Lecture 19: Vector fields and line integrals in the plane
In this lesson, we learn about vector fields and line integrals in the plane. A vector field is a vector that varies with x and y, while a line integral is the work done by a force along a trajectory. We learn how to draw vector fields and how to compute the work done by a force using a line integral. The physical significance of the work done by a force is the amount of energy provided by the force to perform a motion, and the total work done along a trajectory is given by an integral of the force along that trajectory.
Learn about vector fields and line integrals, what they mean, what they look like, 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
- Lecture Notes - Lecture Notes from this lesson.
- Transcript of Lesson - Written transcript from this lesson.
- Problem Set - A problem set with some problems from this lecture.
- What are vector fields and how do you find one?
- What is a line integral and how do you find them?
- What does a vector field look like and what are some examples of vector fields?
- What does a velocity field for uniform rotation look like?
- How do you find the work done by a vector field?
- How do you find the work done by a force F = -yi + xj if the particle is moving along x = t, y = t^2 from t = 0 to 1?
- Why does the value of a line integral depend on the path taken?
- How do you simplify a line integral in 2 variables down to one variable?
- What is a geometric approach to line integrals?
-
Staff Review
Vector fields and line integrals, two very important concepts in this course, are discussed in this lecture. They are explained theoretically, and examples are also shown. This is a great lecture on some difficult conceptual topics. This is a very complete and descriptive lesson.
