Lecture 31: Stokes' theorem

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Lesson Summary:

In Lecture 31, you will learn about Stokes' Theorem and its applications. The curl of a vector field in space measures the rotation of a velocity field, and a vector field is conservative if its curl is zero. Stokes' Theorem states that the work done by a vector field along a closed curve can be replaced by a double integral of the curl of the vector field over a suitably chosen surface bounded by the curve. However, the orientations of the curve and surface need to be compatible, and there are conventions to determine their orientations. The right-hand rule can be used to determine the orientations, and the examples provided will help you understand the concept better.

Lesson Description:

Learn about Stokes' Theorem and why it is useful.

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 Stokes' Theorem?
  • How do you use the right-hand rule in three dimensions?
  • How are Stokes' Theorem and Green's Theorem similar?
  • Why is Stokes' Theorem true?
  • Staff Review

    • Currently 4.0/5 Stars.
    This lesson starts with a quick review of 3-dimensional curl from last lecture, and moves into Stokes’ Theorem. The physical, geometrical interpretation of the regions and the normal vectors to the region are shown. A flat region, a cone, and a cylinder are used as example objects to understand concepts. We find out the Green’s Theorem was simply a special case of Stokes’ Theorem.