Geneseo Math 223 01 Greens Theorem 1

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The Circulation Form of Green’s Theorem

Based on “Extending the Fundamental Theorem of Calculus” and “Circulation Form of Green’s Theorem” in section 5.4 of the textbook.

Questions

Why use Green’s Theorem? It’s sometimes an easier alternative to other ways of evaluating line integrals. Or, less often, integrals over 2-dimensional regions. Green’s Theorem is also theoretically interesting because it extends a theme in calculus of evaluating integrals in terms of their boundaries.

Key Ideas

Green’s Theorem gives an equivalence between line integrals around a region and double integrals over it.

There’s a formula that describes the equivalence, and a proof.

Integral around C of P D X plus Q D Y is double integral over region of Q sub X minus P sub Y

Green’s Theorem only applies in 2 dimensions (unlike much of the rest of what we’ve seen this semester, which has natural generalizations to any number of dimensions).

There’s also a flux form of the theorem (which is coming up tomorrow).

Green’s Theorem has applications, e.g., to computing work.

Examples

Evaluate the integral of sin(x²) dx + (4x - y) dy counterclockwise around the triangle with vertices (-1,1), (1,1), and (1,3).

Graph with triangle, and integral around it of sine of X squared D X plus the quantity 4 X minus Y D Y

The “P dx + Q dy” notation in our textbook’s statement of Green’s Theorem is basically a form of “F(r(t)) • r′(t)”:

Integral around C of F of X and Y dot R prime of T become P D X plus Q D Y

Recalling the formula from Green’s Theorem, the first thing to do in order to use it is to identify what P and Q are. In this case, they’re the expressions integrated relative to dx and dy, respectively:

The parts of a line integral match P and Q from Green's Theorem

Once you identify P and Q, you take their derivatives and plug the derivatives into the formula. Notice that in this case the derivatives were so simple that the double integral turned out to just be a multiple of the area of the triangle inside the path:

Line integral around a triangle turns into a multiple of the triangle's area with Green's Theorem

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

This time, get P and Q from the equation for the vector field (this is a place where knowing where the notation comes from is helpful). Then find their derivatives and plug the derivatives into the formula, like in the example above. This time the double integral isn’t so simple that it’s just an area though:

Evaluating a line integral around a semicircle as a double integral over the semicircle with Green's Theorem

The flux form of Green’s Theorem.

Please read “Flux Form of Green’s Theorem” in section 5.4 of the textbook.