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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.

MaplePrimes Activity

These are replies submitted by rlopez

@The function 

The help systems in either Maple 2021 or 2022 contain the Calculus Study Guide and the Multivariate Calculus Study Guide. These are ebooks that demonstrate how to solve more than 1200 exercises in the calculus using the tools of Maple. Are you aware of these resouces?


Thanks for the extended exposition. My aim for my earlier response was simply to call attention to the paradigm: Use the most direct set of tools available in Maple. Apparently, the OP might not have been familiar with the tools in the VectorCalculus packages. To my mind, knowledge of such tools allows for a more conceptual thinking about mathematics.


The ConicsTutor in the Student Precalculus package will graph and interpret a conic section not given in standard form. The results are reported in floating-point form, and no computational details are provided.

This tutor is available from the Tools/Precalculus menu as "Conic Sections" or by executing the command Student:-Precalculus:-ConicsTutor().


Yes, the Atomic Variables will look like ordinary Maple notation (i.e., not magenta) by simply unchecking "Atomic Variables" in the View menu.


A simple extension, nicely implemented. Thanks.


The "vertical" line is not an asymptote. It is an artifact of Maple's connect-the-dots plotting algorithm. It looks like an asymptote, but it really isn't. There are ways to suppress it, and there are Maple commands and tools to draw rational functions and their asymptotes, both vertical and horizontal. I suspect you probably know all this, but for the sake of anyone else reading your comment, I thought a clarification might be in order.

@Carl Love 

I initially passed on this question but Carl's post provokes me to respond.

If w=f(x,y,z), then w=0 and w=1 are two level surfaces defined, as in the example function provided, implicitly. An earlier response says to use implicitplot3d. Yes.

If in the same breath we are asked about level curves on a function of two variables, then that breath should not speak of f(x,y), but rather, should use a different function name. There can be no logical connection between the level surfaces of f(x,y,z) and the level curves of f(x,y). Doesn't make much sense to me, and I would give this same lecture to any student who posed the question asked by the OP using the same confusion between the name "f" for a function of three variables and for a function of two variables.

The level curves of g(x,y) are obtained in Maple with the implicitplot command (or even the contourplot command), as pointed out by tomleslie.


Sorry about that. It's not the letter V, but the keystrokes for a shilled fraction. Maple typesets a fraction as a-over-b, but to get fractions to take just 1 horizontal line, as in a/b, you need to escape the forward (/) slash. The escape character is the backslash (\). Hence, when put together, it looks like the letter V, but that appeared only because I copied/pasted some typeset 2D math, and it pasted in the two characters instead of just the one (/).

(I learned "shilled fraction" from the editor of my Advanced Engineering Math book more than 2 decades ago. I pretty much write all my fractions in Maple that way.)

I should have looked at what I pasted into my response before I sent it off.


The usage is d1:=Distance(..., not d1:=distance(...

Maple is a symbolic language. The line L0 is held in Maple's memory with the unknown parameters a,b,c, and the distance between the lines is an expression containing those unknowns.

Thus, d1 and d2 are two expressions both containing a,b,c as unknowns. Maple's solve command does not insist on explicitly setting an expression to zero. It assumes the "right-hand-side" is to be take as zero. Hence, when solve is applied to d1 and d2, the values for a, b, c that make the two distances zero are computed.

Does that help?


There's a third way: the PlotVector command in the VectorCalculus packages.

The advantage of this command: Simply define the vector as a vector or as a RootedVector, and hit it with the PlotVector command. The two arrow commands in Maple require a lot of re-writing of the vector to conform to the differing syntax of these two commands.


The TangentLine command in the VectorCalculus packages accepts the equation of the curve in implicit form, obviating the need to solve for y=f(x).


(If indeed it is an objection.) It takes 4 keys to make one of these conversions: ALT, R, V, and one of 2, M, or i.

Math can be entered in one of 3 ways, and for each method of entry, there are two other forms. Hence, there are 6 transformations possible. How can this be done with one key?


@Carl Love 

The Context Panel's "simplify" includes the option "With Side Relations." A pop-up dialog then has a place for the side relation.

Hence, It takes just two visits (simplify and solve) to the Context Panel to implement the equivalent of the coded solution.

@Rouben Rostamian  

What if the equations f=0, diff(f,p)=0 are so entangled that there is no explicit solution for x(p) and y(p)? These functions would have to be determined numerically, making the whole computation more difficult.


An internet search for "method of lines" brings up many links. In particular, Wikipedia has a substantial article on the method.

One surprising result I found in Dr. Dao's work was the use of this method in conjunction with Ritz/Galerkin and collocation techniques.

In none of my readings on MoL did I find reference to the method of characteristics.


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