rlopez

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12 years, 110 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.

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These are answers submitted by rlopez

The example given is an explicit function of two variables: z = z(x,y). Why  not just apply the solve command to the two equations obtained by differentiating with respect to x and y? There are Maple commands for such problems, but why not just take a simple approach?

The "also" is not clear. Is it a question about solving a constrained optimization problem in two variables? If so, I suggest the LagrangeMultipliers command in the Student MultivariateCalculus package. Of course, the Lagrange multiplier method could easily be implemented from first principles, of via the task template in Tools/Tasks/Browse: Calculus-Multivariate/Optimization.

Regression analysis is only available in Maple's Statistics package. See ?Statistics,Regression,Solution for the help pages where this material is detailed.

Functions of matrices are obtained by computations involving eigenvalues and eigenvectors of the matrix. It isn't likely that one will know the exact eigenpairs of a large matrix. I'm not sure what Maple's MatrixFunction command does with a matrix containing at least one float so that the eigenpair calculation switches to numeric algorithms. The OP is welcome to experiment. Put a decimal point after any one of the entries in the 9x9 matrix and see what happens.

The help page for pdsolve,numeric indicates that Maple can solve (numerically) a "single or set or list of time-dependent partial differential equations in two independent variables", What is deceptive is that one of the two independent variables must be the evolution variable, namely, time. That leaves just one spatial variable. The pde provided by the OP has two spatial variables. Hence, it is outside the scope of what Maple can solve.

If I understand correctly:

y:=x^3;

plot([y,diff(y,x,x),x=0..1])

 

mtaylor(P,[x,y],3) - mtaylor(P,[x,y],2)

The first call to mtaylor creates a polynomial whose terms are of degree 2 or less; the second, a polynomial of degree 1 or less. The difference should be a polynomial of exactly degree 2.

I'm sure there is a more elegant way to do this, but I think my approach works works.

plots:-shadebetween(2,x^2+y^3,x=0..1,y=-1..1,style=surface,color=khaki,changefill=[color=pink,transparency=.5]);

To see all three solutions in Maple, include the option "implicit" in the dsolve command.

with(geometry):

point(cent,2,3);

circle(C,[cent,5]);

This will define a circle named C, center at (2,3), and radius 5.

Because the Help icon is mentioned, I infer that the question is about the Maple icons shown in a launched copy of Maple.

If that inference is correct, then note that the Tools/Options dialog contains a setting for large icons under the Interface tab.

The old linalg package required the use of the evalm command. The new LinearAlgebra package does not. The old linalg package use the "matrix" command but the new LinearAlgebra package uses the "Matrix" command. The noncommutative multiplication operator in LinearAlgebra is the period, not &*. So, it would be helpful to work within the confines of the newer LinearAlgebra package.

The simplest way to equate structures elementwise is to use the Equate command.

A complete solution to the stated problem is given in the attached worksheet.Matrix_calculation.mw

First, when you use a[i] as a variable (i.e., a Maple name), the element a[i] is actually a member of a table whose name is also "a". So, when you assign each equation to the same name "a" each time the loop cycles, you end up with just one equation, namely, the last one. And you have really clobbered your table.

So, rather than name each equation and then later feed all the names to another command, generate a sequence of unnamed equations. For example, write eqns := seq(a[i]+a[i+1]=i^2, i=0..99). Thus, you can send these equations to the solve command with the syntax solve({eqns}), However, you will have 100 equations in 101 unknowns (a[0], a[1], ..., a[100]), so Maple will pick the indeterminate variable. Either supply one more equation or provide the set of names you want solved for.

Perhaps you could use solve({eqns},{seq(a[k],k=1..100)}). This will return a[1],...,a[100] in terms of a[0], provided solve can obtain the solutions. If you have to resort to a numeric solution via the fsolve command, then there can be no indeterminates, and you definitely would have to provide that one more equation to have the same number of equations as variables.

Carl Love has provided a complete solution based on first principles. There are three built-in tools that would have drawn the region of integration and provided an appropriate integral for the volume inside the sphere but outside the cylinder.

The VolumeOfRevolution tutor in Student Calculus1 will implement the method of shells for any vertical axis of rotation.

There are two relevant Task Templates in Tools/Tasks/Browse: Calculus-Multivariate/Integration/Visualizing Regions of Integration/3D, one for integration in cylindrical coordinates, and one for integration in spherical coordinates.

See the attached worksheet for details.Tools_for_Volume.mw

The CenterOfMass command in the Student MultivariateCalculus package will provide the CM for both 2D and 3D regions that can be swept by a single mulltiple integral. Give the density function and the ranges of integration as equations of the form x=a..b, y=c..d, etc, and Maple returns either the unevaluated integrals, of the values for the coordinates of the CM.

The dchange command in the PDEtools package will also do it.

PDEtools:-dchange(_Z1=n,symbolic,[n])

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