Lecture 30: Line integrals in space, curl, exactness and potentials
In this lesson on line integrals in space, students learn about computing line integrals and work in 3D, testing whether a vector field is a gradient field, and the concepts of curl, exactness, and potentials. The criterion for determining whether a vector field is a gradient field involves checking three conditions, which is different from the criterion for the two-variable case. The lesson includes examples and shows how to find the potential when there is one.
A lesson about line integrals, curl, exactness, and potentials in 3-dimensional space.
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.
- How do you find line integrals, work, curl, exactness, and potentials in 3-dimensional space?
- How do you test if a field is a gradient field in 3-dimensional space?
- What is a test for gradient fields in three dimensions?
- For which a and b is axy dx + (x^2 + z^3)dy + (byz^2 - 4z^3)dz exact?
- How do you find curl F in 3D?
- What does curl mean geometrically for three dimensional objects?
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Staff Review
The first few minutes of this lesson finish off some topics from last lecture. Then, line integrals, work, curl, exactness, and potentials are found for various objects in three dimensions now, instead of two. Again, this lesson builds on the analogous computations that we have seen so far as we move from two to three dimensions. Many examples are shown to make the ideas more concrete.
