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Q 12.

Expert-verified

Calculus

Found in: Page 1131

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Give an example of a field with positive divergence at (1, 0, π).

$An example of field with positive divergence at (1, 0, π) is, F (x, y, z) = x^{2} i + 3 \sin (y) j + \cos (z) k$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information is

Point is (1,0,π)

Step 2. Example of field

$If F (x, y, z) = F_{1} (x, y, z) i + F_{2} (x, y, z) j + F_{3} (x, y, z) k is a vector field, then the divergence of the vector field is defined as follows : \nabla . F = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z} Note that value of \nabla . F is scalar . Consider the following vector field : F (x, y, z) = x^{2} i + 3 \sin (y) j + \cos (z) k The divergence of this vector field will be, \nabla . F = \frac{\partial (x^{2})}{\partial x} + \frac{\partial (3 \sin (y))}{\partial y} + \frac{\partial (\cos z)}{\partial z} = 2 x + 3 cosy - sinz At the origin, that is, at (x, y, z) = (1, 0, π), the divergence of this vector field will be, \nabla . F = 2.1 + 3 \cos 0 - \sin π = 2 + 3 - 0 = 5, which is positive . Therefore, an example of field with positive divergence at (1, 0, π) is, F (x, y, z) = x^{2} i + 3 \sin (y) j + \cos (z) k$

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