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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


You are correct about there being no curl in 2 dimensions. The difference of the "cross partials" is really one component of a curl. If you give your vector field a fictitious third component of zero, then you can take the curl.

And how quickly those 50 years have gone by.


Thanks for the clarification. It's fortunate that you have access to Maple 18, access that I didn't have when I tested Carl's recipe. Your analysis is therefore far more comprehensive and complete.

@Carl Love 

I believe this works if you first select the section. Use Control+Alt+Shift+S to select the section from anywhere within the section. Then press Enter or Return as appropriate.


Isn't about time that dsolve acquired an option equivalent to the "free=..." option in LinearAlgebra's LeastSquares command? The _C1 of dsolve is ugly and tedious to deal with when working further with a solution containing same.


Invoke the Plot Builder on x^2+y^2+z^2=1 and you get an empty frame because the defaults are -10 to 10 on all three variables. The default grid is too coarse to capture enough points on the sphere. Change the ranges to something smaller, like -1 to 1 for each dimension.


The eliminate command in the corrected worksheet I attached to my last reply returns a list of two sets. The first set contains an equation for the eliminated variable. The second set contains the expression that results from using the first equation. This expression is the left side of an equation whose right-side is an unwritten zero.

In my usage of the command, I appended [], open and close square brackets. This causes the list brackets from the equate command to be dropped, so that the return is now just a sequence of two sets.

I then wanted to solve for z in the second "equation" returned by "eliminate." To access this expression, I used q[2], the second set in the returned seqence. But this referenced the second set, so to get rid of the set braces around the desired expression, I used q[2][1], in other words, the first thing inside the second member of the sequence q.

The isolate command is like a weak solve command that returns an equation whose left-hand side is the variable solved for.

These are operations that become familiar to users of 1D math input, the linear form advocated by so many of the respondants to this forum. This is the kind of thing I had to teach my students back when Maple had only the worksheet interface. They didn't like having to learn a computer language just to implement their math. That's why I see so much utility and value in the Typeset (2D math) input mode. It eliminates the need to teach students the minutiae of Maple manipulations that are only in service of Maple itself, not of the math you're trying to get Maple to do.

I could have done all the steps in the Corrected Worksheet in a syntax-free mode, but that requires explanatory text to describe what I would have clicked on, etc. I chose not to do that because it is indeed faster to use 1D input. But given your confusion over the syntax used, I have to ask "Was it really faster for the learner to have used what was faster for me?"


The green surface is defined parametrically with parameters x and z: x=x, y=2*x-2, z, with z=2*x+y^2 and y=2*x-2, so z=0..2*x+(2*x-2)^2 = 4*x^2-6*x+4.

These calculations have been added to your worksheet along with some corrections and the calculation of the path integral.



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.

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