Maple Questions and Posts

How can I calculate a cutting point in a function ...

Asked by: gabita310 5

November 09 2017

1 4

hello everyone!!

I need to plot the function x^3-x^2-x-1, and in the plot I need to calculate de point X that make the function equals to cero (the cutting point with the x axis).

Thanks you!!

Want to solve the system of algebraic equations...

Asked by: Muhammad Usman 240

November 09 2017

1 1

Hello dear!

Hope everyone is fine. I want to solve the system of algebraic equations. The answer of c[i]'s should be in term of q. Please see the attachment and fix the problem for any value of M. I am waiting your positive response. Thanks in advance.

Help.mw

Solution to the Simultaneous Equations DO NOT matc...

Asked by: uomcsg 40

November 08 2017

0 1

Dear All,

I wanted to check the calcultations carried out in a Journal Article I was reading as I was not very convinced.

The objective is to find q1i, q2i, q3i in terms of px, py, and pz and constant parameters φ, a, b, h, and r.

I have realised that the results given by the article do not match with the solution to the dsolve command.

Could you please spot where the mistake is?

My Solution: IK_equations.mw

how to draw table/graph for different t on f'' or ...

Asked by: faisal 15

November 08 2017

1 0

help me on thispde_solutionalpha.mw

pde1 := diff(u(y, t), t)+S.(diff(u(y, t), y))-2*k^2*u(y, t) = diff(u(y, t), y, y)+theta(y, t)+N.C(y, t)+M.u(y, t)+u(y, t)/K

proc (Pr) options operator, arrow; {C(y, t)+t*(diff(C(y, t), t))+.78*(diff(C(y, t), y)) = 5.000000000*(diff(diff(C(y, t), y), y))-.3*C(y, t), diff(u(y, t), t)+.78*(diff(u(y, t), y))-.50*u(y, t) = diff(diff(u(y, t), y), y)+theta(y, t)+.6*C(y, t)+3.000000000*u(y, t), theta(y, t)+t*(diff(theta(y, t), t))+.78*(diff(theta(y, t), y)) = (diff(diff(theta(y, t), y), y))/Pr-.7*theta(y, t)} end proc

(1)

PrList := [.71, 7, 10, 100]; Colours := table(`~`[`=`](PrList, ["blue", "red", "green", "cyan"])); for Pr in PrList do pds := pdsolve(PDE(Pr), IBC, numeric); Plots[Pr] := pds:-plot[display](u(y, t), t = .5, linestyle = "solid", colour = Colours[Pr], legend = sprintf("Pr=%2.2f", Pr), title = "Velocity Profile", labels = ["y", "theta"]) end do; plots:-display([seq(Plots[Pr], `in`(Pr, PrList))])

Download pde_solutionalpha.mw

Solving a matrix equation ...

Asked by: Math Pi Euler 30

November 08 2017

1 2

Dear All,

I'm stuck in a question, perhaps somebody can help. I have to solve the following equation:

restart;
with(LinearAlgebra);
A := Matrix(3, 3, [[1, 3, 2], [4, 5, 1], [3, 7, 2]]);

M := IdentityMatrix(3);

eq := A^2*a+A*b+M*c;
K:= A^-2
Set := {seq(eq[i] = K[i], i = 1 .. nops(eq))};

But now I want to solve for a, b and c. I tried this with the solve function, but I gives an error. What do I have to do now?

Math

Quantum Commutation Rules Basics

by: ecterrab 14540 Product: Maple

November 08 2017

11 0

Quantum Commutation Rules Basics

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab²

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

In Quantum Mechanics, in the coordinates representation, the component of the momentum operator along the x axis is given by the differential operator

The purpose of the exercises below is thus to derive the commutation rules, in the coordinates representation, between an arbitrary function of the coordinates and the related momentum, departing from the differential representation

These two exercises illustrate how to have full control of the computational process by using different elements of the Maple language, including inert representations of abstract vectorial differential operators, Hermitian operators, algebra rules, etc.

These exercises also illustrate a new feature of the Physics package, introduced in Maple 2017, that is getting refined (the computation below requires the Maplesoft updates of the Physics package) which is the ability to perform computations algebraically, using the product operator, but with differential operators, and transform the products into the application of the operators only when we want that, as we do with paper and pencil.

>

Start setting the problem:

–	all of are Hermitian operators

–	all of commute between each other

–	tell the system only that the operators are the differentiation variables of the corresponding (differential) operators but do not tell what is the form of the operators

>	$Setup(mathematicalnotation = true, differentialoperators = {[p_, [x, y, z]]}, hermitianoperators = {p, x, y, z}, algebrarules = {%Commutator(x, y) = 0, %Commutator(x, z) = 0, %Commutator(y, z) = 0}, quiet)$

$[algebrarules = {%Commutator(x, y) = 0, %Commutator(x, z) = 0, %Commutator(y, z) = 0}, differentialoperators = {[p_, [x, y, z]]}, hermitianoperators = {p, x, y, z}, mathematicalnotation = true]$

(1.1)

Assuming is a smooth function, the idea is to apply the commutator to an arbitrary ket of the Hilbert space , perform the operation explicitly after setting a differential operator representation for , and from there get the commutation rule between and .

Start introducing the commutator, to proceed with full control of the operations we use the inert form %Commutator

>

(1.2)

>

(1.3)

For illustration purposes only (not necessary), expand this commutator

>

(1.4)

Note that , and the ket are operands in the products above and that they do not commute: we indicated that the coordinates are the differentiation variables of . This emulates what we do when computing with these operators with paper and pencil, where we represent the application of a differential operator as a product operation.

This representation can be transformed into the (traditional in computer algebra) application of the differential operator when desired, as follows:

>

(1.5)

Note that, in , the application of is not expanded: at this point nothing is known about , it is not necessarily a linear operator. In the Quantum Mechanics problem at hands, however, it is. So give now the operator an explicit representation as a linear vectorial differential operator (we use the inert form %Nabla, , to be able to proceed with full control one step at a time)

>

(1.6)

The expression (1.5) becomes

>

(1.7)

Activate now the inert operator and simplify taking into account the algebra rules for the coordinate operators

>

(1.8)

To make explicit the gradient in disguise on the right-hand side, factor out the arbitrary ket

>

(1.9)

Combine now the expanded gradient into its inert (not-expanded) form

>

(1.10)

Since (1.10) is true for all, this ket can be removed from both sides of the equation. One can do that either taking coefficients (see Coefficients ) or multiplying by the "formal inverse" of this ket, arriving at the (expected) form of the commutation rule between and

>

(1.11)

Tensor notation, "[`X__m`,P[n]][-]=i `&hbar;` g[m,n]"

The computation rule for position and momentum, this time in tensor notation, is performed in the same way, just that, additionally, specify that the space indices to be used are lowercase latin letters, and set the relationship between the differential operators and the coordinates directly using tensor notation.

You can also specify that the metric is Euclidean, but that is not necessary: the default metric of the Physics package, a Minkowski spacetime, includes a 3D subspace that is Euclidean, and the default signature, (- - - +), is not a problem regarding this computation.

>

Setup(mathematicalnotation = true, coordinates = cartesian, spaceindices = lowercaselatin, algebrarules = {%Commutator(x, y) = 0, %Commutator(x, z) = 0, %Commutator(y, z) = 0}, hermitianoperators = {P, X, p}, differentialoperators = {[P[m], [x, y, z]]}, quiet)

[algebrarules = {%Commutator(x, y) = 0, %Commutator(x, z) = 0, %Commutator(y, z) = 0}, coordinatesystems = {X}, differentialoperators = {[P[m], [x, y, z]]}, hermitianoperators = {P, p, t, x, y, z}, mathematicalnotation = true, spaceindices = lowercaselatin]

(2.1)

Define now the tensor

>

${Physics:-Dgamma[mu], P[m], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(2.2)

Introduce now the Commutator, this time in active form, to show how to reobtain the non-expanded form at the end by resorting the operands in products

>

(2.3)

Expand first (not necessary) to see how the operator is going to be applied

>

(2.4)

Now expand and directly apply in one ago the differential operator

>

(2.5)

Introducing the explicit differential operator representation for , here again using the inert to keep control of the computations step by step

>

(2.6)

The expanded and applied commutator (2.5) becomes

>

(2.7)

Activate now the inert operators and simplify taking into account Einstein's rule for repeated indices

>

(2.8)

Since the ket is arbitrary, we can take coefficients (or multiply by the formal Inverse of this ket as done in the previous section). For illustration purposes, we use Coefficients and note hwo it automatically expands the commutator

>

(2.9)

One can undo this (frequently undesired) expansion of the commutator by sorting the products on the left-hand side using the commutator between and

>

(2.10)

And that is the result we wanted to compute.

Additionally, to see this rule in matrix form,

>

Matrix(%id = 18446744078261558678)

(2.11)

In the above, we use equation (2.10) multiplied by -1 to avoid a minus sign in all the elements of (2.11), due to having worked with the default signature (- - - +); this minus sign is not necessary if in the Setup at the beginning one also sets

For display purposes, to see this matrix expressed in terms of the geometrical components of the momentum , redefine the tensor explicitly indicating its Cartesian components

>

${Physics:-Dgamma[mu], P[m], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(2.12)

>

Matrix(%id = 18446744078575996430)

(2.13)

Finally, in a typical situation, these commutation rules are to be taken into account in further computations, and for that purpose they can be added to the setup via

>

$[algebrarules = {%Commutator(x, p__x) = I*`&hbar;`, %Commutator(x, p__y) = 0, %Commutator(x, p__z) = 0, %Commutator(x, y) = 0, %Commutator(x, z) = 0, %Commutator(y, p__x) = 0, %Commutator(y, p__y) = I*`&hbar;`, %Commutator(y, p__z) = 0, %Commutator(y, z) = 0, %Commutator(z, p__x) = 0, %Commutator(z, p__y) = 0, %Commutator(z, p__z) = I*`&hbar;`}]$

(2.14)

For example, from herein computations are performed taking into account that

>

(2.15)

>

Download DifferentialOperatorCommutatorRules.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

isolate expresion where variables are matrices...

Asked by: jmr115 15

November 08 2017

3 3

Hello,

I'm trying to find the way to manipulate matrix equalities easily and isolate them as variables like if they were simple real variables but keeping the matrix properties. However, I couldn't find the way to do it efficiently without describing all the parameters of the matrices. I'd like to keep them as variables rather than perform matrix operations when defining the expressions. I've attached a snapshot which describes my problem much better:pdf_kmaple.pdf.

For instance, I'd like to be able to isolate the expression that defines the state-space when I change the discretization method.

I'm wondering if there's a generic way I can use to isolate other matrix systems as well.

Thank you very much for your help!

JMarc

function call does not work...

Asked by: digerdiga 390

November 08 2017

0 3

fgh := proc (x) options operator, arrow; [x, x^2, x^3] end proc;

sol := proc (x) options operator, arrow; min(`~`[Re](select(proc (t) options operator, arrow; is(abs(Im(t)) < 0.1e-19) and is(0 < Re(t)) end proc, fgh(x)))) end proc;

eval('sol(x)', [x = X]);

eval(%, X = 1)

Why does this not work?

Trouble entering a partial differential equation i...

Asked by: trb7074 10

November 08 2017

2 2

Good morning,

I've been having some trouble entering this expression in Maple 17.

My teacher suggested using this format, however, this format is from 2004 and certainly isn't recognized by the current version of Maple, as it keeps spitting back 0=0 no matter how many different ways I try to enter it.

Could anyone provide feedback on the correct syntax of how to enter this expression? Thank you very much for your time.

The command " interface(typesetting = extended)" d...

Asked by: eager0626 55

November 07 2017

2 1

Hello,

I work in maple 2017，but the "interface "command seems doesn't work. But it works in the old version(maple 18). Does anyone know what problem with it?

help the calculation...

Asked by: Wilson Eduardo CAMACHO MAMANI 20

November 07 2017

3 5

problems with Norm.

Download force_net.mw

The OEIS Package

by: Daniel Skoog 1771 Product: Maple 2017

November 07 2017

5 0

The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. Originally founded by Neil Sloane in 1964, OEIS has evolved into a larger community based project that today contains information records on over 280,000 sequences. Each record contains information on the leading terms of the sequence, keywords, mathematical motivations, literature links, and more - there’s even Maple code to reproduce many of the sequences!

You can read more about the OEIS project on the main site, or on Wikipedia. There have also been several articles written about the OEIS project; here’s a (somewhat) recent article on Building a Searching Engine for Mathematics that gives a little more history on the project.

I wrote a simple package for Maple that can be used to search OEIS for a sequence of numbers or a string and retrieve information on various integer sequences. You can interact with the OEIS package using the commands: Get, Search or Print, or using the right-click context menu.

Here are some examples:

with(OEIS):

Search for a sequence of 6 or more numbers (note that you can also just right-click on an integer sequence to run the search):

Search(0, 1, 1, 2, 4, 9, 20);

6 results found
81, 255636, 35084, 58386, 5908, 14267

Retrieve information on the integer sequence A000081. By default, this returns the data as a JSON string and stores the results in a table:

results81 := Get(81):
indices(results81);

["revision"], ["number"], ["formula"], ["offset"], ["example"], ["keyword"], ["id"], ["xref"], ["data"], ["reference"], ["mathematica"], ["maple"], ["created"], ["comment"], ["references"], ["time"], ["link"], ["author"], ["program"], ["name"]

Entries in the table can be accessed as follows:

results81["author"];

"_N. J. A. Sloane_"

I’ve also added some functionality for printing tables containing information on integer sequences. The Print command displays an embedded table containing all or selected output from records. For example, say we search for a sequence of numbers:

Search(0, 1, 1, 2, 4, 9, 20);

6 results found
81, 255636, 35084, 58386, 5908, 14267

Let’s print out selected details on the first three of these sequences (including Maple code if available):

Print([81, 255636, 35084], output = ["author", "name", "data", "maple"]);

The OEIS API is somewhat limited, so I have had to put some restrictions in place for returning results. Any query that returns more than 100 entries will return an error and a message to provide a more precise specification for the query. Also, in order to reduce the number of search results, the Search command requires at least 6 integers.

You can install the OEIS package directly from the MapleCloud or by running:

PackageTools:-Install(5693538177122304);

Comments and feedback are welcome here, or on the project page: https://github.com/dskoog/Maple-OEIS

pde solve in maple...

Asked by: radhika 30

November 07 2017

0 1

restart; with(PDEtools);
U := diff_table(Psi(`ψ_2`, `ψ_1`));
pde := `ψ_2`*(diff(Psi(`ψ_2`, `ψ_1`), `ψ_1`))+`ψ_1`*U[`ψ_1`]+2*U[] = 0;
ψ_2 Psi(ψ_2 + 1, ψ_1)

+ ψ_1 Psi(ψ_2 + 1, ψ_1) + 2 Psi(ψ_2, ψ_1) =

0
pdsolve(pde);
Error, (in pdsolve/info) first argument is not a differential equation

How can i solve it?

How do I find higher eigenvalues for an ODE BVP ...

Asked by: markweitzman 110

November 07 2017

2 12

My Maple worksheet is attached below - where I am solving two quantum mechanics problems, trying to find the eigenvalues for certain potentials. Using a previous post in Mapleprime (https://www.mapleprimes.com/questions/221629-Can-Maple-Find-Solution-To-Eigenvalue), I saw how to get the lowest eigenvalue. My question is how using numeric solutions, I can get the higher eigenvalues and corresponding solutions. Is there a way to specify that e (the eigenvalue) has to be in a certain range. I tried to specify assumptions (e>0.6), but that didn't work. I know that for the second problem (using the shooting method) that the 4 lowest eigenvalues are 0.184358, 0.70747, 1.2065, 1.5625. Thank you very much for you consideration.

ODE_BVP_eigenvalues.mw

Why does solve not find the relative length in thi...

Asked by: Earl 985

November 07 2017

1 6

Why does solve not find the value of xp = sqrt(1-v^2) in this worksheet?

SpecialRelativity.mw

Reference: Special Relativity and Classical Field Theory. Authors: Leonard Susskind and Art Friedman

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