Applying Green's Theorem Question

Question

We've just been introduced to Green's Theorem and I've been working on some homework questions and came across one which I have now been struggling with for a couple of days. And help would be appreciated. I really want to grasp this concept.

The question:

Suppose F(x,y) = P(x,y)i + Q(x,y)j is a vector field

Suppose E is the rectangle E = { (x,y) ∈ R^2: −4 ≤ x ≤ 4, −2 ≤ y ≤ 2 }

Furthermore B1 = {(x,y)∈R^2 : (x−(−2))^2 +y^2 ≤ 1}

and

B2 ={(x,y)∈R^2 : (x−2)^2 +y^2 ≤1}.

Consider the region D={ (x,y) ∈ E : (x,y) ∈/ (Not element of) B1∪B2}.

The question asks:

(a) Formulate the extended version of Green’s Theorem for the given region D.

(b) Use Green’s Theorem to prove the extended version of Green’s Theorem which you had to formulate in part (a) of this question. In- clude one or more sketches and refer to your sketches in your proof.

I've been comfortable with using Green's Theorem on simpler problems - when the region is given or easy to determine. But I struggle with this one as I struggle to determine D. I have tried drawing it out, but still don't quite get it.

Image from book

Answer 1

Unfortunately your question is incomplete since it does not provide the vector field. I will give you the main bullet points to succeed on the Green's theorem computation:

• Draw the region;
• Orient the border;
• See which of both integrals are easier: the surface or curve integral.

The domain you require is below.