Lecture 29: Applications and proof of the Divergence theorem

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Taught by OCW
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Lesson Summary:

In this lesson, we continue exploring the Divergence theorem and its applications. The theorem gives us a way to calculate the flux of a vector field for a closed surface. If we have a closed surface S bounding some region D and a vector field defined in space, we can compute the flux of the vector field from the surface. By reducing this to a calculation of the triple integral of the divergence of F inside, we can understand the physical interpretation of the divergence theorem, which tells us that the divergence of a vector field corresponds to the amount of flux generated per unit volume. The lesson also includes a proof of the Divergence theorem for vertically simple regions.

Lesson Description:

A continuation of the previous lesson; applications and proof of the Divergence Theorem.

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.

Additional Resources:
Questions answered by this video:
  • What is the Divergence Theorem?
  • What is are some applications of the Divergence Theorem?
  • What are some applications of the Divergence Theorem?
  • What does it mean for a region to be vertically simple?
  • What is the diffusion equation?
  • Staff Review

    • Currently 4.0/5 Stars.
    Because the Divergence Theorem was just introduced and squeezed into the last few minutes of the last lecture, it is continued in this lesson with applications and a proof of the theorem. The explanation leads to the diffusion equation, which is a partial differential equation. The physical interpretation of the theorem and equation are explained also.