Learning Materials
Features
Discover
Chapter 12: Problem 16
Find the domain of the following functions. $$f(x, y)=\frac{12}{y^{2}-x^{2}}$$
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the x y-, x z^{-}, \text {and } y z-\text {traces, when they exist. c. Sketch a graph of the surface. $$-\frac{x^{2}}{3}+3 y^{2}-\frac{z^{2}}{12}=1$$
Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\).
A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=e^{a x} \cos a y, \text { for any real number } a$$
The density of a thin circular plate of radius 2 is given by \(\rho(x, y)=4+x y .\) The edge of the plate is described by the parametric equations \(x=2 \cos t, y=2 \sin t,\) for \(0 \leq t \leq 2 \pi\) a. Find the rate of change of the density with respect to \(t\) on the edge of the plate. b. At what point(s) on the edge of the plate is the density a maximum?
Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14 .) In two dimensions, the motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi .\) The velocity components of the fluid, \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi .\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\).
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine