## 2493 Reputation

15 years, 80 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.

## Use the Student MultivariateCalculus pac...

The following three commands will do what you want.

with(Student:-MultivariateCalculus):
P:=Plane([A,B,C],<u,v,w>);
Distance([a,b,c],P);

In fact, these calculations can be implemented syntax-free via the Context Menu, once the Student MultivariateCalculus package has been loaded. See http://www.maplesoft.com/teachingconcepts/detail.aspx?cid=141 for a recorded example of how to do this.

RJL Maplesoft

## The display of steps is not ubiquitous...

By design, the ShowSolution command can be applied only to the evaluation of integrals, derivatives, and limits.

If the laplace transform is written as an integral, the ShowSolution command can display the steps of the integration. However, if the command is applied to the laplace command, no steps will be displayed because the ShowSolution command was never intended for such a case.

RJL Maplesoft

## ChangeOfVariables...

The Student MultivariateCalculus package has the ChangeOfVariables command that applies to inert multiple integrals. A general Cartesian iterated double integral is

q:=Int(F(x,y),[y=g(x)..f(f),x=a..b])

An application of the ChangeOfVariables command:

ChangeOfVariables(q,[cartesian[x,y],polar[r,t]])

The return is an iterated integral in which the integrand is expressed in polar coordinates, but the limits of integration are still the original Cartesian limits. Maple can only change the limits of integration in the simplest of cases.

The problem of changing the limits of integration is like that for changing the order of iteration in a multiple integral. The only way to do that is to draw the region of integration, and from the picture, figure out the limits of integration if the order of iteration is changed. It's the same problem with a change of coordinates. Draw the region of integration, and from the picture, figure out what those bounds would be in the new coordinate system.

In other words, changing the limits of integration when changing coordinates is a hard problem, one that has  not as yet been reduced to an algorithm.

RJL Maplesoft

## Typesetting trick...

No need for extended typesetting mode. In the label option of the plot command, use the following syntax: Typesetting:-Typeset(diff(phi(t),t))

I learned this several years ago from our graphics programmer, and wrote about it in a Tips&Techniques article for the Maple Reporter. The article is stored in the Maple Applications Center here: (http://www.maplesoft.com/applications/view.aspx?SID=141091)

RJL Maplesoft

## It's not as bad as you think...

The particular solution returned by DEtools:-particularsol actually has zero imaginary part if you apply simplify(evalc(Im... to it. The real part is, unfortunately, more complicated than it needs to be since it contains terms that are actually in the homogeneous solution.

The command dsolve(f,u[1](t),output=basis) will return a fundamental set (for the homogeneous solution) and a particular solution. This particular solution is much simpler than the one obtained by the particularsol command, and, after factoring, contains just one term from the homogeneous solution.

I know of no way to force Maple to return the "simplest" particular solution, either here, or in any of the cases I've encountered in my own work.

RJL Maplesoft

## Use Drawing Tools...

Click on the graph. Click on the "Drawing" button in the Plot Toolbar. There are now tools for drawing on the graph, and arrows can be so inserted. The disadvantage is that any re-execution of the graphing command will lose the drawn items. To counteract that difficulty, export the composite graph to some file format such as PNG, and then insert the graph into the worksheet as an image.

It's also possible to combine the graph of an arrow with the other graph. Draw the graph and assign it to a name such as p1. Graph the arrow and assign it to a name such as p2. Use the display command from the plots package to join p1 and p2 into a single image. Since this composite is done with Maple commands, re-execution re-creates the same composite.

Simplest way to draw the arrow with Maple commands is via the Student VectorCalculus package. Use the RootedVector command to create the arrow that has its tail at the root-point of your choice. Draw the arrow with the PlotVector command. (There are more primitive commands for creating and drawing arrows, but I find the tools in the VectorCalculus packages "simplest" to use.)

RJL Maplesoft

The task template at Tools/Tasks/Browse: Algebra/Rational Function - Graph and Asymptotes calls the Rational Function tutor, but in a format that embeds the results in the worksheet so that when the tutor is closed, the information generated by it does not disappear.

RJL Maplesoft

## The devil is in the details...

If this question was posed by a veritable new user of Maple, then probably such details as how one enters the differential equations, etc. are at the heart of the user's needs.

Suppose the user has changed the default setting of Maple so that he/she is in a worksheet using linear (1D math) inpput at the red prompts. Then the DEs would be entered as

eq1 := diff(x(t),t) = 2*x(t)-5*y(t)+t;

eq2 := diff(y(t),t) = 4*x(t)+9*y(t)+sin(t);

The solution of the IVP would be obtained with the command

Q := dsolve({eq1,eq2,x(0)=0,y(0)=0},{x(t),y(t)});

The solution for y(t) is most easily extracted via the syntax

Y:=eval(y(t),Q);

Evaluation at t=1 is then done via Y1 := eval(Y,t=1); Since this gives an exact value of some complexity, the next step would be conversion to a floating-point number via evalf(Y1).

Back when I was still teaching with Maple, only the command-version of the interface existed, so a lot of energy went into instructing students on a workflow and the concomitant syntax needed to accomplish the required calculations. From my 15-years experience using Maple with students in this environment, I think anyone who admits to being a new user deserves more help than simply being pointed to a Maple help page.

A much easier workflow can be implemented in a Document using typeset math input, but unfortunately, while this is easier to implement, it is harder to describe in writing. So, since there is a recorded demo of how to solve a linear system with syntax-free techniques already available, I'll not reproduce that solution here. In the Teaching Concepts section of the Maplesoft web site, one can find more than 150 examples implemented in the syntax-free paradigm Maplesoft has coined as Clickable Calculus.

RJL Maplesoft

## Two alternatives...

In the VectorCalculus package, use %Divergence to apply the unevaluated divergence operator. The value command applied to this expression will cause the divergence to be evaluated. The downside to this approach is that the return of %Divergence is precisely that.

Alternatively, work within the Physics:-Vectors package. The inertization operator is the same, namely, %, but the display is the Nabla (Del) dot the expression. Again, the value command causes evaluation.

The Physics:-Vectors package has one additional advantage. It supports what are called "unprojected" vectors, that is, vectors whose components don't have to be declared. (A "projected" vector is one for which components are explicitly given.) Hence, a certain amount of symbolic vector calculus can be implemented in this package, something available nowhere else in Maple.

RJL Maplesoft

## Limit-Comparison...

F:=evalc(Re(1/(1-x-I*y)));

f:=exp(F);

F^2 < f, as shown by plot3d(f-F^2,x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),view=-1..3);

Since f and F^2 are positive, their integrals are likewise ordered.

q:=int(F^2,y=-sqrt(1-x^2)..sqrt(1-x^2));

int(q,x=-1..1) => Maple returns the infinity symbol.

The integral of a smaller quantity diverges, so the original integral, which is bigger, diverges.

I wonder if this is the "by hand" approach originally taken?

RJL Maplesoft

## Tutor and command...

See the ApproximateInt tutor in the Multivariate Calculus package. Access it from the Tools/Tutors menu.

At the bottom of the tutor you will also see the command upon which the tutor is based. You can copy/paste that to obtain similar results without the tutor.

If your need is for code that accomplishes these tasks from first principles, then you will have to learn a bit more about programming in Maple.

RJL Maplesoft

## Use PlotVector in VectorCalculus...

I believe I would use the tools in the VectorCalculus packages, probably the Student version, which is more lenient with respect to declaring coordinate systems.

To create a vector that knows where it's tail should be, use the RootedVector command with argument "root" that takes the point at which the tail is to reside. Then, use the PlotVector command whose argument can be a list of vectors, RootedVectors, etc.

I find the use of the primitives, either the arrow command in the plottools package, or the arrow command in the plots package, tedious, requiring too many parameters to get the vectors to look right and to be in the right places.

RJL Maplesoft

## VectorCalculus package does not know any...

The VectorCalculus packages in Maple do not treat tensors. Use either the Physics package or the DifferentialGeometry package to deal in any way with tensors.

Moreover, the M in your calculation is not a tensor in Maple. It is seen in Maple as a matrix. Mathematically, one can blur the lines between structures but in a computer algebra system, a structure is what it is. And M is just a matrix as far as Maple knows.

RJL Maplesoft

## Try restarting Maple...

Solve works find in Maple 18.02. I suspect that you had previously assigned something that caused the error messages. Try a restart.

RJL Maplesoft

## Homogeneous solutions mixed in...

It appears that Maple uses variation of parameters to obtain a particular solution. This method has the propensity to provide a particular solution that contains solutions of the homogeneous equation.

I am not aware of any simple way to filter out the homogeneous solutions from such a particular solution.

Since the ode is linear, constant-coefficient, and the right-hand side yields to the method of undetermined coefficients for finding a particular solution, that might be the best way to obtain the "simpler" particular solution.

RJL Maplesoft

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