Question

# \text { 72. Let } f(x, y)=\left\{\begin{array}{ll} x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}}, & \text { if }(x, y) \neq 0, \\ 0, & \text { if }(x, y)=0 . \end{array}\right. \text {

a. Show that } \frac{\partial f}{\partial y}(x, 0)=x \text { for all } x, \text { and } \frac{\partial f}{\partial x}(0, y)=-y \text { for all } y . \text { b. Show that } \frac{\partial^{2} f}{\partial y^{2} x}(0,0) \neq \frac{\partial^{2} f}{\partial x \partial y}(0,0) \text { . } The graph of f is shown on page 800. The three-dimensional Laplace equation \frac{\partial^{2} f}{d x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}=0 is satisfied by steady-state temperature distributions T = f(x, y, z)in space, by gravitational potentials, and by electrostatic potentials. The two-dimensional Laplace equation \frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0 obtained by dropping the ð^f /ðz^2 term from the previous equation,describes potentials and steady-state temperature distributions in a plane (see the accompanying figure). The plane (a) may be treated as a thin slice of the solid (b) perpendicular to the z-axis.

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7

Fig: 8

Fig: 9

Fig: 10