Lecture 31: Stokes' theorem
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.
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.
- Problem Set - A problem set with some problems from this lecture.
- Transcript of Lesson - Written transcript from this lesson.
- Lecture Notes - Lecture Notes from this lesson.
- 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?
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Staff Review
This lesson starts with a quick review of 3-dimensional curl from last lecture, and moves into Stokesâ 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 Greenâs Theorem was simply a special case of Stokesâ Theorem.
