Green's Theorem: Evaluate & Sketch R

In summary, Green's Theorem is a mathematical formula that relates the line integral of a two-dimensional vector field over a closed curve to the double integral of the curl of the field over the region enclosed by the curve. It can be used to evaluate line integrals by converting them into double integrals, and the curl plays a significant role in providing a measure of the net circulation of the vector field around the curve. Green's Theorem is only valid for closed curves and can be applied to any smooth, closed curve. It is also a special case of Stokes' Theorem, which is a generalization to higher dimensions.
  • #1
geft
148
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Question: Evaluate using Green's Theorem and sketch R.

The question (excluding the sketch) and the attempted solution are on the attached image. I may have gotten the solution, but the numbers seem funny. Where did I go wrong?

9jCnX.jpg
 
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  • #2
I don't see anything humorous in those numbers :-p

Your work is correct.
 
  • #3
Well, it's just that I seldom get something like sinh 1 or cosh 1. Anyway, thanks! :biggrin:
 

FAQ: Green's Theorem: Evaluate & Sketch R

What is Green's Theorem?

Green's Theorem is a mathematical formula that relates the line integral of a two-dimensional vector field over a closed curve to the double integral of the curl of the field over the region enclosed by the curve.

How is Green's Theorem used to evaluate integrals?

Green's Theorem can be used to evaluate line integrals over a closed curve by converting it into a double integral over the region enclosed by the curve. This makes it easier to evaluate complex integrals using known techniques for double integrals.

What is the significance of the curl in Green's Theorem?

The curl represents the rotational behavior of a vector field. In Green's Theorem, it is integrated over the region enclosed by the curve, providing a measure of the net circulation of the vector field around the curve.

Can Green's Theorem be used to evaluate integrals over any type of curve?

Green's Theorem is only valid for closed curves, meaning that the starting and ending points of the curve must be the same. It can be used to evaluate integrals over any type of smooth, closed curve, such as circles, ellipses, and polygons.

How is Green's Theorem related to Stokes' Theorem?

Green's Theorem is a special case of Stokes' Theorem, which is a generalization of Green's Theorem to higher dimensions. They both relate line integrals and surface integrals, but Green's Theorem is specifically for two-dimensional vector fields while Stokes' Theorem applies to three-dimensional vector fields.

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