Lecture 22: Green's theorem
In this lesson, we learned about Green's Theorem and how it can be used to compute line integrals along closed curves. By computing the double integral of the curl of a vector field over the region enclosed by the curve, we can avoid calculating the line integral directly. If the curl is zero, then the vector field is conservative, and we can use Green's Theorem to verify this property. However, if the vector field is not defined everywhere, then we cannot apply Green's Theorem, and the vector field may not be conservative, even if its curl is zero everywhere it is defined.
Learn about Green's Theorem and how it is used.
Denis Auroux. 18.02 Multivariable Calculus, Fall 2007. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (Accessed March 21, 2009). License: Creative Commons Attribution-Noncommercial-Share Alike.
- Transcript of Lesson - Written transcript from this lesson.
- Lecture Notes - Lecture Notes from this lesson.
- Problem Set - A problem set with some problems from this lecture.
- What is Green's Theorem?
- What is a way to avoid calculating line integrals?
- How do you compute the line integral along a closed curve of M dx + N dy?
- What is the line integral of ye^-x dx + 1/2x^2 - e^-x dy?
- What happens to Green's Theorem if curl F equals 0?
- How do you prove Green's Theorem?
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Staff Review
This video explains Greenâs Theorem and how it helps make line integral computations easier. This method is a very interesting and helpful way of making calculations. Of course, this only works if you have a closed curve. Some good examples are shown of how Greenâs Theorem actually works in practice. The geometry of these theorems is examined in much of this lecture to help understand what is really going on. This video also serves to justify and prove parts of Greenâs Theorem.
