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Q. 20

Expert-verified

Calculus

Found in: Page 1119

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Give a formula for a normal vector to the surface S determined by y = g(x,z), where g(x,z) is a function with continuous partial derivatives.

$Therefore the required normal vector is = - g_{x} i + j - g_{z} k .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given

g(x, z) is a function with continuous partial derivatives.

Step 2. Find rx and rz

$B y a b o v e s t e p s, t h e s u r f a c e S i s p a r a m e t r i z e d b y a s f o l l o w s r (x, z) = <g (x, z), x, z> T h e n \begin{aligned} r_{x} = \frac{\partial}{\partial y} r (x, z) \\ = \frac{\partial}{\partial y} <g (x, z), x, z> \\ = <\frac{\partial g}{\partial y}, 1, 0> \\ = <g_{y}, 1, 0> \end{aligned} a n d \begin{aligned} r_{z} = \frac{\partial}{\partial z} r (y, z) \\ = \frac{\partial}{\partial z} <g (x, z), x, z> \\ = <\frac{\partial g}{\partial z}, 0, 1> \\ = <g_{z}, 0, 1> \end{aligned}$

Step 3. Find n=rz×rx.

$\begin{aligned} n = r_{z} \times r_{x} \\ = <0, g_{z}, 1> \times <1, g_{x}, 0> \\ = |\begin{array}{ccc} i & j & k \\ 0 & g_{z} & 1 \\ 1 & g_{x} & 0 \end{array}| \\ = [g_{z} \cdot 0 - 1 \cdot g_{x}] i - [0 \cdot 0 - 1 \cdot 1] j + [0 \cdot g_{x} - g_{z} \cdot 1] k \\ = - g_{x} i + j - g_{z} k . \end{aligned} Therefore the required normal vector is = - g_{x} i + j - g_{z} k .$

Most popular questions for Math Textbooks

Find the flux of the given vector field through a permeable membrane described by surface S.

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