91Ó°ÊÓ

Show that the function satisfies the wave equation \(\partial^{2} z / \partial t^{2}=c^{2}\left(\partial^{2} z / \partial x^{2}\right)\). \(z=\cos (4 x+4 c t)\)

Consider the function \(w=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta .\) Prove each of the following. (a) \(\frac{\partial w}{\partial x}=\frac{\partial w}{\partial r} \cos \theta-\frac{\partial w}{\partial \theta} \frac{\sin \theta}{r}\) \(\frac{\partial w}{\partial y}=\frac{\partial w}{\partial r} \sin \theta+\frac{\partial w}{\partial \theta} \frac{\cos \theta}{r}\) (b) \(\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}=\left(\frac{\partial w}{\partial r}\right)^{2}+\left(\frac{1}{r^{2}}\right)\left(\frac{\partial w}{\partial \theta}\right)^{2}\)

Consider the function defined by $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0) \end{array}\right. $$ (a) Find \(f_{x}(x, y)\) and \(f_{y}(x, y)\) for \((x, y) \neq(0,0)\) (b) Use the definition of partial derivatives to find \(f_{x}(0,0)\) and \(f_{y}(0,0)\) $$ \left[\text { Hint }: f_{x}(0,0)=\lim _{\Delta x \rightarrow 0} \frac{f(\Delta x, 0)-f(0,0)}{\Delta x} .\right] $$ (c) Use the definition of partial derivatives to find \(f_{x y}(0,0)\) and \(f_{y x}(0,0)\). (d) Using Theorem 11.3 and the result of part \((\mathrm{c}),\) what can be said about \(f_{x y}\) or \(f_{y x}\) ?