## 2670 Reputation

16 years, 165 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

## First-order linear ODE...

If m(t) and S(t) are known functions, and h, Cp,  Te and Tw are constant, then your ODE is of the form y'+p(t)*y=r(t), where prime denotes differentiation with respect to t. There is a known form for the solution involving an integrating factor. Maple knows this form. So, the simplest way to proceed is to write the ODE with m and S appropriately supplied, then simply submit the whole thing to Maple's dsolve command. I suspect that the integrations needed for an explicit solution cannot be carried out i closed form, and you might have to resort to a numeric solution. If so, the t in front of T' might make it difficult to prescribe an initial condition at t=0. The behavior of the factor multiplying T' will also depend on m', and only you know what m(t) is.

## PlotPositionVector command...

This graph is easily drawn with the PlotPositionVector command in the Student VectorCalculus package.

Since r(t) is a planar curve, the graph would also be planar, not 3D.

There is no command that returns either the tangential or normal components of the acceleration vector. However, these are easily obtained since, if t is time, then the second derivative of r(t) is a linear combination of the tangent and principal normal vectors. The coefficient of the unit tangent vector is the derivative of the speed; the coefficient of the (unit) principal normal is kappa*speed^2, where kappa is the curvature of r(t) and the speed is the length of dr/dt.

Since r(t) defines a circle, the curvature is constant. So is the speed constant. So, the acceleration vector lies along the principal normal. There is no tangential component of the acceleration in this case.

The speed is 72, kappa=1/36, so the length of the acceleration vector is 144. The radius of the circle is 36, so each acceleration vector points inward, passes through the center of the circle, and goes beyond the circumference by a considerable amount. The unit tangent vector is so small in comparison, that it is nearly invisible on a graph of the circle.

Some command that might be helpful:

with(Student:-VectorCalculus):

R:=PositionVector([36*cos(2*t),sin(2*t)]);

kappa:=Curvature(R,t);
v:=Norm(diff(R,t))

PlotPositionVector(R,t=0..Pi,pvdiff=[t\$2])  (Draws circle with the acceleration vector that necessarily lies along the principal normal)

PlotPositionVector(R,t=0..Pi,normal=true) (Draws circle with principal normal vectors. They are more visible than tangent vectors, but barely so.)

Change "normal" to "tangent" and you get a graph of the circle with unit tangent vectors, but they are so small as to be just about invisible. The visualization problem stems from taking a circle with such a large radius. Change that 36 to something smaller like 5 and both the tangent and normal vectors along the circle will assume more reasonable dimensions.

## Operating-system dependent...

Maple 10 will not install under Windows7. Only Maple 11 onwards. Also, Maple 2017 would not install under Windows XP. Hence, even if you had the source-code for Maple 6, I doubt that you would have an operating system old enough to allow its installation.

I have an old laptop (more than 20 years old) with Maple 6 running under Windows 95. I'm not sure how things work under other operating systems and/or other platforms, but probably your best bet is to contact Maple's Tech Support group for definitive answers.

## solve vs. evaluate...

Please reserve the word "solve" for the solution of equations. Use "evaluate" when merely making a substitution or obtaining the value of an integral, derivative, or limit. There are no equations to solve when t=2. At that time, both vectors are identical, so the angle between is zero.

If you are new to Maple in a Calc 3 class, might I recommend the Multivariate Calculus Study Guide

https://www.maplesoft.com/products/studyguides/multivariatecalculus/index.aspx

## Lagrange multiplier technique...

The problem yields immediately to the Lagrange multiplier technique as embodied in the Maple command

Student:-MultivariateCalculus:-LagrangeMultipliers(obj, [lhs(cnsts)-rhs(cnsts)], [x, y])

## There is no finite max...

In the objective function, replace either x or y with its value from the constraint. In either case an upward-oening parabola results. Each such parabola has a minimum, but no finite maximum.

Using Maple's built-in LagrangeMultiplier command, the single extrema is at (4/3, 2/3).

## evaluation rules are complicated...

Maple's evaluation rules are complicated. In addition to the issues raised, note that Maple will immediately apply the distributive law to 2*(x+1)  but not to a*(x+1). Learn to live with these design decisions.

For your second question, the evalf command converts an exact expression to a numeric (floating-point, or deciman) one. Hence, with evalf, you get the equivalent of 1.0, a floating point number. Without evalf, you get 1, the exact value.

And by the way, there's one other way to get the exponential e: If you are entering in typeset (math) mode, type e and press the Esc key to bring up a list of all things beginning with the letter e. The very first one is the exponential e. Select that and you have the exponential e. (The Escape key performs command completion, which can also be implemented from the Tools menu.

## Arc Length tutor...

In the Tools menu, select Tutors/Calculus-Single Variable/Arc Lengths. In this pop-up tool (housed in the Student Calculus1 package) simply enter the expression for the curve and the x-coordinates of the endpoints between which the arc length is to be calculated. If Maple can't produce a closed-form solution, it will return a numeric value of the arc-length integral.

At the bottom of the tutor you can see the actual command that produces the graph shown. To use this command, copy it and paste it into your worksheet. Be sure to install the Student[Calculus1] package. If the output option in the command is changed to "integral" the arc-length integral is returned. If this option is not there, the value of the integral is returned. (Or use output=value.)

## General case vs specific instance...

If a combine/trig is applied to the integrand in the general case, four cosine terms appear. The arguments in each case are x times one of (K-L-M), (K-L+m), (K+L-M), (K+L+M). When K=14, L-2, M=12, that first term is cos(0)/4, and hence the integral is Pi/4. If none of these factors is zero, and each integer is even, the integral will be zero.

In the general case, add the option AllSolutions=true to the int command and this anomaly will be completely clarified by the piecewise result that is returned.

## It works for me...

I wrote this Task Template, so I'm instantly on alert if someone claims it doesn't work. I just tried it in Maple 2017.3 and it works.

According to the second bullet point on the left, press the Initialize button to make this template work.

To see the code, right-click on the Initialize button and select the option "Edit Click Action."

If there's a problem with this task template, please let me know.

## Use Drawing Tools...

Faced with this task, I'd probably use the Drawing Tools accessed by selecting Drawing in the plotting toolbar after clicking on the graph. It's tedious, but one could enter into a Drawing-Tools text box one letter at a time, hitting the return key after each letter. Clearly, one has to be desperate to go this route, but it would work in a pinch.

The unhappy thing about using the interactive drawing tools is impermanence. If the code that produced the graph is re-executed, the drawings on it are not preserved. I get around this by exporting the annotated graph as a PNG file, then importing that back into Maple. Of course, this gives me a graph that can't be re-executed, and could be lost if accidentally deleted.

## Defaults to the complex field...

In the LinearAlgebra package, the DotProduct command defaults to the complex number field. Maple chooses to conjugate the second vector. Include the option conjugate=false to have the DotProduct calculate over the reals.

There is no DotProduct command in the Student LinearAlgebra package; use the period as an infix operator. This operation defaults to the reals.

There is a DotProduct command in the Student MultivariateCalculus package that defaults to the reals.

Finally, since you ask why conjugation for the dot product when the vectors can be complex: without conjugation, the dot product of a complex vector with itself can be zero. This would then violate the definition of that operator (zero only if the vector is the zero vector). It requires conjugation of one of the vectors to prevent this. Some books will define the dot product with the first vector conjugated, some with the second. Maple conjugates the second.

For the sake of completeness: the norm of <1,I> without conjugation would be zero. With conjugation, it is sqrt(2), as expected.

## testplot...

Maple's algorithm for axis-label placement is a percentage of the axis length. Changing the graphing ranges shifts the position of the label. I tried it with your graph. It works, but the distortion might not be acceptable.

When faced with similar challenges where it really mattered, I suppressed the labels from the plot command and placed my own labels with the textplot command. As far as I know, Maple has never addressed the issue of user-control of axis-label placement.

## Control+Shift+R...

This works in every version of Maple, Maple 15 included.

The first "How do I" question answered in the Maple Portal's Student version is "How do you set a piecewise function in Maple?" The detailed example that gives the answer shows how to add an additional "row" or rule to the piecewise template inserted from the Expression palette.

## Typesetting Rules Assistant...

Select "Typesetting Rules" in the View menu to launch the Typesetting Rules Assistant. In the lower left corner make the two changes shown in the following figure.

The exact syntax needed for this is somewhere in my Little Red Book of Maple Magic, but that's no longer handy; so I tend to use the Assistant rather than go look up the syntax.

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