32. Line Integrals in Vector Fields

Recall that in physics, work is a force times a distance. Specifically, $\large W = \vec{F}\cdot \vec{d}$, where the dot product captures the component of the work that points in the direction of motion:

work dot product

^{^{Image from: https://www.youtube.com/watch?v=rkqCiu6pQCY}}

Now imagine an object is travelling on a curve in the midst of a force vector field. The force varies - both in magnitude and direction - with position. To compute the work done in moving the object through the field, we want to break the path up into small pieces where we can reasonably approximate the force as a constant and then add up the work done over each piece of the curve:

lineInt Vector Field line Int Path

^{^{Images from: https://math24.net/line-integrals-vector-fields.html and https://profoundphysics.com/dot-product-in-physics-what-is-the-physical-meaning-of-it/}}

If that sounds and looks like integration, it's because it is integration! This is a line integral over a curve in a vector field. Here's a gif illustrating the idea:

line Int Vector Field Gif

^{^{Image from: https://en.wikipedia.org/wiki/Line_integral}}

Line Integral in a Vector Field (Work Integral)

To find the work done in moving an object along a curve in a vector field, we compute a line integral as follows:

$$\large \int_C\vec{F}\cdot d\vec{s} = \int_a^b \vec{F}(x(t),y(t),z(t))\cdot \vec{r'}(t)dt$$

Just like our scalar line integrals from before, we end up with a single integral via the parameterization of the curve. Instead of a magnitude $\large |\vec{r'}(t)|$, we have a dot product with $\large \vec{r'}(t)$.

Important notes:

Since we are working with $\large \vec{r'}$ instead of $\large |\vec{r'}|$, the direction for the curve's parameterization now matters! Things like clockwise, counterclockwise, from point $\large A$ to $\large B$, etc. didn't matter with scalar line integrals. Now they matter up to a sign difference.
If the starting and ending point of the curve are the same (like for a full circle), we have a closed line integral, and we use the notation $\large \displaystyle \oint_C\vec{F}(x,y,z)\cdot d\vec{s}$. This is sometimes called the circulation of $\large \vec{F}$ around the curve $\large C$.
If $\large \vec{F} = \left< P(x,y),Q(x,y)\right>$, it is common to use the following notation, which takes the dot product as part of the notation: $\large \int_C\vec{F}\cdot d\vec{s} = \int_C Pdx+Qdy$. We will use both notations in class.

Steps to find a line integral in a vector field

The steps to compute a work integral are:

Find the parameterization $\large \vec{r}(t)$ of the curve $C$. See https://github.com/drnickmines/MinesCalc3/wiki/6.-Vector-Valued-Functions to review the curves we may encounter in problems.
Find the associated parameter domain $\large a \le t \le b$.
Get the derivative $\large \vec{r'}(t)$.
Set up the integral, using the parameter domain for the endpoints, and replacing $\large x$, $\large y$, and $\large z$ in the integrand with their values along the curve from $\large \vec{r}(t)$.
Integrate.

This video shows an example:

Tomorrow's class:

Practice work integrals

Section Outcomes:

I can:

Recognize and use notations for line integrals and closed line integrals.
Interpret line integrals over vector fields and work.
Set up and evaluate a work integral (a line integral of a function along a curve in a vector field)