Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.5 The Gradient and Directional Derivatives - Exercises - Page 801: 8

Answer

$$\nabla r =\left\langle z e^{y w}, xw z e^{y w},x e^{y w}, xy z e^{y w} \right\rangle$$

Work Step by Step

Given $$ r(x, y, z, w)=x z e^{y w}$$ Since \begin{align*} \frac{\partial r}{\partial x}&=z e^{y w}\\ \frac{\partial r}{\partial y}&=xw z e^{y w}\\ \frac{\partial r}{\partial z}&= x e^{y w}\\ \frac{\partial r}{\partial w}&= xy z e^{y w} \end{align*} Then \begin{align*} \nabla r&=\left\langle\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial r}{\partial z}, \frac{\partial r}{\partial w}\right\rangle\\ &=\left\langle z e^{y w}, xw z e^{y w},x e^{y w}, xy z e^{y w} \right\rangle \end{align*}
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