SUNY Geneseo Department of Mathematics

More about Integrals of Vector Valued Functions

Monday, February 27

Math 223
Spring 2023
Prof. Doug Baldwin

Anything You Want to Talk About?

Scalar definite integrals can be understood as areas under curves. What, if anything, is the analogous understanding of definite integrals of vector-valued functions?

There isn’t as nice a geometric understanding of vector-valued integrals, but there are physical interpretations that take advantage of vector-valued definite integrals being Riemann sums, just as scalar integrals are.

For example, in physics, the work done (basically energy expended) by a force moving a weight is the dot product of that force with the displacement vector that describes how much the weight moves. This is a fine formula as long as force is constant, but if force varies with position you need to think of the work as a sum of infinitely many distinct forces moving the object infinitely small distances. This is a vector-valued Riemann sum, or an integral:

Force moving weight along vector delta X takes work F dot delta X; becomes integral for varying force

(The “dot dx” form of this integral is actually a little different than we’ve seen up to now; we’ll come back to look at such integrals more carefully at the end of the semester.)

Vectors and Vector-Valued Functions in Mathematica

Write vectors by enclosing their components in curly braces, { and }.

Then addition, subtraction, and scalar multiplication are written just like scalar addition and multiplication.

The Norm function calculates magnitudes.

Dot products are represented by a period used as an operator, or by the Dot function.

The Cross function calculates cross products.

All of these work on vector-valued functions as well as on vectors of constants.

To take derivatives, of either vector functions or scalar functions, use the D function.

Integrals, either definite or indefinite, are calculated by the Integrate function.

For examples, download this notebook.

Integrals of Dot and Cross Products

As a general rule, do the product first, then integrate the result.

For example, find the general antiderivative of ⟨t, 1, t²⟩ × ⟨1, t³, √t⟩

Evaluating the cross product gives an ordinary vector-valued function, which you can then integrate like any other:

Integrating a cross product by evaluating the product and integrating the resulting vector

But you can derive forms of integration by parts for vector-valued functions if you want. The inspiration for this is that integration by parts is basically the antiderivative rule that reverses the product rule for derivatives:

Rule integral of U D V equals U V minus integral of V D U comes from product rule for derivatives

For example, can you derive an integration-by-parts-like rule for evaluating integrals of the form r(t)•du?

Start with the product rule for dot products, then integrate both sides and isolate the resulting integral of u•dv. But realize that in order for this to make sense, all products are dot products.

Working out a formula for integral of vector U dot D V from product rule for dot product derivatives

How long is a curve in space?

Please read “Arc Length for Vector Functions” and “Arc-Length Parameterization” in section 2.3 of the textbook for Wednesday.

Remember there’s no class tomorrow, on account of the Diversity Summit.

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