Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.3 More Derivatives 2.3.3 Hessian Determinant 2.3.4 Taylor Series 2.3.6 Equality of Mixed Partials 2.4 Integration Search

Equality of Mixed Partials

Text

If f(x, y) is a twice continuously differentiable function then fxy(x, y) = fyx(x, y) for all (x, y). If f(x, y) is not twice continuously differentiable, then this is not necessarily true.

Demos

 Geometric Intrepretation of Mixed Partials 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.

Exercises

• 1. For the function in the demonstration, calculate the mixed partial derivative fuv(0,v) as v approaches 0 and then fuv(u,0) as v approaches 0. What can we say about the the value of fuv(u,v) at the origin?
• 2. Now use the demo and enter the function . Again, look at the mixed partial derivative fuv(u,v) along the u- and v-axes. What is the limit of f(u,v) as we approach the origin? Notice that this surface is just a rotation of the graph in exercise 1.
• 3. Show that if f(u,v) is a polynomial, then fuv(u,v) = fvu(u,v).