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Chapter 12: Problem 1
Suppose \(z=f(x, y),\) where \(x\) and \(y\) are functions of \(t .\) How many dependent, intermediate, and independent variables are there?
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Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+y^{2}+4 z^{2}+2 x=0$$
Recall that Cartesian and polar coordinates are related through the transformation equations $$\left\\{\begin{array}{l} x=r \cos \theta \\ y=r \sin \theta \end{array} \quad \text { or } \quad\left\\{\begin{array}{l} r^{2}=x^{2}+y^{2} \\ \tan \theta=y / x \end{array}\right.\right.$$ a. Evaluate the partial derivatives \(x_{r}, y_{r}, x_{\theta},\) and \(y_{\theta}\) b. Evaluate the partial derivatives \(r_{x}, r_{y}, \theta_{x},\) and \(\theta_{y}\) c. For a function \(z=f(x, y),\) find \(z_{r}\) and \(z_{\theta},\) where \(x\) and \(y\) are expressed in terms of \(r\) and \(\theta\) d. For a function \(z=g(r, \theta),\) find \(z_{x}\) and \(z_{y},\) where \(r\) and \(\theta\) are expressed in terms of \(x\) and \(y\) e. Show that \(\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}\)
Show that the following two functions have two local maxima but no other extreme points (therefore, there is no saddle or basin between the mountains). a. \(f(x, y)=-\left(x^{2}-1\right)^{2}-\left(x^{2}-e^{y}\right)^{2}\) b. \(f(x, y)=4 x^{2} e^{y}-2 x^{4}-e^{4 y}\)
Find an equation of the plane passing through (0,-2,4) that is orthogonal to the planes \(2 x+5 y-3 z=0\) and \(-x+5 y+2 z=8\)
Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. $$u(x, t)=5 \cos (2(x+c t))+3 \sin (x-c t)$$
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