91Ó°ÊÓ

In Exercises \(27-30,\) describe the level surfaces of the given functions of three variables. $$ f(x, y, z)=x^{2}+y^{2}+z^{2} $$

Describe the level surfaces of the given functions of three variables. $$ f(x, y, z)=z-x^{2}+y^{2} $$

Form a function \(z=f(x, y)\) such that \(f_{x}\) and \(f_{y}\) match those given. $$ f_{x}=x+y, \quad f_{y}=x+y $$

A function \(w=F(x, y, z),\) a vector \(\vec{v}\) and a point \(P\) are given. (a) Find \(\nabla F(x, y, z)\). (b) Find \(D_{\vec{u}} F\) at \(P,\) where \(\vec{u}\) is the unit vector in the direction of \(\vec{v}\). $$ F(x, y, z)=\frac{2}{x^{2}+y^{2}+z^{2}}, \vec{v}=\langle 1,1,-2\rangle, P=(1,1,1) $$

Describe the level surfaces of the given functions of three variables. $$ f(x, y, z)=\frac{x^{2}+y^{2}}{z} $$

In Exercises \(27-30\), find \(\frac{d z}{d t},\) or \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t},\) using the supplied information. $$ \begin{array}{l} \frac{\partial z}{\partial x}=-4, \quad \frac{\partial z}{\partial y}=9 \\ \frac{\partial x}{\partial s}=5, \quad \frac{\partial x}{\partial t}=7, \quad \frac{\partial y}{\partial s}=-2, \quad \frac{\partial y}{\partial t}=6 \end{array} $$

Form a function \(z=f(x, y)\) such that \(f_{x}\) and \(f_{y}\) match those given. $$ f_{x}=6 x y-4 y^{2}, \quad f_{y}=3 x^{2}-8 x y+2 $$

In Exercises \(27-30\), find \(\frac{d z}{d t},\) or \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t},\) using the supplied information. $$ \begin{array}{l} \frac{\partial z}{\partial x}=2, \quad \frac{\partial z}{\partial y}=1 \\ \frac{\partial x}{\partial s}=-2, \quad \frac{\partial x}{\partial t}=3, \quad \frac{\partial y}{\partial s}=2, \quad \frac{\partial y}{\partial t}=-1 \end{array} $$

Describe the level surfaces of the given functions of three variables. $$ f(x, y, z)=\frac{z}{x-y} $$

Form a function \(z=f(x, y)\) such that \(f_{x}\) and \(f_{y}\) match those given. $$ f_{x}=\frac{2 x}{x^{2}+y^{2}}, \quad f_{y}=\frac{2 y}{x^{2}+y^{2}} $$