rlopez

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14 years, 263 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.

MaplePrimes Activity


These are replies submitted by rlopez

@Vee 

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.

@Kitonum 

If V is a vector of complex numbers, the command

plots:-complexplot(convert(V,list),style=point);

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

Explore(plots:-implicitplot3d(u=c,r=0..1,theta=0..2*Pi,z=0..1,coords=cylindrical),parameters=[c=0..3.]);

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.

@Alektas 

The expressed purpose of atomic variables is that these collections of symbols be recognized as a valid name to which an assignment can be made. If you are getting an error when you assign to whatever it is you created, then you did not properly create an atomic variable.

It sounds like your input mode is text (1D math) at a prompt in a worksheet. The atomic variable has to be created in a document in Typeset (2D) math. By using the lprint command you can obtain the underlying command by which the atomic variable is created. If this is copied and pasted into a worksheet at a text-input prompt, it can be followed by the assignment operator (:=) and a value to be assigned to this name. It works. If it didn't work for you, you didn't do it correctly.

RJL Maplesoft

Given a surface defined implicitly by an equation of the form f(x,y,z)=0, you can pick anything "convenient" for x=x(u,v), y=y(u,v) and solve f(x(u,v),y(u,v),z)=0 for z=z(u,v). Determining what the "convenient" choices are is an art, based on previously accumulated knowledge. There is no recipe by which you can determine what is going to be most "convenient."

Maple's ability to "do" algebraic manipulations would certainly be helpful, but without the insight into what functions to pick for x(u,v) and y(u,v), Maple is just a servant waiting for instructions.

RJL Maplesoft

Maple makes provision for grouping symbols to form a name. Such groupings are called atomic variables (earlier called atomic identifiers). These objects are simplest to construct using Typeset math. Form the grouping of symbols (in Typeset math), select all of it, and convert to atomic variable. There is a keyboard shortcut for this conversion: Control+Shift+A. Otherwise, use the Format menu and select "Convert to".

There is a complicated string of symbols created behind the scenes by this process. If you apply the lprint command to the atomic variable, you will see the ascii code that is associated. You could use that code in text-mode (1D math in a worksheet) if necessary, but it can be pretty ugly stuff, depending on what atomic variable was created.

In a document, if the option "Atomic Variables" in the View menu is selected, atomic variables will appear in color (purple?) in the document. Unfortunately, every time you need to re-use the atomic variable, you have to re-create it, or copy/paste it.

If an assignment is made to an atomic variable, the associated string will show up in the Variables palette. It might be possible to create a "snippets palette" containing the atomic variables, but clicking on an item in such a palette inserts it at the left margin of a new line. Again, you have to copy/paste to put it where you want it. Perhaps the simplest thing to do is to copy/paste to the bottom of the worksheet or to another parallel worksheet.

@leiniu Are you using Maple 2016.1 or just 2016? Have you added something to your Maple library that might be interfering with the calculation? Have you done a restart? Do you have access to Maple 2015 just to see if the problem is in your particular installation of Maple 2016?

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