In this demonstration, we draw a function graph X(u,v) = (u, v, f(u,v)). In the window labeled "Domain", we can choose a point (u0, v0) in the domain, which maps to the point X(u0, v0) on the surface. We then draw the u and v slice curves through this point, which appear as red and green curves respectively.
The purpose of this demo is to provide a geometric interpretation of the mixed partial derivatives of a function f(u,v). We already know what it means to differentiate twice with respect to a single coordinate. For example, the iterated partial derivative fuu(u0, v0) is just the second derivative of the v slice-curve, f(u, v0), evaluated at u = u0. A similar observation can be made about fvv(u0, v0). To attach some meaning to the terms fuv(u0, v0) and fvu(u0, v0) we consider a point X(u0, v0) and the red u-slice curve, X(u0, v), through this point.
Suppose we are walking along the surface along the red slice curve in the positive v-direction. We look to our right and see that the slope of the surface in the positive u-direction changes as we go. That is, we are looking at fu(u0, v) while moving along the v-direction.
This can be seen in the demonstration by using the tapedeck to animate the variable labeled animateV. In the 2-D window, we graph fu(u0, v) (the magenta curve) as a function of v and indicate the slope of this curve at the point v = v0. The magnitude of the slope is equal to the mixed partial derivative fuv(u0, v0).
To get the other mixed partial derivative, we can look at how fv(u,v0) changes as we walk along the green slice curve X(u,v0) in the positive u-direction. In the 2-D window, we also graph f(u,v0) (the cyan curve) as a function of u and show the slope of the curve at the point u = u0. The slope of this curve is equal to the mixed partial derivatve fvu(u0, v0). With the given function, notice that the two slopes are identical at any point except the origin.