Section 3.1 Vector-Valued Functions and Space Curves
References.
- OpenStax Calculus Volume 3, Section 3.1.
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openstax.org/books/calculus-volume-3/pages/3-1-vector-valued-functions-and-space-curves - Calculus, Early Transcendentals by Stewart, Section 13.1.
- The Desmos 3D Calculator.
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www.desmos.com/3d
Introduction.
Since a position in space can be described by a vector \(\vec{r}=\vector{x,y,z}\) and position can be a function of time, it is natural to consider vector functions like
\begin{equation*}
\vec{r}(t) = \vector{f(t),g(t),h(t)} = f(t)\veci + g(t)\vecj + h(t)\vec{k},
\end{equation*}
functions whose value is a vector (or loosely, a vector each of whose components is a function). Many familiar concepts of calculus like limits, continuity, derivatives and integrals extend simply to such vector functions, and they also have the nice geometrical significance of describing curves in space.
Topics.
Subsection 3.1.1 Limits of Vector-Valued Functions
A limit of a vector function \(\vec{r} = \vector{f(t),g(t),h(t)}\) is built up from limits of its components:
\begin{equation*}
\lim_{t \to a} \vec{r}(t) = \vector{\lim_{t \to a} f(t),\lim_{t \to a} g(t),\lim_{t \to a} h(t)}
\end{equation*}
This limit thus exists if and only if all of the three component limits exist.
Subsection 3.1.2 Continuity of Vector-Valued Functions
With this notion of a limit, it makes sense to say that vector function \(\vec{r}(t)\) is continuous at \(a\) if the limit exists there and equals the value
\begin{equation*}
\lim_{t \to a} \vec{r}(t) = \vec{r}(a)
\end{equation*}
which is true if and only if all of the component functions are continuous at \(a\text{.}\)
Subsection 3.1.3 Space Curves
A continuous vector valued function describes a space curve. A space curve \(C\) can be described as the set of points in space whose position vectors are given by the values of a function \(\vec{r}(t)=\vector{f(t),g(t),h(t)}\) for values of \(t\) in some interval \(I\text{.}\) The interval can be finite like \(I=[a,b]\) or infinite such as \(I=(-\infty,\infty)\text{.}\)
The component equations
\begin{equation*}
x = f(t), y=g(t), z=h(t)
\end{equation*}
are parametric equations for \(C\), with \(t\) the parameter.
Subsection 3.1.4 The Visual Meaning of Continuity
Continuity has a familiar geometrical meaning: a vector function \(\vec{r}(t)\) is continuous if the space curve it describes has no breaks.
Note: when drawing space curves, arrowheads are typically used on the curve to indicate the direction of motion as the parameter increases.
Study Guide.
Study Section 3.1 of Calculus Volume 3; in particular
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openstax.org/books/calculus-volume-3/pages/3-1-vector-valued-functions-and-space-curves- All the Definitions, Theorems, Examples and Checkpoints.
- One or several exercises from each of the following ranges: 1, 2, 3, 5–7, 8–13, 15–17, 22–26, 27–32, 33, 34, 35 and 36 (this last four go together).
For the graphing examples and exercises, I suggest trying the (new) Desmos 3D graphing tool.
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www.desmos.com/3d/
