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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

@Zahid Shareef 

I appreciate your kind words. Thanks.




In '91, I would have been teaching at the Rose-Hulman Institute of Technology in Terre Haute, Indiana. If you took your calculus at some other institution, then perhaps I have a clone I never knew about.

In any event, the real me will continue participating in the discussions on MaplePrimes.

Thanks for the kind words and encouragement.


Is the essential question "Can Maple be made to solve all the equations at once?" or is it "How can the separate sets of solutions be joined into one set?"

I don't know the answer to the first possiblity, but for the second, pdsolve returns a set of solutions, so if you want one big set, use the union command. The syntax is `union`(ans1,ans2,ans3,ans4) for forming the union of the individual sets.

@Carl Love 

As a reminder to all readers: The % refers to the CHRONOLOGICALLY last command executed, so if the worksheet is then executed out of the original order, the % will point to the wrong item. When I was teaching, my policy was to forbid the use of %, and to refuse to even look at a student's worksheet if it has a % in it. I took this draconian approach because of the many entanglements I had to sort out when students used this device.


If x is real and positive, x^(1/3) is the "cube root of x" and one expects the principal value to be real. However, if x is real but negative, the principal value is complex. In fact, there are three "roots", two complex, one real. Unfortunately, the principal root is the one with the smallest argument in the complex plane, and that one is not the real root. In Maple, the real root can be obtained by writing x^(1/3) as surd(x,3). This returns the real cube root of x when x is negative. The surd structure was in Maple before the RealDomain package was written.

Roots and powers of real and complex numbers are studied in the precalculus realm under the heading of DeMoivre's law. That's where you will learn about taking roots and powers, and will learn the adage "roots before powers". In other words, for something like x^(5/3), one computes the root first, then raises that root the the 5th power. Do it the other way around and you can end up with the wrong value. Try some experiments with things like x^(2/3) and x^(5/3). Let x be real positive, and then real negative. Try root-before-power, and then reverse the calculation. See how it all plays out.

The expression in blue would be zero for even integers n, but the factor in black that you claim the answer should be is zero for odd integers n. I don't see how the blue expression could ever become the "right answer", so something deeper is going on.


If V is a vector of complex numbers, the command


will graph the points in the "complex plane". No need to separate into real and imaginary parts.

Given a set of linear equations, the GenerateMatrix command in the LinearAlgebra package will put the equations into matrix/vector form. However, to pass from the vector x that you have defined to a set of equations on which the GenerateMatrix command will work requires the use of the Equate command, a command which takes two lists or two vectors and equates corresponding components. So, define a vector X with components X1..Xn, and equate X with x to form a list of equations. Convert this list to a set and apply the GenerateMatrix command. Look up the help page for GenerateMatrix to determine the precise options that fit your needs.

Use the orientation option in the plot3d command. It takes a list of three numbers, the angles (in degrees) that you can see in the plot toolbar.

@Matt C Anderson 

The polynomial (1/24)*x^4-(1/4)*x^3+(23/24)*x^2+(1/4)*x+2 reproduces (for x=1,...5) the five given values, and for x=6, produces 38. Why do you ask for THE next number?

RJL Maplesoft

A command such as


will produce a slider-controlled graph, where the value of the slider will determine c, the constant value of the function u. This gives a view of the various level surfaces. Carl's animation shows values of u on concentric cylinders of varying radii.

RJL Maplesoft

Consider the function w=w(x,y,z), where w is the temperature in a room endowed with Cartesian coordinates. What would a graph of w be? There is a value of w at each point in the room. Perhaps at each point in the room you could write the value of w on a ping-pong ball and fill the room with such balls. Not a very practical solution. Alternatively, you could color each ping-pong ball with the same value with the same color. Then, the room would be filled with layers of colored sheets of balls.

From here, you should see that what would work for you would be a graph of some of the level sets (iso-surfaces) obtained via graphs of u(r,theta,z)=constant. You could color the surfaces according to value if you like.

Maple has no command for drawing a set of level surfaces. The implicitplot3d command graphs a single surface given implicitly. You would have to generate several such surfaces and bring them together with the display command.

So, is this going in the right direction?

RJL Maplesoft

The VectorField command in the Student VectorCalculus package supports the option "Output=plot", in which case your order of definition of the components of the vector field and any constants will work. Also note that the command plots:-fieldplot(x*y,x=0..1,y=0..1) results in the same error message you display. As pointed out in the other responses, fieldplot looks for some sort of representation of a vector, not a scalar.

RJL Maplesoft

If you want the symbol dy/dx to be the differentiation operator applied to y, and have it be "active" so that it evaluates the derivative, then the best you are going to get is d/dx y. If you want just the notation dy/dx as notation, you can type it in Typeset math and set it to non-executable.

For executable math, there is also the option to use y' as the differentiation operator.

RJL Maplesoft

The polar curve given is actually the circle (x-3/2)^2+(y+1)^2=13/4. Hence, the area and circumference are knowable without integration. That should provide a check on your work.

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