In this demonstration, we draw a function graph X(r,t) = (r*cos(t), r*sin(t), f(r,t)). In the window labeled "Domain", we can choose a point (r₀, t₀) in the domain, which maps to the point X(r₀, t₀) on the surface. We then draw the r and t slice curves through this point, which appear as green and red curves respectively.

The purpose of this demo is to provide a geometric interpretation of the mixed partial derivatives of a function f(r,t). We already know what it means to differentiate twice with respect to a single coordinate. For example, the iterated partial derivative f_rr(r₀, t₀) is just the second derivative of the t slice-curve, f(r, t₀), evaluated at r = r₀. A similar observation can be made about f_tt(r₀, t₀). To attach some meaning to the terms f_rt(r₀, t₀) and f_tr(r₀, t₀) we consider a point X(r₀, t₀) and the green r-slice curve, X(r₀, t), through this point.

Suppose we are walking along the surface along the green slice curve in the positive t-direction. We look to our right and see that the slope of the surface in the positive r-direction changes as we go. That is, we are looking at f_r(r₀, t) while moving along the t-direction.

In the 2-D window, we graph f_r(r₀, t) (the cyan curve) as a function of t and indicate the slope of this curve at the point t = t₀. The magnitude of the slope is equal to the mixed partial derivative f_rt(r₀, t₀).

To get the other mixed partial derivative, we can look at how f_t(r, t₀) changes as we walk along the green slice curve X(r, t₀) in the positive r-direction. In the 2-D window, we also graph f(r, t₀) (the magenta curve) as a function of r and show the slope of the curve at the point r = r₀. The slope of this curve is equal to the mixed partial derivatve f_tr(r₀, t₀). With the given function, notice that the two slopes are identical at any point except the origin.