geometric meaning of $\partial_{xy}f(x,y)?$

Question

I know that $\partial_x f(x,y)$ gives us the slope of the tangent line at the point (x,y) in the x direction. And $\partial_{xx}f(x,y)$ gives us the rate of change of that same slope. The same happens with $\partial_yf(x,y)$ and $\partial_{yy}f(x,y)$ but this time in the x direction. But what is the geometric meaning of $\partial_{xy}f(x,y)?$

Answer 1

$\partial_{xy} f = \partial_x(\partial_y f)$ measures how the slope in the direction of $y$ changes while moving in the direction of $x$, i.e the rate of the slope in the direction $y$ if you move perpendicularly to $y$.

Answer 2

By considering $\partial_x f(x,y)$, you are thinking about the slope of a tangent line which is tangent to the graph of $f(x,y)$ at the point $(x,y)$ and the line is parallel with the positive $x$ direction.

By further considering $\partial_y( \partial_xf(x,y))$, you are asking how the slope of this line changes if you vary $y$ slightly.

By doing this local analysis, you recover a small rectangular area around the point $(x,y)$ which is swept out by the turning tangent lines. This is a hand-wavy argument as to why you might expect that $\partial_{xy} f = \partial_{yx} f$.