Calculus (3rd Edition)

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

Chapter 17 - Line and Surface Integrals - Chapter Review Exercises - Page 970: 14

Answer

$div\left( {{{\bf{e}}_r}} \right) = \frac{2}{{\sqrt {{x^2} + {y^2} + {z^2}} }}$ $curl\left( {\bf{F}} \right) = {\bf{0}}$

Work Step by Step

We have ${{\bf{e}}_r} = {r^{ - 1}}\left( {x,y,z} \right)$, where $r = \sqrt {{x^2} + {y^2} + {z^2}} $. So, ${{\bf{e}}_r} = \left( {{F_1},{F_2},{F_3}} \right) = \left( {\frac{x}{{\sqrt {{x^2} + {y^2} + {z^2}} }},\frac{y}{{\sqrt {{x^2} + {y^2} + {z^2}} }},\frac{z}{{\sqrt {{x^2} + {y^2} + {z^2}} }}} \right)$ 1. $div\left( {{{\bf{e}}_r}} \right) = \frac{{\partial {F_1}}}{{\partial x}} + \frac{{\partial {F_2}}}{{\partial y}} + \frac{{\partial {F_3}}}{{\partial z}}$ $div\left( {{{\bf{e}}_r}} \right) = \frac{{{y^2} + {z^2}}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}} + \frac{{{x^2} + {z^2}}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}} + \frac{{{x^2} + {y^2}}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}}$ $div\left( {{{\bf{e}}_r}} \right) = \frac{{2{x^2} + 2{y^2} + 2{z^2}}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}}$ $div\left( {{{\bf{e}}_r}} \right) = \frac{{2\left( {{x^2} + {y^2} + {z^2}} \right)}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}} = \frac{2}{{\sqrt {{x^2} + {y^2} + {z^2}} }}$ 2. $curl\left( {\bf{F}} \right) = \left| {\begin{array}{*{20}{c}} {\bf{i}}&{\bf{j}}&{\bf{k}}\\ {\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\ {\frac{x}{{\sqrt {{x^2} + {y^2} + {z^2}} }}}&{\frac{y}{{\sqrt {{x^2} + {y^2} + {z^2}} }}}&{\frac{z}{{\sqrt {{x^2} + {y^2} + {z^2}} }}} \end{array}} \right|$ $curl\left( {\bf{F}} \right) = \left( { - \frac{{yz}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}} + \frac{{yz}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}}} \right){\bf{i}}$ $ - \left( { - \frac{{xz}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}} + \frac{{xz}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}}} \right){\bf{j}}$ $ + \left( { - \frac{{xy}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}} + \frac{{xy}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{3/2}}}}} \right){\bf{k}}$ $curl\left( {\bf{F}} \right) = {\bf{0}}$
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