Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - Review - Concept Check - Page 1188: 11


(a) See the explanation below. (b) See the explanation below. (c) See the explanation below.

Work Step by Step

a) A parametric surface let us say $S$ is known to be a surface in $R^3$ with two parameters $(m,n)$ represented as: $r(m,n)=p(m,n)i+q(m,n)j+r(m,n)k$ Here, $p,q,r$ are scalar functions. The grid curves of a parametric surface $S$ are the curves that develop with keeping one of the parameters$(m,n)$ as constant. b) Area of a surface parametric surface $S$ can be defined as: $\iint_D|r_m \times n_v|$ dA where $(m,n) \in D$ c) Area of the surface $S$ of equation $z=g(x,y)$ can be calculated as: $\iint_D\sqrt {1+(g_x)^2+(g_y)^2}dA=\iint_D\sqrt {1+(\dfrac{\partial z}{\partial x})^2+(\dfrac{\partial z}{\partial y})^2}dA$
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