Calculus (3rd Edition)

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

Chapter 17 - Line and Surface Integrals - 17.5 Surface Integrals of Vector Fields - Exercises - Page 968: 31

Answer

We prove the following: $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} {\bf{F}}\cdot{\rm{d}}{\bf{S}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( { - {F_1}\dfrac{{\partial g}}{{\partial x}} - {F_2}\dfrac{{\partial g}}{{\partial y}} + {F_3}} \right){\rm{d}}x{\rm{d}}y$

Work Step by Step

A graph $z = g\left( {x,y} \right)$ can be parametrized by $G\left( {x,y} \right) = \left( {x,y,g\left( {x,y} \right)} \right)$ From here we obtain the tangent vectors: ${{\bf{T}}_x} = \left( {1,0,{g_x}} \right)$, ${\ \ \ \ \ }$ ${{\bf{T}}_y} = \left( {0,1,{g_y}} \right)$ And the normal vector: ${\bf{N}}\left( {x,y} \right) = {{\bf{T}}_x} \times {{\bf{T}}_y} = \left| {\begin{array}{*{20}{c}} {\bf{i}}&{\bf{j}}&{\bf{k}}\\ 1&0&{{g_x}}\\ 0&1&{{g_y}} \end{array}} \right| = - {g_x}{\bf{i}} - {g_y}{\bf{j}} + {\bf{k}}$ Write ${\bf{F}}\left( {x,y} \right) = {F_1}{\bf{i}} + {F_2}{\bf{j}} + {F_3}{\bf{k}}$. By Eq. (3): $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} {\bf{F}}\cdot{\rm{d}}{\bf{S}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} {\bf{F}}\left( {G\left( {x,y} \right)} \right)\cdot{\bf{N}}\left( {x,y} \right){\rm{d}}x{\rm{d}}y$ $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} {\bf{F}}\cdot{\rm{d}}{\bf{S}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( {{F_1}{\bf{i}} + {F_2}{\bf{j}} + {F_3}{\bf{k}}} \right)\cdot\left( { - {g_x}{\bf{i}} - {g_y}{\bf{j}} + {\bf{k}}} \right){\rm{d}}x{\rm{d}}y$ $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} {\bf{F}}\cdot{\rm{d}}{\bf{S}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( { - {F_1}{g_x} - {F_2}{g_y} + {F_3}} \right){\rm{d}}x{\rm{d}}y$ Hence, $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} {\bf{F}}\cdot{\rm{d}}{\bf{S}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( { - {F_1}\dfrac{{\partial g}}{{\partial x}} - {F_2}\dfrac{{\partial g}}{{\partial y}} + {F_3}} \right){\rm{d}}x{\rm{d}}y$
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