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 `(u`_{0}, v_{0}) in the domain, which maps to the point `X(u`_{0}, v_{0}) 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 `f`_{uu}(u_{0}, v_{0}) is just the second derivative of the `v` slice-curve, `f(u, v`_{0}), evaluated at `u = u`_{0}. A similar observation can be made about `f`_{vv}(u_{0}, v_{0}). To attach some meaning to the terms `f`_{uv}(u_{0}, v_{0}) and `f`_{vu}(u_{0}, v_{0}) we consider a point `X(u`_{0}, v_{0}) and the red `u`-slice curve, `X(u`_{0}, 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 `f`_{u}(u_{0}, 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 `f`_{u}(u_{0}, v) (the magenta curve) as a function of `v` and indicate the slope of this curve at the point `v = v`_{0}. The magnitude of the slope is equal to the mixed partial derivative `f`_{uv}(u_{0}, v_{0}).

To get the other mixed partial derivative, we can look at how `f`_{v}(u,v_{0}) changes as we walk along the green slice curve `X(u,v`_{0}) in the positive `u`-direction. In the 2-D window, we also graph `f(u,v`_{0}) (the cyan curve) as a function of `u` and show the slope of the curve at the point `u = u`_{0}. The slope of this curve is equal to the mixed partial derivatve `f`_{vu}(u_{0}, v_{0}). With the given function, notice that the two slopes are identical at any point except the origin.