This page has information about

- Derivatives of Vector Functions
- The Unit Tangent Vector
- Arc Length
- The Arc Length Function
- Parameterization with Respect to Arc Length

**The Derivative of a Vector Function**

The derivative of a vector function **r**=<f(t),g(t),h(t)> is
**r**'(t)=<f'(t),g'(t),h'(t)>. Note one differentiates each
component independently. For example, consider the 2-dimensional space curve
defined by
**r**(t)=<2cos(t),sin(t)>. The derivative is
**r**'(t)=<-2sin(t),cos(t)>. If **r**(t) is the position
function of a particle, then **r**'(t)
is the velocity function, analogous to the velocity function in one
dimension.

Here are various rules for differentiation of vector functions:

- [
**u**(t)+**v**(t)]'=**u**'(t)+**v**'(t) (addition rule)

- [c
**u**(t)]'=c**u**'(t) (scalar multiplication)

- [f(t)
**u**(t)]'=f'(t)**u**(t)+f(t)**u**'(t) (product rule)

- [
**u**(t)o**v**(t)]'=**u**'(t)o**v**(t)+**u**(t)o**v**'(t)(dot product rule) - [
**u**(t)x**v**(t)]'=**u**'(t)x**v**(t)+**u**(t)x**v**'(t)(cross product rule) - [
**u**(f(t))]'=f'(t)**u**'(f(t)) (chain rule)

There is a nice geometric description of the derivative
**r**'(t). The derivative **r**'(t) is tangent to the space
curve **r**(t). This is shown in the figure below, where
the derivative vector **r**'(t)=<-2sin(t),cos(t)> is plotted
at several points along the curve **r**(t)=<2cos(t),sin(t)>
with 0<=t<=2*pi.

The unit tangent vector, denoted **T**(t), is the derivative
vector divided by its length:

Suppose that the helix **r**(t)=<3cos(t),3sin(t),0.25t>,
shown below, is a piece of string. If we straighten
out the string and measure its length we get its *arc length*.

To compute the arc length, let us assume that the vector function
**r**(t)=<f(t),g(t),h(t)> represents the position function of a
particle. In an infinitesimal time interval from t to t+dt
the particle essentially travels in a straight line. The distance
traveled is

From the discussion above, the velocity of the particle is
**r**'(t). Hence,

The distance traveled is

If the initial time is t=a and the final time is t=b, then the total distance traveled is

For the helix above **r**'(t)=<-3sin(t),3cos(t),0.25> and
0<=t<=7*pi. Hence,

Let **r**(t) for a<=t<=b be a space curve. The arc length function
s(t) measures the length of the curve from a to t. Based on our
discussion above,

For the helix above, where a=0, the arc length function is given by

Note that

**Parameterization with Respect to Arc
Length**

There is not a unique way to define a space curve. For example, the vector functions

and

both trace out the helix in the figure above.
Think of **r**(t) as representing the position of a particle.
The difference between the two descriptions (or parameterizations)
of the helix is the speed at which the particle travels. The particle
travels twice as fast in the second parameterization than in the
first parameterization.

A natural parameterization for a space curve is with respect to arc length. The distance along a space curve is independent of parameterization. This simply means that the total distance traveled along a curve is independent of the speed.

To parameterize a space curve **r**(t) we compute the
we invert the arc length function s(t) to get t(s).
The new parameterization for the curve is **r**(t(s)).
For the helix **r**(t)=<3cos(t),3sin(t),0.25t> (0<=t<=7*pi),
we have s(t)=sqrt(9.0625)t. Hence t=s/sqrt(9.0625). Let
c=sqrt(9.0625). We have

Here is the significance of parameterization with respect to
arc length: s=0 corresponds to the initial point on the helix;
**r**(s=1) corresponds to the point one unit of length along the curve
from the initial point; and, in general **r**(s=s_1) is the point s_1
units of length along the curve from the initial point.

**Copyright **© **1996 Department
of Mathematics, Oregon State
University**

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