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Q0

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Calculus
Found in: Page 1131
Calculus

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Problem Zero: Read the section and make your own summary of the material.

The detailed summary of the section is as follows :-

The chapter begins with the basic definitions of Del operator, the divergence and the curl of a vector field. After the we studied the one theorem about divergence of Curl and Curl of a Gradient. The we studied the important theorem Green's Theorem. This theorem is about curl and divergence expressions. Then the section ended with some examples

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1. Given Information

We have to write a detailed summary of section 14.4. Green’s Theorem of chapter 14 Vector Analysis.

Step 12. Detailed summary of the Section

The title of this section is Green's Theorem. As from the title it is clear that in this section we studied about Green's Theorem. The section begins with some basic fundamentals of Green's Theorem.

In this section, we firstly studied about Del Operator ∇, the Divergence and the Curl of a vector field.

After that we studied one of the most important theorem, about divergence of Curl and Curl of a Gradient. Then we move to the most important theorem of the chapter Green's Theorem. This theorem is about Curl and divergence expressions. After that we solved some examples related to these topics.

Most popular questions for Math Textbooks

The current through a certain passage of the San Juan Islands in Washington State is given by


F=F1(x,y),F2(x,y)=0,1.152-0.8x2.


Consider a disk R of radius 1 mile and centered on this region. Denote the boundary of the disk by localid="1650455155947" ∂R.

  • (a) Compute localid="1650455166518" ∬R∂F2∂x-∂F1∂ydA.
  • (b) Show that ∫∂RF·nds=∫02π1.152-0.8cos2θsinθdθ. Conclude that Green's Theorem is valid for the current in this area of the San Juan Islands.
  • (c) What do the integrals from Green's Theorem tell us about this region of the San Juan Islands?

S is the portion of the saddle surface determined by z = x2 − y2 that lies above and/or below the annulus in the xy-plane determined by the circles with radii

32 and 2

and centered at the origin.

Generalize your answer to Exercise 12 to give a parametrization and a normal vector for the extension of any differentiable plane curve y = f(x) through a ≤ z ≤ b.

Compute dS for your parametrization in Exercise 9.

F(x,y,z)=z cos yzj+z sin yzk, where S is the portion of the plane with equation 2x−8y−10z=42 that lies on the positive side of the rectangle with corners 0,-π,0(0,π,0),(0,π,π),(0,−π,π) in the yz-plane.
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