SUNY Geneseo Department of Mathematics

Introduction to Vector Line Integrals

Friday, November 18

Math 223 01
Fall 2022
Prof. Doug Baldwin

Anything You Want to Talk About

Problem set 11, problem 3 asks you to evaluate a line integral along a semi-circular path:

Coordinate axes and path in semicircle from point on negative Y axis counterclockwise to point on positive Y

(It’s not an integral over a half-circle area, even though it’s easy to mistake for that.) It shouldn’t come out trivial to evaluate, but it shouldn’t be super hard either.

End-of-Semester Grading

I plan to give out one more problem set, assigned right after break and with a grade-by date on or just before the last day of classes.

But appointments for grading are getting snapped up very quickly, mostly as people who are behind try to get caught up. So you might want to try to schedule your last appointments now, and if you do need to catch up, definitely schedule now.

If you have appointments you know you won’t need, please cancel them as soon as you know, so other people can use the time. And if you can conveniently make appointments that don’t make short gaps before or after others, that’s helpful too.

Appointments through finals are fine.

Vector Line Integrals

Based on “Vector Line Integrals” in section 5.2 of the textbook.

Key Ideas

Definition: The vector line integrals seen so far in the book are integrals of the magnitudes of the parts of vectors in the field that are parallel to the curve. As such, they are integrals of scalars (the magnitudes), and so are ultimately scalar line integrals. It’s also possible to integrate components of the vectors in other directions, e.g., perpendicular to the path.

Path curving through a group of vectors; vectors project onto path and perpendicular to it

Just as with scalar line integrals, there’s an abstract view (an integral of components of vectors parallel to a curve) and a view that’s better suited to calculating with (the integral of F(r(t)) • r’(t)).

Vector line integrals have lots of physical applications, e.g., if the field represents a force and the path represents the path some object affected by that force follows, then the integral is the work done by the force on the object.

Unlike scalar line integrals, the “orientation” of the path matters. In other words, it matters what direction motion along the path is in, because that affects what direction the tangent to the path points. Paths can also be closed, i.e., start and end at the same place.

Vector line integrals have many algebraic properties in common with other integrals, e.g., they have a similar constant multiple law, similar sum and difference laws, etc.

Examples

Integrate F(x,y) = ⟨y, x²⟩ along the path following the curve y = 1 - x² from (-1,0) to (1,0).

F of X and Y is vector Y comma X squared; path is 1 minus X squared from X equals negative 1 to 1

The first step is to figure out what a parameteric form of the curve is. One good way to find parametric equations for explicitly defined curves y = f(x) is to let x(t) just be t, and y(t) be f(t). Using this idea on this problem we get…

Path Y equals 1 minus X squared in parametric form vector T comma 1 minus T squared.

Now we can plug r(t) and r′(t) into the formula for a vector line integral:

Integrating F of R of T dot R prime of T eventually yields 4 thirds.

Flux and circulation.

Please read “Flux” and “Circulation” in section 5.2 of the textbook.

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