# Lecture 19: Vector fields and line integrals in the plane

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Lesson Summary:

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.

Lesson Description:

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.

• 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

• Currently 4.0/5 Stars.
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.