A function is called a * vector function* in . We'll focus on vector functions in the plane and space , but everything
goes over to without any difficulty.

* Example.* Find and for given by

It is no accident that the function in the last example looked like a parametric curve, because that's what it is. A vector function in is a curve in .

Our main concern is the calculus of vector functions, and the basic idea is that we do everything component-by-component, and so many of the things you learned in single-variable calculus carry over with just minor adjustments.

* Definition.* Suppose is a vector function, , and . Then means: For every , there is a , such that

(" " means the length of , regarded as a vector in .)

* Proposition.* Suppose has components

Then

This means that the limit on the left exists if and only if all the component limits on the right exist, and in that case the two sides are equal.

In other words, you take the limit of a vector function by taking the limit of each component. All of the usual rules for computing limits work, one component at a time.

* Example.* Find the value of if it exists.

The component functions of are , , and . You can see that the component functions are ordinary one-variable functions of the kind you see in a first-term calculus course.

Note that

So

* Definition.* Let be a vector function in , and let . Then f is continuous
at c if .

* Proposition.* Suppose has components

Then f is continuous at c if and only if , , ... are continuous at c, considered as functions .

If that looks a bit technical, don't worry. The meaning is that a vector function is continuous at a point if its component functions are.

* Example.* Define by

Prove or disprove: f is continuous at .

Since , the function is not continuous at .

* Definition.* Let be a vector function in . The * derivative* of f is the
vector function given by

I'll often write or for .

* Proposition.* Let be vector functions in , and let . Then:

(a) If is a constant, then .

(b) .

(c) .

(d) (* Dot product*) .

Note: In (d), all the products are dot products.

(e) (* Cross product*) Suppose . Then

* Proof.* The proofs amount to proving the
results component-wise. For example, consider (d). Suppose

Then

Using the Product Rule for functions of one variable, I have

The other results are proved in similar fashion.

* Example.* Let

Compute and .

Thinking of as a curve, is a * tangent vector* to the
curve.

* Example.* Find parametric equations for the
tangent line to

The point of tangency is . Now

Thus, is a vector tangent to the curve, so it's parallel to the tangent line to the curve. The tangent line is

You can integrate vector functions component-by-component.

* Definition.* Suppose has components

Then

A similar definition holds for definite integrals.

* Proposition.* Let be vector functions in , and let . Then:

(a) .

(b) .

* Example.* Compute the integral .

* Example.* Compute the integral .

Copyright 2018 by Bruce Ikenaga