rlopez

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14 years, 290 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.

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These are replies submitted by rlopez

@Kitonum 

This solution is easy for the woodworker because all pieces can be cut "straight through." And it is an amazing piece of mathematics.

Unfortunately, provision for the saw kerf, at least 1/8", has not been made. Perhaps one should add the dimension of the kerf to all measurements, but there are pieces that would then end up oversized. Some woodworkers discount the squareness of 4x8 sheet goods and waste the factory edges as a matter of course. The stock-cutting problem is not easy!

 

Perhaps this post from December of 2010 is relevant.

https://www.mapleprimes.com/maplesoftblog/100265-Subscripts-As-Partial-Differentiation-Operators

It summarizes a Reporter article from July of 2010.

Because you say the equation in Word has to be editable, you can't just copy/paste - that results in an uneditable graphic.

Look into the program MathType that sets equations in an editable form in Word. I think the connection is that MathType understands LaTeX, and Maple can export the LaTeX form of its equations.

It's been about 20 years since I operated in that world, so forgive me if I'm in error on any of these points.

@Nia Dutta 

Since the Maximize command is part of the Optimization package, the usage in your check.mw worksheet should be

Optimization:-Maximize(...

If you first load the package via the with(Optimization) command, then the Maximize command will work as it appears in the check.mw worksheet.

We're just guessing here. Perhaps the last expression (x+4-2) lost it's "executable" tag. So, right-click on the expression and inspect the pop-up that results. There's a line "Executable Math" in the pop-up. If there's no check at the left of this line, then what was clicked on is not executable math. Also, executable math will be in a blue-tinged box but non-executable math will be in a gray-tinged box.

Next time you have a difficulty, post the worksheet in which the difficulty occurred. We'd all have a better chance of diagnosing an error if we could test the worksheet, not just an image of it. Use the green upwards-pointing arrow in the toolbar to upload a worksheet.

@one man 

When I authored the Maple application just referenced, I was not aware that the correct usage is "Draghilev" and not "Dragilev." Shortly after my article was published, I was admonished about the spelling. I have since made it a point to use the correct spelling, but the original post in the Maplesoft Application Center unfortunately still retains the incorrect spelling. To correct this, I would have to revise the article and then induce Maplesoft to replace the original with an update. Not impossible, but tedious. I will put this on my to-do stack, which, in retirement, seems to grow faster than it can be diminished.

@Mariusz Iwaniuk 

If q is Laplace's equation, then the following pdsolve command returns a weak solution to a BVP that has discontinuous boundary data.

pdsolve([q,u(0,y)=0,u(x,0)=0,u(x,Pi)=0,u(Pi,y)=1]);

Whether this is by design or by accident, I don't know.

If the design is to catch all BVPs with discontinuous BCs, then this example points to a bug that I would hate to see fixed. I would not want Maple to stop returning a weak solution to such a problem. This issue of pdsolve and weak solutions really needs clarification. It appears that pdsolve presently solves some BVPs with discontinuous BCs, but not all. I would rather see it solve more such problems rather than fewer.

 

Similar problems have been solved in this forum with Draghilev's method and with the DirectSearch package. This package is not built into Maple. It can be downloaded from the cloud.

Do a search on "Draghilev" in this forum and find examples that have been solved by both methods. Be advised that some posts in this forum have been saved with the tag "Dragilev".

If there is sufficient smoothness, etc., your particular example defines a curve parametrically. In principle, three equations in four unknowns can be solved for three of the unknowns in terms of the fourth. Hence, x=x(a), y=y(a), z=z(a). The nonlinearity in your equations might make the algebra of solving intractable, but a numeric solution should be possible. Note that for a given value of "a", there may be multiple points (x,y,z) that satisfy the equations. That would mean the curve so defined loops back over itself or has branches. The devil is always in the details.

@mmcdara 

The evolute of the ellipse x^2+4*y^2=4 is given parametrically by x=3/2*cos(t)^3,y=-3*sin(t)^3. Call this evolute E. An involute of this evolute is given vectorially by E-s*T, where T is a unit tangent vector and s is arc length. In an upcoming webinar (A Tale of Two Involutes, prepared but not yet scheduled), I will show that the arc-length function must look essentiallylike the graph drawn by the OP. A strictly increasing s(t) will not yield any correct involute. I found that the indefinite integral rather than the definite integral worked. The evolute E does not have a continuous tangent vector, and the points of discontinuity are the cause of the anomalous behavior.

The Maple command for obtaining a numeric solution of an equation is fsolve. While there is an implementation of Newton's method in the Student Calculus1 package, it is there as a pedagogic tool. The command is NewtonsMethod, and there is a Tutor that implements it. But I imagine that you need the fsolve command for what you have seemed to describe.

@ira044 

csgn first appears with argument I*kappa^4+((2*I)*n^2+12*I)*kappa^2-24*kappa*n+I*n^2*(n^2-12). The value of csgn hinges on the signs of the real and imaginary parts of this argument. If you give values to n and kappa, you will find that the signs of the real and imaginary parts will change depending on what values you give. Maple does not know what these values might be, so no simplification is possible unless the assumptions on n and kappa are more specific.

I tried simplify(the product) assuming n>0, kappa>0 and got a much simpler expression that appears to be strictly real.

 

@bridgit_teds 

Couldn't stop thinking about this question. Here's what I would have done to examine this vector field.

with(Student:-VectorCalculus):
BasisFormat(false):

Explore(VectorField(<((rho/rho0)^2*exp((rho/rho0)^2), rho*sin(phi)^2/rho0, 1>, coords = cylindrical[rho, phi, z], output = plot, fieldoptions = [arrows = SLIM, fieldstrength = fixed]), rho0 = 1 .. 10.)

Notice that F0 has been set to 1 because it is a simple multiple of each coordinate. It can be included if it really matters.

The Explore command generates a graph controlled by a slider that varies the value of rho0.

The BasisFormat command controls the display format for vectors. I prefer seeing column vectors and not a sum of components multiplied by basis vectors.

In the Student VectorCalculus package, the VectorField command is modified to accept the option "output=plot". A graph of the vector field's arrows is returned. This avoids the need for invoking the fieldplot or fieldplot3d commands. See the help page for the VectorField command for the options that modify the look of the graph drawn with the VectorField command.

https://www.maplesoft.com/demo/streaming/ClickableCalculusSeries6-VectorCalculus.aspx

 

The problem you want to view is the third, and it starts at approximately 21 minutes.

It appears that the Enter key was pressed after each of the three lines of the do-loop. The loop has to be in one single execution group, and this is done by use of Shift-Enter, not just Enter.

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