91Ó°ÊÓ

Show that each function in Exercises \(83-90\) satisfies a Laplace equation. $$f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}$$

Show that each function in Exercises \(83-90\) satisfies a Laplace equation. $$f(x, y, z)=e^{3 x+4 y} \cos 5 z$$

Does a function \(f(x, y)\) with continuous first partial derivatives throughout an open region \(R\) have to be continuous on \(R ?\) Give reasons for your answer.

If a function \(f(x, y)\) has continuous second partial derivatives throughout an open region \(R,\) must the first-order partial derivatives of \(f\) be continuous on \(R ?\) Give reasons for your answer.

The heat equation An important partial differential equation that describes the distribution of heat in a region at time \(t\) can be represented by the one-dimensional heat equation $$\frac{\partial f}{\partial t}=\frac{\partial^{2} f}{\partial x^{2.}}$$ Show that \(u(x, t)=\sin (\alpha x) \cdot e^{-\beta t}\) satisfies the heat equation for constants \(\alpha\) and \(\beta .\) What is the relationship between \(\alpha\) and \(\beta\) for this function to be a solution?

Let \(f(x, y)=\left\\{\begin{array}{ll}{0,} & {x^{2} < y < 2 x^{2}} \\ {1,} & {\text { otherwise. }}\end{array}\right.\) Show that \(f_{x}(0,0)\) and \(f_{y}(0,0)\) exist, but \(f\) is not differentiable at \((0,0) .\)

The Korteweg-deVries equation This nonlinear differential equation, which describes wave motion on shallow water surfaces, is given by $$4 u_{t}+u_{x x x}+12 u u_{x}=0.$$ Show that \(u(x, t)=\operatorname{sech}^{2}(x-t)\) satisfies the Kortweg-deVries equation.

Show that \(T=\frac{1}{\sqrt{x^{2}+y^{2}}}\) satisfies the equation \(T_{x x}+T_{y y}=T^{3}.\)