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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


If you are asking about the red and green surfaces in the worksheet I posted, then note that the line integral of the scalar function is equal to the surface area of the green surface. The red surface is just a graph of the scalar f(x,y). It's there to bound the green surface at the top. The green surface goes from the plane curve C up to the red surface f(x,y). There is no claim about the area of the red surface. That does not enter the discussion at all.

The code for the streams procedure invokes the transform command, which is in the plottools package. But nowhere is that package referenced. Hence, I don't think it gets used at all. This app goes back to 2008, perhaps before the "use" command was available. Not sure on that, but the failure of plottools to appear anywhere raises a red flag for me.

Another red flag for me is the value of the function V at (x,y)=(1.5,1.5). It's -289.15, and an implicitplot of V=-289 results in a graph that in no way resembles flow past a cylinder. So, the coordinate system is, to my mind, not well defined. Is V(x,y) the stream function in Cartesian coordinates, or should this be mapped to polar coordinates? And what kind of laminar flow pst a cylinder does this V purport to describe, anyway?

I'd like to see multiple equations placed one below the other align vertically on the equal signs. When Scientific Workplace was an iterface to Maple back in the 90s, that ability was available in that interface. Oh, that it could be found in Maple itself.


All my Tips and Techniques articles are available in MapleApps. If you click the link in Acer's reply, you get to the article and can download it as a Maple worksheet or as a pdf.

There's some relevance for you in the comparison between a vector defined in LinearAlgebra, and a Vector defined in VectorCalculus. I don't think that at this point you want to be looking at the DifferentialGeometry package.


Here is the worksheet I used for the Student VectorCalculus package webinar. Perhaps having it to read will help.



In addition to the links provided by Acer, try the following three.

Teaching Concepts (see the examples on vector calculus)

Webinar on the package itself

Webinar on some Clickable VC examples


The VectorCalculus Gradient command returns a VectorField. If you use eval on a vector field, you just make substitutions into the components, and not the moving basis vectors. The result of the eval is still a VectorField, and not just a single vector in that field. If you graph this result, you see you have the graph of a constant vector field.

Hence, there's the evalVF command for VectorFields, which takes care of evaluating the basis vectors as well as the components. The evalVF command returns a RootedVector that remembers the point of evaluation, and hence can be graphed (via PlotVector) as an arrow with tail at the point of evaluation.


The "Teaching Concepts" recordings on the Maplesoft website would be one source of information about using Maple to do interesting things in math. Another would be either (or both) of the Study Guides. One is for single-variable calculus, and one is for multivariate calculus. Also, the Maple YouTube channel has recorded webinars of the Clickable Calculus series: Single-variable calculus, multivariate calculus, differential equations, linear algebra, vector calculus. These would give you some idea of the tools and resources in Maple for doing mathematics.


Would it not be more precise to say that while p(t) has an antiderivative, its representation can't be given in terms of any functions known at this time?


I responded by sending the worksheet as an attachment to a private response to your email address, then I sent a second email stating that the attachment was on its way. I asked that you respond to me directly if you did not get the email with the attachment. What went wrong with this process?

There's no need for me to upload a worksheet. Take Acer's last worksheet, execute it, then append the following commands.



As I understand your problem, you need not only the values of lambda, but the algorithm by means of which they were found. The fsolve command uses mostly a Newton iteration, but if you use the NewtonsMethod command in the Student Calculus1 package, you will get the same values that Acer got, but you can confidently state that they were obtained by Newton's method. I actually did this, and the values are the same as those produced by fsolve as implemented by Acer.


Unfortunately, the "Large Operators" are not operators, at least not the contour and surface integral symbols. To behave as operators, the symbols would need a way to capture all the information needed for constructing line and surface integrals. For example, the differential area element in a surface integral needs a description of the surface before it can be meaningful. Maybe in another lifetime? 

The implicitplot command is applied to the expression in (11). It contains L[2], L[3], and mu. At best you would need to use implicitplot3d, but the command you did use has ranges for L[1] and mu. There is a big disconnect here.

Are the functions f(x) = (x^2-1)/(x-1) and g(x) = x+1 the same? Are the equations y/x = 1 and y = x the same?

In the first case, what is the role of domains in defining a function? In the second, what is the role of solution sets in defining equivalent equations?

Mathematica's second "solution," namely, Y2=(x+2)^2 does not satisfy the ode for x>-2.

Indeed, substitution into the ode yields on the left, -x*(x+2), but on the right, x*abs(x+2). (Please note that the symbol sqrt(u) is a single positive number. It does not mean the pair of numbers +/-.)

A graph of the left and right sides shows that the two sides match for x<= -2, and for x=0. And nowhere else. 

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