SUNY Geneseo Department of Mathematics
Tuesday, November 29
Math 223 01
Fall 2022
Prof. Doug Baldwin
(No.)
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Based on “Extending the Fundamental Theorem of Calculus” and “Circulation Form of Green’s Theorem” in section 5.4 of the textbook.
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.
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.
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.
Evaluate the integral of sin(x2) dx + (4x - y) dy counterclockwise around the triangle with vertices (-1,1), (1,1), and (1,3).
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)”:
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:
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:
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:
The flux form of Green’s Theorem.
Please read “Flux Form of Green’s Theorem” in section 5.4 of the textbook.