A Physics Student’s Guide to Line Integrals

Image: Rhett Allain

There are two kinds of line integrals — those you see in your math course and then there are real line integrals that we use in physics. OK, I’m just joking. But really, let’s do some line integrals.

Definition of Work

Let me start off with the most common example of a line integral in physics — the work. Work is defined as:

The work is important as it is part of the Work-Energy Principle:

But let’s just focus on the work part. This is a path integral of some force F over some path from a to b with dr being a short displacement along the path. Maybe I should give a quick refresher for dot products. Assume that I can write F and dr as the following vectors.

Then the dot product (also called the scalar product) for these two vectors is just the sum of the product of their components.

There is an alternative calculation for the dot product. If you know the angle between the two vectors, then you get:

So in a sense, the work due to some force is the force in the direction of the path added up along that path. Let’s do some examples. We can…