Lecture 28: Divergence theorem
In Lecture 28, we learn about the Divergence Theorem and how to use it to compute the flux of a vector field for a given surface. The formula for NDS is explained in detail and we see how it applies to various surfaces, including the graph of a function. An example problem is presented in which we find the flux of a vector field through a portion of a paraboloid above the unit disk, using the Divergence Theorem and polar coordinates. This theorem is a powerful tool for computing flux integrals and is a crucial concept in vector calculus.
Learn what the Divergence Theorem is, what it means, and how to use it.
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.
- Lecture Notes - Lecture Notes from this lesson.
- Transcript of Lesson - Written transcript from this lesson.
- Problem Set - A problem set with some problems from this lecture.
- What is the Divergence Theorem in three dimensions?
- How do you find the normal vector in three dimensions?
- Why does n hat dS = +- <-fx, -fy, 1> dx dy?
- How do you find the flux F = zk through a portion of paraboloid z = x^2 + y^2 above the unit disk?
- What is the Gauss-Green Theorem?
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Staff Review
More about flux and surface integrals are covered in this lecture. Example problems are computed and explained in the video. The Divergence Theorem (or the Gauss-Green Theorem) is finally introduced and explained quickly in the final six minutes of this lecture. An example is shown and the geometric idea is explained.
