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Chapter 14: Q. 7 (page 1131)

Explain how your answer to Exercise 6 is relevant to the discussion of Green’s Theorem.

Short Answer

Expert verified

It has been explained how the answer to Exercise 6 is relevant to the discussion of Green’s Theorem.

Step by step solution

01

Step 1. Given information.

The objective is to explain how the answer to Exercise 6 is relevant to the discussion of Green’s Theorem.

02

Step 2. Green's Theorem

Green's Theorem states that,
"Let be a region in the plane with a smooth boundary curve C oriented counterclockwise by

r(t)=⟨(x(t),y(t))⟩fora≤t≤b

If a vector field F(x,y)=F1(x,y),F2(x,y)is defined on R, then

∫C F⋅dr=∬R ∂F2∂x−∂F1∂ydA⋅"

03

Step 3. Explanation

Notice that, if you set, G(x,y)=∂F2∂x−∂F1∂y, then you can use Green's Theorem to evaluate the integral ∬R G(x,y)dA.

Hence, on setting G(x,y)=∂F2∂x−∂F1∂y, allows the use of Green’s Theorem to evaluate the integral.

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