A space curve is any function \(f:\R \to \R^n\) for \(n \ge 2\text{.}\) These are one dimensional structures (one parameter) existing in 2 dimensions or more. The 2D case were previously studied as parametric form. Most of ours will be in 3D. We will use them as a first opportunity to convert definitions from single variate calculus to definitions in multi-variable calculus.
Subsection1.4.1Defining a Limit of Vector Valued Functions
Our goal is to consider how limits apply in higher dimensions, specifically how they apply to vector valued functions. We will start with a graphical experiment.
Figure1.4.1.Demonstration of Graphing Space Curves by Hand
Activity18.
(a)
Watch the video on graphing 3D curves (Figure 1.4.1) as needed then graph the following curves by hand. Graphing by hand should help you estimate the limits in the next task; it is not a useful skill for daily life.
Based on your graphing estimate the following limits.
(i)
\(\lim_{t \to 5} C(t) \)
(ii)
\(\lim_{t \to \infty} D(t) \)
(iii)
\(\lim_{t \to \infty} F(t) \)
Activity19.
All limit definitions thus far are for scalar valued functions rather than vector valued functions. The following steps lead to a generalized definition. Note it is often convenient to reference individual elements of vectors. The notation \(C_x\) refers to the \(x\) (or first) element of the point (or vector) \(C.\)
(a)
Calculate the following three limits. \(\lim_{t \to 5} C_x(t), \)\(\lim_{t \to 5} C_y(t), \)\(\lim_{t \to 5} C_z(t). \)
(b)
How do these calculations match your estimate for the limit for \(C(t)\) in Activity 18?
(c)
Copy the formal definition (\(\epsilon\) and \(\delta\) expressions) of \(\lim_{t \to a} f(t) = L. \) You may wish to look up the Illustrated Limit available online.
(d)
Determine which parts of the definition must change for a vector valued function and write this definition.
Example1.4.2.
Consider the following proof of a limit. \(lim_{x \to 5} 7x-9 = 26.\)
\begin{align*}
\amp |7x-9-26| \amp \amp \lt \epsilon \amp \text{From the definition of a limit.} \\
\amp |7x-35| \amp \amp \lt \epsilon \amp \text{Algebra} \\
-\epsilon \lt \amp 7x-35 \amp \amp \lt \epsilon \amp \text{From the definition of absolute value.} \\
-\epsilon \lt \amp 7(x-5) \amp \amp \lt \epsilon \amp \text{Algebra, because we want } |x-5| \\
-\frac{\epsilon}{7} \lt \amp x-5 \amp \amp \lt \frac{\epsilon}{7} \amp \text{Algebra} \\
\amp |x-5| \amp \amp \lt \frac{\epsilon}{7} \amp \text{From the definition of absolute value.} \\
\amp \delta \amp \amp = \frac{\epsilon}{7} \amp \text{From the definition of a limit.}
\end{align*}
Thus for all \(\epsilon>0\) there exists a \(\delta>0\) namely \(\delta=\epsilon/7\) that satisfies \(|x-5| \lt \delta\) implies \(|f(x)-26| \lt \epsilon.\)
Activity20.Vector Valued Limit Proof.
Use the example from Example 1.4.2 and the definition from Activity 19 to prove the limit \(\lim_{t \to 5} C(t) \) using the formal definition and this example.
Subsection1.4.2Defining a Derivative of Vector Valued Functions
Definition1.4.3.Derivative of a Scalar Valued Function.
The derivative of the function \(f:\R \to \R\) at a value \(a\) denoted \(f^\prime(a)\) is defined by
This definition is for scalar rather than vector valued functions. The following steps lead to a generalized definition.
(a)
Note that \(f(x)-f(a)\) provides the distance from \(a\) to \(x\text{.}\) How does it also include direction?
(b)
This is direction and distance which we call what?
(c)
Can the same calculation as in the first step be performed for a vector valued function?
(d)
Can the division (by \(x-a\)) still be performed? If so, how?
(e)
Write a definition for the derivative of a vector valued function.
(f)
If \(f(x)\) represented the position of an object at time \(x\text{,}\) then \(f^\prime(x)\) represented the velocity of the object. Does this still work with vector valued functions?
(g)
How are the vector valued function definitions of limit and derivative similar?
If \(P\) represents Guido’s position at time \(t,\) what is \(P^\prime\text{?}\)
Subsection1.4.3Integrals of Vector Valued Functions
Just as limits and derivatives can be extended to vector valued functions, so can integrals. The first, arclength, is an extension of the arclength calculated in parametric form, the second is an introduction to later topics this semester.
Subsubsection1.4.3.1Arclength
In this section consider how integrals can be applied when vector valued functions output points. It is possible to add the areas beneath curves like in single variate calculus, but that will be done later. Note in the following examples the functions provide points, but you must calculate distances.
Activity22.
(a)
As a big ball is pushed around it ends up at the following locations in order. It starts at \((0,0),\) then moves to \((3,1),\)\((5,2),\)\((9,3),\)\((12,5).\) How far has the ball traveled?
(b)
A car is spinning across an icy surface. At time \(t\) the car is at position \((t, \sqrt{2t-t^2}).\) At time \(t=2\) how far has the car spun?
Because we are calculating distances, these problems can be considered to be summing the lengths of the vectors from one point to the next (if only finitely many points are considered). Thus arclength of a vector valued function \(P(t)\) over \(t \in [a,b]\) can be calculated by
