More Examples of Green’s Theorem

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Examples of Green’s Theorem

Continuing the examples from Friday and “Circulation Form of Green’s Theorem” from section 5.4 in the textbook.

A Multi-Part Curve

Find the integral of F(x,y) = ⟨ 3x², xy ⟩ counterclockwise around the semicircle bounded by the curve y = √(1-x²) and the x axis.

This is mainly another example of Green’s Theorem avoiding the need to evaluate multiple line integrals by evaluating one relatively simple double integral instead. But unlike Friday’s similar example, this time the double integral doesn’t just simplify to some multiple of the region’s area.

Start by turning the double integral from Green’s Theorem into an iterated integral based on this specific region and field. Evaluating that integral uses techniques we learned for integrals over general regions:

Finding Area

Green’s Theorem seems more useful for turning a time-consuming line integral into a simpler double integral, but you can also go the other way. The book has an example that turns an area integral over an ellipse (hard to evaluate) into a simple line integral; here’s another example that illustrates the process, although the area problem is easier to solve without Green’s Theorem:

Use Green’s Theorem to turn a double integral for the area of the circle x² + y² = 1 into a line integral. Show that the line integral does evaluate to the area of the circle, namely π.

The trick to Green’s Theorem as a way of finding areas is to treat the integrand in an area integral (1) as a difference of two things that can be interpreted as the derivatives of the Q and P components of a vector field, respectively. It does not at all matter what two things you take the difference of, or what vector field you arrange to have those things be the derivatives of, except maybe in influencing how simple or ugly the resulting line integral is. You’ll get the right area in any case. So we decided to express 1 as 5/4 minus 1/4, and then use the simplest vector field with those numbers as the derivatives of its components:

Once we had a vector field, we set up its line integral around the circle in the usual way, and proceeded to integrate. The so-called “half-angle” trig identities were helpful for simplifying the integrand, and they set things up for a u-substitution that finished the problem:

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Another way of looking at Green’s Theorem: the flux form.

Please read “Flux Form of Green’s Theorem” in section 5.4 of the textbook through Example 5.4.5 (water across a rectangle).

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