Why should the equality of mixed partials be "intuitively obvious"? I am reading Ted Shifrin's excellent book Multivariable Mathematics. It claims that the equality of mixed partials is "an intuitively obvious result, but the proof is quite subtle". However, I guess I must be thinking in the wrong way, because I do not see the intuition behind this result. This is how I think about it:
Let $f:\mathbb{R}^2 \to \mathbb{R}$. I think of $f_x$ as a "field of slopes" in the $x$-direction. If we analyze the movement in the $y$ direction in this field of slopes, we get $f_{xy}$. Now $f_y$ is a "field of slopes" in the $y$-direction. If we analyze movement in the $x$ direction here, we get $f_{yx}$. 
It's unclear to me why movement in the $x$-direction in the "field of $y$-slopes" should be the same as movement in the $y$-direction in the "field of $x$-slopes".

If you write the difference quotient for a small change $\Delta x$ in $x$ and then the difference quotient for that when you change $y$ by $\Delta y$ the result is the symmetric expression
$$\frac{
f(x + \Delta x, y + \Delta y)
-f(x + \Delta x, y )
-f(   x, y + \Delta y)
+f(x,y)
}
{\Delta x \Delta y} .
$$
