because we're taking with respect to y first and then respect to x second, instead of partial x then partial y! Thanks!Thu, 31 Jul 2014 11:26:17 -0500http://calc3.askbot.com/question/211/partial-derivatives/?answer=214#post-id-214http://calc3.askbot.com/question/211/partial-derivatives/?answer=213#post-id-213It's not the same as $\nabla f$ because you have $\vec F = (P,Q)$ (pretend this is vector notation which isn't showing up for me) and to find if $\vec F$ is conservative, we are taking $\frac{\partial P}{\partial y}$ and $\frac{\partial Q}{\partial x}$. Basically, use the same set up as $\nabla f$ and swap the $\partial x$ with the $\partial y$.Thu, 31 Jul 2014 11:24:46 -0500http://calc3.askbot.com/question/211/partial-derivatives/?answer=213#post-id-213http://calc3.askbot.com/question/211/partial-derivatives/?answer=215#post-id-215I 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 $f$).Thu, 31 Jul 2014 12:43:04 -0500http://calc3.askbot.com/question/211/partial-derivatives/?answer=215#post-id-215