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Chapter 11: Problem 27
Describe the domain and range of the function. $$ g(x, y)=\frac{1}{x y} $$
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In Exercises \(39-42,\) find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(r\) and \(\boldsymbol{\theta}\) before differentiating. \(w=x^{2}-2 x y+y^{2}, x=r+\theta, \quad y=r-\theta\)
Find \(d w / d t\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(t\) before differentiating. \(w=x y+x z+y z, \quad x=t-1, \quad y=t^{2}-1, \quad z=t\)
Find \(d w / d t\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(t\) before differentiating. \(w=x y z, \quad x=t^{2}, \quad y=2 t, \quad z=e^{-t}\)
Find a function \(f\) such that \(\nabla f=e^{x} \cos y \mathbf{i}-e^{x} \sin y \mathbf{j}+z \mathbf{k}\).
The temperature at the point \((x, y)\) on a metal plate is \(T=\frac{x}{x^{2}+y^{2}}\). Find the direction of greatest increase in heat from the point (3,4) .
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