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Chapter 11: Q39E (page 624)

Use a Computer to Graph the Function using various domains and viewpoints.

\(f\left( {x,y} \right) = {e^{\frac{{ - \left( {{x^2} + {y^2}} \right)}}{3}}}\left( {\sin {x^2} + \cos {y^2}} \right)\)

Short Answer

Expert verified

Plotted a Contour Map for the Function.

Step by step solution

01

Sketch The Curve From A Number Of Viewpoints.

02

The Curve Is Sketched Below From A Number Of Viewpoints.

03

The Curve Is Drawn Here From Different Viewpoints.

04

Plotting Contour Lines

A Contour Map for the Function looks like this:

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Most popular questions from this chapter

Find and sketch the domain of the function.\(f(x,y) = \frac{{\sqrt {y - {x^2}} }}{{(1 - {x^2})}}\).

Determine the set of points at which the function is continuous.

\(f(x,y) = \left\{ \begin{aligned}{l}\frac{{xy}}{{{x^2} + xy + {y^2}}}, if(x,y) \ne (0,0)\\0, if(x,y) = (0,0)\end{aligned} \right.\)

Find the value of \(\frac{{\partial w}}{{\partial r}}\) and\(\frac{{\partial w}}{{\partial \theta }}\),using the chain rule if \(w = xy + yz + zx,x = r\cos \theta ,y = r\sin \theta \) and \(z = r\theta \)when \(r = 2\) and \(\theta = \frac{\pi }{2}.\)

Determine the values of the derivatives \({g_r}(1,2)\)and \({g_r}(1,2).\)The functions are \(g(r,s) = f\left( {2r - s,{s^2} - 4r} \right),x = x(r,s)\) and \(y = y(r,s).\)

(a) Find the values of \(\frac{{\partial z}}{{\partial r}}\) and \(\frac{{\partial z}}{{\partial \theta }}\) if \(z = f(x,y)\), where \(x = r\cos \theta \) and \(y = r\sin \theta \).

(b) Show the equation\({\left( {\frac{{\partial z}}{{\partial x}}} \right)^2} + {\left( {\frac{{\partial z}}{{\partial y}}} \right)^2} = {\left( {\frac{{\partial z}}{{\partial r}}} \right)^2} + \frac{1}{{{r^2}}}{\left( {\frac{{\partial z}}{{\partial \theta }}} \right)^2}\).
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