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: 12


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

Work Step by Step

a) The surface integral of a scalar function $f(x,y,z)$ over the surface $S$ can be defined as: $\iint_Sf(x,y,z)dS$ b) 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. Area of a surface parametric surface $S$ can be defined as: $\iint_Sf(x,y,z)dS=\iint_Df(r(m,n))|r_m \times n_v|$ dA where $(m,n) \in D$ c) Area of the surface $S$ of equation $z=g(x,y)$ can be calculated as: $\iint_Sf(x,y,z)dS=\iint_Sf(x,y,g(x,y))\sqrt {1+(g_x)^2+(g_y)^2}dA=\iint_Sf(x,y,g(x,y))\sqrt {1+(\dfrac{\partial z}{\partial x})^2+(\dfrac{\partial z}{\partial y})^2}dA$ d) Mass of the thin sheet has the shape of a surface $S$ can be expressed as: $m=\iint_S \rho(x,y,z) dS$ Center of mass of the sheet can be defined as:$(\bar{x},\bar{y},\bar{z})$ Here, $\bar{x}=\frac{1}{m}\iint_Sx \rho(x,y,z) dS$; $\bar{y}=\frac{1}{m}\iint_Sy \rho(x,y,z) dS$; $\bar{z}=\frac{1}{m}\iint_Sz \rho(x,y,z) dS$
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