Labware - MA35 Multivariable Calculus - Two Variable Calculus



Equality of Mixed Partials


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.



  • 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).