TITLE: Geometric interpretation of mixed partial derivatives?
QUESTION [54 upvotes]: I'm looking for a geometric interpretation of this theorem:

My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation.
Thanks.

REPLY [7 votes]: Here's the best way I've found to think of it.
First, the background material:
Take a function $f(x,y)$.  
$f_x$ = the partial derivative measuring the rate of change (the slope) of $f$ in the $x$-direction.
$f_y$ = the partial derivative measuring the rate of change (the slope) of $f$ in the $y$-direction.
$f_{xx}$ = the rate of change of $f_x$ as one moves in the $x$-direction. Similar to the second derivative for a one-variable function, it shows if the slope is increasing (concave up) or decreasing (concave down).
Now, the mixed part:
$f_{xy}$ = the mixed partial derivative measuring the rate of change of the slope in the $x$-direction as one moves in the $y$-direction.
$f_{yx}$ = the mixed partial derivative measuring the rate of change of the slope in the $y$-direction as one moves in the $x$-direction.
The original poster's theorem says that these mixed partial derivatives are equal (given appropriate function behavior):   $f_{xy}=f_{yx}$
Here's my reference with a nice figure: http://www.math.harvard.edu/archive/21a_fall_08/exhibits/fxy/

SUPPLEMENT:
- For consistency, I've used the idea from Ted Shifrin's comment and Rahul's answer/figure.  I've included some colorized figures, which might help with the geometric interpretation for some individuals.




Perhaps, a good way to think of the equality of mixed partial derivatives is like this:  the change in slope (in the y-direction) of the changes in slope (in the x-direction) is the same as the change in slope (in the x-direction) of the changes in slope (in the y-direction).  Either way, we're effectively finding the change in slope between the high point(C) and the low point (A), with consideration to the intervening points (B and D), for the particular function we're working with.  Whether we go via the x-direction first or the y-direction first doesn't matter.