Lecture 20: Path independence and conservative fields
In Lecture 20, we learn about path independence and conservative fields. The fundamental theorem of calculus for line integrals tells us that if we integrate a vector field that's a gradient along a curve, we'll get the value of the potential function at the endpoints. This means that we can avoid computing line integrals by finding a function whose gradient is the given vector field. We also see an example where we use the potential function to easily find the work done by a vector field along a curve, without having to compute line integrals.
Learn about path independence and conservative fields, what they mean and how to find them.
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.
- Curves and Vector Fields Applet - A Java applet for curves and vector fields
- Problem Set - A problem set with some problems from this lecture.
- Lecture Notes - Lecture Notes from this lesson.
- Transcript of Lesson - Written transcript from this lesson.
- What is path independence for gradient fields?
- What are conservative fields, what do they look like, and how do you find them?
- How do you find the line integral around a portion of a unit circle centered at the origin?
- How can you avoid computing line integrals?
- What is the Fundamental Theorem of Calculus for line integrals?
- When is a field conservative or a path independent?
- What are properties of a gradient field?
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Staff Review
This lecture on path independence and conservative fields help sum up the ideas from the previous lecture and push into a slightly different way of looking at a similar topic. Mostly, example problems are shown in this lecture. Sometimes, fields are path independent and conservative, and other times, they are not. This is discussed in some depth along with gradient fields.
