Ex 105

Let
\[ f(x, y) = \begin{cases} \frac{x^3 y - x y^3}{x^2 + y^2}, & \text{if } (x, y) \ne (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases} \]

  1. Use a computer to graph \(f\).
  2. Find \(f_x(x, y)\) and \(f_y(x, y)\) when \((x, y) \ne (0, 0)\).
  3. Find \(f_x(0, 0)\) and \(f_y(0, 0)\) using Equations 2 and 3.
  4. Show that \(f_{xy}(0, 0) = -1\) and \(f_{yx}(0, 0) = 1\).
  5. Does the result of part (d) contradict Clairaut’s Theorem? Use graphs of \(f_{xy}\) and \(f_{yx}\) to illustrate your answer.