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Chapter 14: Q0 (page 1131)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information

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

02

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.

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Most popular questions from this chapter

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?

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

F(x,y,z)=yi+xj+k, where S is the paraboloid with equation role="math" localid="1650299986090" z=4x2+y2+1that lies above the annulus determined by 1≤x2+y2≤4 in the XY-plane.

Write two different normal vectors for a smooth surface S given by (x, y, g(x, y)) at the point(x0,y0,g(x0,y0)).

Why do surface integrals of multivariate functions not include an n term, whereas surface integrals of vector fields do include this term?

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.
See all solutions

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