Green’s Theorem and Polygons

Two nights ago, I assigned my multivariable calculus class a problem from our textbook (Anton, Section 15.4, Problem 38). Even though I’ve stopped using Anton for my non-AP Calculus class, I have found that Anton does a good job treating the multivariable calculus material. I think the problems are quite nice.

Anyway, the problem was in the section on Green’s theorem, and stated:

(a) Let C be the line segment from a point (a,b) to a point (c,d). Show that:

\int_C -y\text{ }dx+x\text{ }dy=ad-bc

(b) Use the result in part (a) to show that the area A of a triangle with successive vertices (x_1,y_1),\text{ }(x_2,y_2), and (x_3,y_3) going counterclockwise is:

A=\frac{1}{2}[(x_1y_2-x_2y_1)+(x_2y_3-x_3y_2)+(x_3y_1-x_1y_3)]

(c) Find a formula for the area of a polygon with successive vertices (x_1,y_1),\text{ }(x_2,y_2),...,(x_n,y_n) going counter-clockwise.

Today we started talking about our solutions. We all were fine with part (a). But part (b) was the exciting part, because of the variation in approaches. We had five different ways we were able to get the area of the triangle.

  • There was the expected way, which one student got using part (a). This was the way the book intended the students to solve the problem — and I checked using the solution manual to confirm this. What was awesome was that even though we as a class understood the algebra behind this answer, a student still asked for a conceptualgeometric understanding of what the heck that line integral really meant. I knew the answer, but I left it as an exercise for the class to think about. So we’re not done with this problem.

  • There was a way where a student made a drawing of an arbitrary triangle and then used three line integrals of the form \int_C y\text{ }dx to solve it. In essence, this student was taking the area of a large trapezoid (calculated by using a line integral) and subtracting out the area of two smaller trapezoids (again calculated by using line integrals). Another student astutely pointed out that even though we had an arbitrary triangle, the way we set up the integral was based on the way we drew the triangle — and to be general, we’d have to draw all possibilities. You don’t need to understand precisely what this means — because I know I”m not being clear. The point is, we had a short discussion about what would need to be done to actually have a rigorous proof.

  • There was a way where a student translated the triangle so that the three vertices weren’t (x_1,y_1),\text{ }(x_2,y_2),\text{ }(x_3,y_3) anymore… but instead (0,0),\text{ }(x_2-x_1,y_2-y_1),\text{ }(x_3-x_1,y_3-y_1). Then he used something we proved earlier, that the area of a triangle defined by the origin and two points would involve a simple determinant (divided by 2). And when he did this, he got the right answer.

  • Another two students drew the triangle, put it in a rectangle, and then calculated the area of the triangle by breaking up the rectangle into pieces and subtracting out all parts of the rectangle that weren’t in the triangle. A simple geometric method.

  • My solution involved noticing that \frac{1}{2}(ad-bc) is the area of a triangle with vertices (0,0),\text{ }(a,b),\text{ }(c,d). And so I constructed a solution where a triangle is the sum of the areas of two larger triangles, but then with subtracting out another triangle.

The point of this isn’t to share with you the solutions themselves, or how to solve the problem. The point is to say: I really liked this problem because it generated so many different approaches. We ended up spending pretty much the whole period discussing it and it’s varied forms (when I had only planned 10 or 15 minutes for it). I liked how these kids made a connection between a previous problem we had solved (#28) and used that to undergird their conceptual understanding. I loved how these approaches gave rise to some awesome questions — including “what the heck is the physical interpretation of that line integral in part (a)?” In fact, at the end of class, we were drawing on paper, tearing areas apart, trying to make sense of that line integral. All because a student suggested that’s what we do. (Again, I have made sense of it… but I wanted the kids to go through the sense making process themselves… their weekend work is to understanding the meaning of this line integral.)

I don’t know the real point of posting this — except that I wanted to archive this unexpectedly rich problem. Because it’s not that it is algebraically intensive (though some approaches did get algebraically intensive). Rather, it’s because it is conceptually deep.

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5 thoughts on “Green’s Theorem and Polygons

  1. I have always LOVED those days where I did not get through half of what I planned due to a rich detour.
    Thanks for sharing

  2. Dear Sam,
    The point of this post is you took time to share with us a lovely conversation in your class, allowing students to construct meaning for themselves.
    Thank you,
    Fawn

  3. Hi Sam, I really love this problem. I used it in a unit I am writing on “beautiful problems.”

    In our class a student used the idea that \frac{1}{2}\cdot\vec{r}\times{d}\vec{r} = d\vec{A}_{\triangle} and an extension to the Riemann sum idea: \sum_i d\vec{A_i}_{\triangle} = \Delta \vec{A} to solve the problem beautifully. We hadn’t introduced the idea of an integral that produced a vector, but his work inspired me to flesh out his idea some more. It resulted in some of the most beautiful work we have done in class all year.

      • Yes! I can send you a sample of the beautiful problems I started with.

        I want to re-imagine the entire multivariable course as a seminar where students present beautiful multivariable problems and their solutions. A necessary component of their seminar will be to expand the working definition of “beautiful” each week. By the end of the year, we will compile our work on from a blog as a digital text. I expect it to be comprised of their work and solutions. They will edit and coordinate it as the final project. I am pretty excited for next year already.

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