- Purpose
- Background
- Defining vector valued functions simply in Maple
- Getting Started
- Plotting curves of vector defined functions in the plane
- Derivatives and the slope
- The parametric form of a line
- Arc Length, Speed, and Curvature
- Exercises

\\storage\academics\math\calclab\MA1023\Vector_start_A14.mwYou can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read through the exercises, start up Maple, load the worksheet, and go through it carefully. Then you can start working on the exercises.

>r:=t->[3*sin(t),3*cos(t)];You can evaluate this function at any value of t in the usual way.

>r(0);This is how to access a single component.

>r(t)[1]; >r(t)[2];

>with(CalcP7): >VPlot([t^2,t^3-t],t=-1.5..1.5); >VPlot(r(t),t=0..2*Pi);

In Maple, the slope of a parametric curve can be calculated using the command as in the example below.

> r := -> [3*sin(t),3*cos(t)]; > slope := diff(r(t)[2],t)/diff(r(t)[1],t);

Below is an example in Maple using this parametric form of a line that is tangent to the curve defined above at . The plot of the curve and the line on the same graph verifies that the line is tangent at the given point.

>eval(slope,t=Pi/4);Since the slope at is -1, we want the line through the point , parallel to the vector .

>line:=t->r(Pi/4)+[t,-t]; >with(plots): >a:=VPlot(r(t),t=-Pi..Pi): >b:=VPlot(line(t),t=-Pi..Pi): >display(a,b);

While the concept of arc length is very useful for the theory of parametric curves, it turns out to be very difficult to compute in all but the simplest cases. The maple command below shows that the arclength of a circle with the parametrization has constant arclength.

> evalf(int(sqrt(diff(3*sin(t),t)^2+diff(3*cos(t),t)^2),t=0..2*Pi));

For this lab, we will also assume that we have a vector-valued function that gives the position at time of a moving point in the plane. The velocity of this point is given by the derivative and the acceleration is given by the second derivative, . In many applications of curvilinear motion, we need to know the magnitude of the velocity, or the speed. This is easy to compute - just take the magnitude . If you think of the speed as the rate of change of distance along the curve, and recall that arc length is distance measured along the curve, then you have the following interpretation of the speed

where is arc length. If the speed is not zero for any value of in the interval , then it is possible to define a unit vector, that is tangent to the curve as follows.

Using this definition, you can write the velocity in the following form.

This is not the most useful form for calculating the velocity, but it does lead to a useful way of thinking about the acceleration experience by a particle moving in a curvilinear path. If the path is a straight line, acceleration depends only on whether the particle is speeding up or slowing down. In a curve, however, there is an additional acceleration, called the centripetal acceleration, that is needed to keep the particle moving on the curve. The magnitude of this acceleration depends on the speed of the car and how much the path is curving. It turns out that you can quantify this with an intrinsic property of the curve called the curvature, usually denoted , defined by the following equation.

That is, the curvature is the magnitude of the rate of change of the tangent vector with respect to arc length. For example, the curvature of a straight line is zero and it can be shown that the curvature of a circle of radius is the same for every point on the circle and is given by . The Maple

> r:=t->[3cos(t),3sin(t)]; > v:=diff(r(t),t); > sqrt(v[1]^2+v[2]^2); > Speed(r(t),t); > Speed(r(t),t=Pi); > Curvature(r(t),t=Pi); > int(sqrt(v[1]^2+v[2]^2),t=0..2*Pi);

- Consider the vector function
for
.
- Calculate the coordinate point on the curve at and the slope of the curve at .
- Define the vector equation of the line through the point above tangent to the curve at that point.
- Plot the graph of and this tangent line on the same graphover the interval .

- Given the vector function
- Plot the vector function over the interval .
- Calculate the speed of the particle at
using the magnitude of the velocity and verify using Maple's
`Speed`command. Calculate the curvature at using Maple's`Curvature`command. - Calculate the arclength over the interval .

2014-09-22