3.1 Vector-valued functions and space curves  (Page 3/12)

You may notice that the graphs in parts a. and b. are identical. This happens because the function describing curve b is a so-called reparameterization    of the function describing curve a. In fact, any curve has an infinite number of reparameterizations; for example, we can replace t with $2t$ in any of the three previous curves without changing the shape of the curve. The interval over which t is defined may change, but that is all. We return to this idea later in this chapter when we study arc-length parameterization.

As mentioned, the name of the shape of the curve of the graph in [link] c. is a helix    ( [link] ). The curve resembles a spring, with a circular cross-section looking down along the z -axis. It is possible for a helix to be elliptical in cross-section as well. For example, the vector-valued function $\text{r}\left(t\right)=4\text{cos}t\text{i}+3\text{sin}t\text{j}+t\text{k}$ describes an elliptical helix. The projection of this helix into the $x,y\text{-plane}$ is an ellipse. Last, the arrows in the graph of this helix indicate the orientation of the curve as t progresses from 0 to $4\pi .$

At this point, you may notice a similarity between vector-valued functions and parameterized curves. Indeed, given a vector-valued function $\text{r}\left(t\right)=f\left(t\right)\text{i}+g\left(t\right)\text{j},$ we can define $x=f\left(t\right)$ and $y=g\left(t\right).$ If a restriction exists on the values of t (for example, t is restricted to the interval $\left[a,b\right]$ for some constants $a<b),$ then this restriction is enforced on the parameter. The graph of the parameterized function would then agree with the graph of the vector-valued function, except that the vector-valued graph would represent vectors rather than points. Since we can parameterize a curve defined by a function $y=f\left(x\right),$ it is also possible to represent an arbitrary plane curve by a vector-valued function.

Limits and continuity of a vector-valued function

We now take a look at the limit of a vector-valued function    . This is important to understand to study the calculus of vector-valued functions.

This is a rigorous definition of the limit of a vector-valued function. In practice, we use the following theorem:

In the following example, we show how to calculate the limit of a vector-valued function.