Product Tips & Techniques

2D Input operator assignment syntax

by: acer 32293 Product: Maple

September 16 2020

5 11

Caution, certain kinds of earlier input can affect the results from using the 2D Input syntax for operator assignment.

>

The following now produces a remember-table assignment,
instead of assigning a procedure to name f, even though by default
Typesetting:-Settings(functionassign)
is set to true.

>

With the previous line of code commented-out the following
line assigns a procedure to name f, as expected.

If you uncomment the previous line, and re-execute the whole
worksheet using !!! from the menubar, then the following will
do a remember-table assignment instead.

>

Download operator_assignment_2dmath.mw

Maple 2020.1.1 update

by: eithne 2072 Product: Maple 2020

August 26 2020

1 11

We have just released an update to Maple, Maple 2020.1.1. This update includes the following:

Correction to a problem that occurred when printing or exporting documents to PDF. If the document included a 3-D plot, nearby text was sometimes missing from the printed/exported document.
Correction to an issue that prevented users from installing between-release updates to the Physics package

This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2020.1.1 download page. If you are a MapleSim user, you can obtain this update from the corresponding MapleSim menu or MapleSim 2020.1.1 download page.

In particular, please note that this update fixes the problems reported on MaplePrimes in Maple 2020.1 issue with print to PDF and installing the Maplesoft Physics updates. As always, we appreciate the feedback.

Circumscribed ellipse

by: one man 1355 Product: Maple 17

July 31 2020

0 2

One way to find the equation of an ellipse circumscribed around a triangle. In this case, we solve a linear system of equations, which is obtained after fixing the values of two variables ( t1 and t2). These are five equations: three equations of the second-order curve at three vertices of the triangle and two equations of a linear combination of the coordinates of the gradient of the curve equation.
The solving of system takes place in the ELS procedure. When solving, hyperboles appear, so the program has a filter. The filter passes the equations of ellipses based on by checking the values of the invariants of the second-order curves.
FOR_ELL_ТR_OUT_PROCE_F.mw ( Fixed comments in the text 01, 08, 2020)

Creating Help - using HelpTools

by: dharr 8200 Product: Maple

July 19 2020

5 3

@ianmccr posted here about making help for a package using makehelp. Here I show how to do this with the HelpTools package.

The attached worksheet shows how to create the help database for the Orbitals package available at the Applications Center or in the Maple Cloud. The help pages were created as worksheets - start using an existing help page as a model - use View/Open Page As Worksheet and then save from there. The topics and other information are entered by adding Attributes under File/Document Properties - for example for the realY help page these are:

Active=false means a regular help page; Active=true means an example worksheet.

There may be several aliases, for example the cartesion help page also describes the fullcartesian command and so Alias is: Orbitals[fullcartesian],cartesian,fullcartesian and Keyword is: Orbitals,cartesian,fullcartesian

Once a worksheet is created for each help page they are assembled into the help database with the attached file. More information is in the attached file

Orbitals_Make_help_database.mw

Creating Help - An Exploration of makehelp

by: ianmccr 170 Product: Maple 2019

July 18 2020

3 0

MacDude posted a worksheet for creating help files using the makehelp command that I found very useful. However, his worksheet created a table of contents that organized the command help pages under a single folder. I wanted to create a help file table of contents with the commands organized into sub-folders under the main application folder. ie. Package Name,Folder Name, command. The help page for makehelp is not very informative; in particular the example that shows (purportedly) how to override the existing help pages with your own help file is very misleading. Also, the description of the parameters to the browser option of the makehelp command is too vague. I needed an error message to tell me that the browser option expects parameters in the form List(Name,String). In the end, I was able to get a folder/sub-folder structure using a structure [`Folder Name`,`Subfolder Name`,"Command"]. Sub-folders are sorted alphabetically. If anyone has a need, the attached worksheet shows how I created the table of contents structure.

helpcode.mw

Plot styling - experimenting with Maple's plotting...

by: Samir Khan 2116 Product: Maple

July 13 2020

11 4

I like tweaking plots to get the look and feel I want, and luckily Maple has many plotting options that I often play with. Here, I visualize the same data several times, but each time with different styling.

First, some data.

restart:
data_1 := [[0,0],[1,2],[2,1.3],[3,6]]:
data_2 := [[0.5,3],[1,1],[2,5],[3,2]]:
data_3 := [[-0.5,3],[1.3,1],[2.5,5],[4.5,2]]:

This is the default look.

plot([data_1, data_2, data_3])

I think the darker background on this plot makes it easier to look at.

gray_grid :=
 background      = "LightGrey"
,color           = [ ColorTools:-Color("RGB",[150/255, 40 /255, 27 /255])
                    ,ColorTools:-Color("RGB",[0  /255, 0  /255, 0  /255])
                    ,ColorTools:-Color("RGB",[68 /255, 108/255, 179/255]) ]
,axes            = frame
,axis[2]         = [color = black, gridlines = [10, thickness = 1, color = ColorTools:-Color("RGB", [1, 1, 1])]]
,axis[1]         = [color = black, gridlines = [10, thickness = 1, color = ColorTools:-Color("RGB", [1, 1, 1])]]
,axesfont        = [Arial]
,labelfont       = [Arial]
,size            = [400*1.78, 400]
,labeldirections = [horizontal, vertical]
,filled          = false
,transparency    = 0
,thickness       = 5
,style           = line:

plot([data_1, data_2, data_3], gray_grid);

I call the next style Excel, for obvious reasons.

excel :=
 background      = white
,color           = [ ColorTools:-Color("RGB",[79/255,  129/255, 189/255])
                    ,ColorTools:-Color("RGB",[192/255, 80/255,   77/255])
                    ,ColorTools:-Color("RGB",[155/255, 187/255,  89/255])]
,axes            = frame
,axis[2]         = [gridlines = [10, thickness = 0, color = ColorTools:-Color("RGB",[134/255,134/255,134/255])]]
,font            = [Calibri]
,labelfont       = [Calibri]
,size            = [400*1.78, 400]
,labeldirections = [horizontal, vertical]
,transparency    = 0
,thickness       = 3
,style           = point
,symbol          = [soliddiamond, solidbox, solidcircle]
,symbolsize      = 15:

plot([data_1, data_2, data_3], excel)

This style makes the plot look a bit like the oscilloscope I have in my garage.

dark_gridlines :=
 background      = ColorTools:-Color("RGB",[0,0,0])
,color           = white
,axes            = frame
,linestyle       = [solid, dash, dashdot]
,axis            = [gridlines = [10, linestyle = dot, color = ColorTools:-Color("RGB",[0.5, 0.5, 0.5])]]
,font            = [Arial]
,size            = [400*1.78, 400]:

plot([data_1, data_2, data_3], dark_gridlines);

The colors in the next style remind me of an Autumn morning.

autumnal :=
 background      =  ColorTools:-Color("RGB",[236/255, 240/255, 241/255])
,color           = [  ColorTools:-Color("RGB",[144/255, 54/255, 24/255])
                     ,ColorTools:-Color("RGB",[105/255, 108/255, 51/255])
                     ,ColorTools:-Color("RGB",[131/255, 112/255, 82/255]) ]
,axes            = frame
,font            = [Arial]
,size            = [400*1.78, 400]
,filled          = true
,axis[2]         = [gridlines = [10, thickness = 1, color = white]]
,axis[1]         = [gridlines = [10, thickness = 1, color = white]]
,symbol          = solidcircle
,style           = point
,transparency    = [0.6, 0.4, 0.2]:

plot([data_1, data_2, data_3], autumnal);

In honor of a friend and ex-colleague, I call this style "The Swedish".

swedish :=
 background      = ColorTools:-Color("RGB", [0/255, 107/255, 168/255])
,color           = [ ColorTools:-Color("RGB",[169/255, 158/255, 112/255])
                    ,ColorTools:-Color("RGB",[126/255,  24/255,   9/255])
                    ,ColorTools:-Color("RGB",[254/255, 205/255,   0/255])]
,axes            = frame
,axis            = [gridlines = [10, color = ColorTools:-Color("RGB",[134/255,134/255,134/255])]]
,font            = [Arial]
,size            = [400*1.78, 400]
,labeldirections = [horizontal, vertical]
,filled          = false
,thickness       = 10:

plot([data_1, data_2, data_3], swedish);

This looks like a plot from a journal article.

experimental_data_mono :=

background       = white
,color           = black
,axes            = box
,axis            = [gridlines = [linestyle = dot, color = ColorTools:-Color("RGB",[0.5, 0.5, 0.5])]]
,font            = [Arial, 11]
,legendstyle     = [font = [Arial, 11]]
,size            = [400, 400]
,labeldirections = [horizontal, vertical]
,style           = point
,symbol          = [solidcircle, solidbox, soliddiamond]
,symbolsize      = [15,15,20]:

plot([data_1, data_2, data_3], experimental_data_mono, legend = ["Annihilation", "Authority", "Acceptance"]);

If you're willing to tinker a little bit, you can add some real character and personality to your visualizations. Try it!

I'd also be very interested to learn what you find attractive in a plot - please do let me know.

Maple 2020.1 update

by: eithne 2072 Product: Maple

June 15 2020

7 0

We have just released an update to Maple, Maple 2020.1.

Maple 2020.1 includes corrections and improvement to the mathematics engine, export to PDF, MATLAB connectivity, support for Ubuntu 20.04, and more. We recommend that all Maple 2020 users install these updates.

This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2020.1 download page, where you can also find more details.

In particular, please note that this update includes a fix to the SMTLIB problem reported on MaplePrimes. Thanks for the feedback!

MapleSim 2020 is now available!

by: CTurnbull 170 Product: MapleSim 2020

June 11 2020

6 0

We’re excited to announce a new version of MapleSim! The MapleSim 2020 family of products lets you build and test models faster than ever, including faster simulations, powerful new tools for machine builders, and expanded modeling capabilities. Improvements include:

Faster results, with more efficient models, faster simulations, and more powerful design tools.
Powerful new features for machine builders, with new components, improved visualizations, and automation-focused connectivity tools that make it faster than ever to build and test digital twins.
Improved modeling capabilities, with an extensive collection of updates to components, libraries, and analysis tools.
More realistic machine visualizations with an expanded Kinematic Cam Generation App.
New product: MapleSim Insight, giving machine builders powerful, simulation-based debugging and 3-D visualization capabilities that connect directly to common automation platforms.
New add-on library: MapleSim Ropes and Pulleys Library for the easy creation of winch and pulley systems as part of your machine development.

See What’s New in MapleSim 2020 for more information about these and other improvements!

Scalable Subplots within Figure

by: Bland3 30 Product: Maple 2019

May 29 2020

0 3

This is my attempt to produce a subplot within a larger plot for magnifying data in a small region, and putting that subplot into the white space of the figure.
Based on the questions: How to insert a plot into another plot? and Inset figure in Maple, I wrote a couple of procedures that create sub-plots and allow the user to place the subplot window as he/she chooses. This avoids the graininess issues mentioned by acer in the second link (and experienced by me).

So far, I only have this completed for point plots, but using acer's method of piecewise functions posted in the plotin2b.mw of the second article, with the subplot function being defined only if it satisfies your conditions, would allow the subplot generating procedure to be generalized easily enough. But the data I'm working with all point plots, so that's the example here.

The basic idea is to use one procedure to create boxes, make tickmarks on the expanded region, and make tickmark labels, combine all of those into one graph. Then create scaled and shifted versions of the data series, then make graphs of those. Lastly, combine them all into one picture.

Hope this helps someone who has to do the same.

Mapleprimes isn't inserting the contents, but here is the download of the file: SubPlotBoxesandVectorDataSeries.mw

The equations of motion in curvilinear coordinates,...

by: ecterrab 14540 Product: Maple

May 21 2020

5 0

The equations of motion in curvilinear coordinates, tensor notation and Coriolis force

The formulation of the equations of motion of a particle is simple in Cartesian coordinates using vector notation. However, depending on the problem, for example when describing the motion of a particle as seen from a non-inertial system of references (e.g. a rotating planet, like earth), there is advantage in using curvilinear coordinates and also tensor notation. When the particle's movement is observed from such a rotating referential, we also see "acceleration" that is not due to any force but to the fact that the referential itself is accelerated. The material below discusses and formulates these topics, and derives the expression for the Coriolis and centripetal force in cylindrical coordinates as seen from a rotating system of references.

The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.681 or newer.

Vector notation

Generally speaking the equations of motion of a particle are easy to formulate: the position vector is a function of time, the velocity is its first derivative and the acceleration is its second derivative. For instance, in Cartesian coordinates

>

(1)

>

(2)

>

(3)

Newton's 2^nd law, that in an inertial system of references when there is force there is acceleration and viceversa, is then given by

>

(4)

where represents the total force acting on the particle. This vectorial equation represents three second order differential equations which, for given initial conditions, can be integrated to arrive at a closed form expression for as a function of .

Tensor notation

In Cartesian coordinates, the tensorial form of the equations (4) is also straightforward. In a flat spacetime - Galilean system of references - the three space coordinates form a 3D tensor, and so does its first derivate and the second one. Set the spacetime to be 3-dimensional and Euclidean and use lowercaselatin indices for the corresponding tensors

>

Physics:-g_[mu, nu] = Matrix(%id = 18446744078329083054)

(5)

The position, velocity and acceleration vectors are expressed in tensor notation as done in (1), (2) and (3)

>

(6)

>

(7)

>

(8)

Setting a tensor to represent the three Cartesian components of the force

>

${Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(9)

Newton's 2^nd law (4), now expressed in tensorial notation, is given by

>

(10)

The three differential equations behind this tensorial form of (4) are as expected

>

${F__x(t) = m*(diff(diff(x(t), t), t)), F__y(t) = m*(diff(diff(y(t), t), t)), F__z(t) = m*(diff(diff(z(t), t), t))}$

(11)

Things are straightforward in Cartesian coordinates because the components of the line element are exactly the components of the tensor

>

(12)

and so, the form factors (see related Mapleprimes post) are all equal to 1.

In the case of no external forces, and the equations of motion, whose solution are the trajectory, can be formulated as the path of minimal length between two points, a geodesic. In the case under consideration, because the spacetime is flat (Galilean) those two points lie on a plane, we are talking about Euclidean geometry, that information is encoded in the metric (the 3x3 identity matrix (5)), and the geodesic is a straight line. The differential equations of this geodesic are thus the equations of motion (11) with , and can be computed using Geodesics

>

(13)

Curvilinear coordinates

Vector notation

The form of these equations in the case of curvilinear coordinates, for example in cylindrical or spherical variables, is obtained performing a change of coordinates.

>

(14)

This change keeps the z axis unchanged, so the corresponding unit vector remains unchanged.

Changing the basis and coordinates used to represent the position vector , it becomes

>

(15)

where since in (1) the coordinates () depend on t, in (15), not just and but also the unit vector depends on t. The velocity is computed as usual, differentiating

>

(16)

The second term is due to the dependency of on the coordinate together with the chain rule diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`(t), t) = (diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) and (diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) = `#mover(mi("φ",fontstyle = "normal"),mo("&and;"))`(t)*(diff(phi(t), t)) . The dependency of curvilinear unit vectors on the coordinates is automatically taken into account when differentiating due to the Setup setting geometricdifferentiation = true.

For the acceleration,

>

(17)

where the term involving comes from differentiating in (16) taking into account the dependency of on the coordinate This result can also be obtained by directly changing variables in the acceleration , in equation (3)

>

(18)

Newton's 2^nd law becomes

>

(19)

In the absence of external forces, equating to 0 the vector components (coefficients of the unit vectors) of the acceleration we get the system of differential equations in cylindrical coordinates whose solution is the trajectory of the particle expressed in the (

>

$`~`[`=`]({coeffs(rhs(diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k), [`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("φ",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("k"),mo("&and;"))`])}, 0)$

${2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}$

(20)

>	$solve({2(diff(rho(t), t))(diff(phi(t), t))+rho(t)(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}, {diff(phi(t), t, t), diff(rho(t), t, t), diff(z(t), t, t)})$

${diff(diff(phi(t), t), t) = -2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t), diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2, diff(diff(z(t), t), t) = 0}$

(21)

In this formulation (21) with , although , no acceleration in the direction, is naturally expected, the same cannot be said about the other two equations for and . Those two equations are discussed below under Coriolis and Centripetal forces. The key observation at this point, however, is that the right-hand sides of both unexpected equations involve , rotation around the z axis.

Tensor notation

The same equations (19) and (21) result when using tensor notation. For that purpose, one can transform the position, velocity and acceleration tensors (6), (7), (8), but since they are expressed as functions of a parameter (the time), it is simpler to transform only the underlying metric using TransformCoordinates. That has the advantage that all the geometrical subtleties of curvilinear coordinates, like scale factors and dependency of unit vectors on curvilinear coordinates, get automatically, very succinctly, encoded in the metric:

>

Physics:-g_[a, b] = Matrix(%id = 18446744078263848958)

(22)

The computation of geodesics assumes that the coordinates () depend on a parameter. That parameter is passed as the first argument to Geodesics

>

(23)

These equations of motion (23) are the same as the equations (21) computed using standard vector notation, differentiating and taking into account the dependency of curvilinear unit vectors on the curvilinear coordinates in (16) and (17). One of the interesting features of computing with tensors is, as said, that all those geometrical algebraic subtleties of curvilinear coordinates are automatically encoded in the metric (22).

To understand how are the geodesic equations computed in one go in (23), one can perform the calculation in steps:

1.	Make be a function of directly in the metric

2.	Compute - not the final form of the equations (23) - but the intermediate form expressing the geodesic equation using tensor notation, in terms of Christoffel symbols

3.	Compute the components of that tensorial equation for the geodesic (using TensorArray)

For step 1, we have

>

Physics:-g_[a, b] = Matrix(%id = 18446744078354237910)

(24)

Set this metric where

>

Physics:-g_[a, b] = Matrix(%id = 18446744078342604430)

(25)

Step 2, the geodesic equations in tensor notation with the coordinates depending on the time t are computed passing the optional argument tensornotation

>

(26)

Step 3: compute the components of this tensorial equation

>

(27)

These are the same equations (23).

Having the tensorial equation (26) is also useful to formulate the equations of motion in tensorial form in the presence of force. For that purpose, redefine the contravariant tensor to represent the force in the cylindrical basis

>

${Physics:-D_[a], Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-Ricci[a, b], Physics:-Riemann[a, b, c, d], Physics:-Weyl[a, b, c, d], Physics:-d_[a], Physics:-g_[a, b], Physics:-Christoffel[a, b, c], Physics:-Einstein[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(28)

Newton's 2^nd law (19)

>

(29)

now using tensorial notation, becomes

>

(30)

>

Array(%id = 18446744078329063774)

(31)

where we recall (see related Mapleprimes post) that to obtain the vector components entering from these tensor components we need to multiply the latter by the scale factors (), the component of in the direction of is given by .

Coriolis force and centripetal force

After changing variables the position vector of the particle got expressed in (15) as

A distinction needs to be made here, according to whether the unit vector depends or not on the time , the former being the general case. When is a constant, the value of the coordinate - the angle between and the x axis - does not change, there is no rotation around the z axis. On the other hand, when , the orientation of and so the coordinate changes with time, so either the force acting on the particle has a component in the direction that produces rotation around the z axis, or the system of references - itself - is rotating around the z axis.

Likewise, the expression (15) can represent the position vector measured in the original Galilean (inertial) system of references, where a force is producing rotation around the z axis, or it can represent the position of the particle measured in a rotating, non-inertial system references. Hence the transformation (14) can also be interpreted in two different ways, as representing a choice of different functions (generalized coordinates) to represent the position of the particle in the original inertial system of references, or it can represent a transformation from an inertial to another rotating, non-inertial, system of references.

This equivalence between the trajectory of a particle subject to an external force, as observed in an inertial system of references, and the same trajectory observed from a non-inertial accelerated system of references where there is no external force, actually at the root of the formulation of general relativity, is also well known in classical mechanics. The (apparent) forces observed in the rotating non-inertial system of references, due only to its acceleration, are called Coriolis and centripetal forces.

To see that the equations

diff(rho(t), t, t) = (diff(phi(t), t))^2*rho(t), diff(phi(t), t, t) = -2*(diff(phi(t), t))*(diff(rho(t), t))/rho(t)

that appeared in (27) when in the inertial system of references , are related to the Coriolis and centripetal forces in the non-inertial referencial, following [1] introduce a vector representing the rotation of that referencial around the z axis (when, in the inertial system of references, the non-inertial system rotates clockwise, in the non-inertial system increases in value in the anti-clockwise direction)

>

(32)

According to [1], (39.7), the acceleration in the inertial system is expressed in terms of the quantities in the non-inertial rotating system by the sum of the following three vectorial terms.

First, the components of the acceleration measured in the non-inertial system are given by the second derivatives of the coordinates () multiplied by the scale factors, which in this case are given by () (see this post in Mapleprimes)

>

(33)

Second, the Coriolis force divided by the mass, by definition given by

>

(34)

Third the centripetal force divided by the mass, defined by

>

(35)

Adding these three terms,

>

a_(t)+2*Physics:-Vectors:-`&x`(diff(r_(t), t), omega_)+Physics:-Vectors:-`&x`(omega_, Physics:-Vectors:-`&x`(r_(t), omega_)) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

(36)

So that

>

(37)

and where the right-hand side of (36) is, actually, the result computed lines above in (18)

>

(38)

>

(39)

From (37), in the absence of external forces and so the acceleration measured in the rotating system is given by the sum of the Coriolis and centripetal accelerations

>

(40)

In other words: in the absence of external forces, the acceleration of a particle observed in a rotating (non-inertial) system of references is not zero.

Expressing this equation (40) in terms of () we get

>

(41)

resulting in the three equations

>

(42)

>

(43)

>

(44)

which are the equations returned by Geodesics in (23)

>

(45)

>

References

[1] L.D. Landau, E.M. Lifchitz, Mechanics, Course of Theoretical Physics, Volume 1, third edition, Elsevier.

Download The_equations_of_motion_in_curvilinear_coordinates.mw

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

How to create an interactive Math App in Maple

by: TechnicalSupport 790 Product: Maple 2020

May 13 2020

9 2

Since we are getting many questions on how to create Math apps to add to the Maple Cloud. I wanted to go over the different GUI aspects of how you go about creating a Math App in Maple. The following Document also includes some code examples that are used in the the Math App but doesn't go into them in detail. For more details on the type of coding you do in a Math App see the DocumentTools package help page.

Some of the graphical features of the Math app don't display on Maple Primes so I'd recommend downloading this worksheet from here: HowToMathApp.mw to follow along.

How to make a Math App (An example of using the Document Tools).

This Document will provide a beginners guide on one way to make a Math app in Maple.

It will contain some coding examples as well as where to find different options in the user interface.

Step 1 Insert a Table

•	When making a Math App in Maple I often start with a table. You can enter a table by going to Insert > Table...

•	I often make the table 1 x 2 to start with as this gives an area for input and an area for the output (such as plots).

Add a plot component to one of the cells of the table

•	From the Components Palette you can add a Plot Component . Add it to the table by clicking and dragging it over.

Add another table inside the other cell

•	In the other cell of the table I'll add another table to organize my use of buttons, sliders, and other components.

Add some components to the new table

•	From the Components Palette I'll add a slider, or dial, or something else for interaction.

•	You may also want a Math region for an area to enter functions and a button to tell Maple to do something with it.

Arrange the Components to look nice

•	You can change how the components are placed either by resizing the tables or changing the text orientation of the contents of the cells.

Write some code for the interaction of the buttons.

•	Using the DocumentTools package there are lots of ways you can use the components. I often will start writing my code using a code edit region as it provides better visualization for syntax. On MaplePrimes these display as collapsed so I will also include code blocks for the code.

Let's write something that takes the value of the slider and applies it to the dial

•	Note that the names of the components will change in each section as they are copies of the previous section.

with(DocumentTools):

with(DocumentTools):
sv:=GetProperty('Slider2',value);
SetProperty('Dial2',value,sv);

•	This code will only execute when run using the button. Change the value of the slider below then run the code above to see what happens.

Move the code 'inside' the slider

•	Instead of putting the code inside the code edit region where it needs to be executed, we'll next add the code to the value changed code of the slider.

•	Right click the Slider then select "Edit Value Changed Code".

•	This will open the code editor for the Slider

•	Enter your code (ensuring you're using the correct name for the slider and dial).

•	Notice that you don't need to use the with(DocumentTools): command as "use DocumentTools in ... end use;" is already filled in for you.

•	Save the code in the Slider and hit the button inside it once.

•	Now move the slider.

•	On future uses of the App you won't need to hit as the code will be run on startup.

Add some more details to your App

•	Let's make this app do something a bit more interesting than change the contents of a dial when a slider moves.

•	The plan in the next few steps is to make this app allow a user to explore parameters changing in a sinusoidal expression.

•	I'm going to add a second Math Component, put the expression into both then uncheck the box in the context panel that says "Editable".

•	To make the Math containers fit nicely I'll check the Auto-fit container box and set the Minimum Width Pixels to 200.

Add code to change the value of phi in the second Math Container when the Slider changes

Note: Maple uses Radians for trigonometric functions so we should convert the value of phi to Radians.

use DocumentTools in

use DocumentTools in 
phi_s:=GetProperty(Slider5,value);
expr:= GetProperty(MathContainer6,expression);
new_expr:=algsubs(phi=phi_s*Pi/180,expr);

SetProperty(MathContainer7,expression,new_expr);
end use:

Make the Dial go from 0 to 360°

•	Click the Dial and look at the options in the context panel on the right.

•	Update the values in the Dial so that the highest position is 360 and the spacing makes sense for the app.

Have the Dial update the theta value of the expression

•	Add the following code to the Dial

use DocumentTools in

use DocumentTools in 
theta_d:=GetProperty(Dial7,value);
phi_s:=GetProperty(Slider7,value); #This is added so that phi also has the value updated

expr:= GetProperty(MathContainer10,expression);
new_expr0:=algsubs(theta=theta_d*Pi/180,expr);
new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0);  #This is added so that phi also has the value updated

SetProperty(MathContainer11,expression,new_expr);
end use:

•	Update the value in the slider to include the value from the dial

use DocumentTools in

use DocumentTools in 

theta_d:=GetProperty(Dial7,value); #This is added so that theta also has the value updated
phi_s:=GetProperty(Slider7,value); 

expr:= GetProperty(MathContainer10,expression);
new_expr0:=algsubs(theta=theta_d*Pi/180,expr); #This is added so that theta also has the value updated
new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0);  

SetProperty(MathContainer11,expression,new_expr);

end use:

Notice that the code in the Dial and Slider are the same

•	Since the code in the Dial and Slider are the same it makes sense to put the code into a procedure that can be called from multiple places.

Note: The changes in the code such as local and the single quotes are not needed but make the code easier to read and less likely to run into errors if edited in the future (for example if you create a variable called dial8 it won't interfere now that the names are in quotes).

UpdateMath:=proc()

UpdateMath:=proc()
local theta_d, phi_s, expr, new_expr, new_expr0;
use DocumentTools in 
theta_d:=GetProperty('Dial8','value'); #Get value of theta from Dial
phi_s:=GetProperty('Slider8','value'); #Get value of phi from slider

expr:= GetProperty('MathContainer12','expression');
new_expr0:=algsubs('theta'=theta_d*Pi/180,expr);  # Put value of theta in expression
new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0);  # Put value of phi in expression
SetProperty('MathContainer13','expression',new_expr); # Update expression
end use:
end proc:

•	Now change the code in the components to call the function using .

•

Since the code above is only defined there it will need to be run once (but only once) before moving the components. Instead of leaving it here you can add it to the Startup code by clicking or going to Edit > Startup code. This code will run every time you open the Math App ensuring that it works right away.

•	The startup code isn't defined in this document to allow progression of these steps.

Make the button initialize the app

•	Since the startup code isn't defined in this document we are going to move this function into the button.

UpdateMath:=proc()

UpdateMath:=proc()
local theta_d, phi_s, expr, new_expr, new_expr0;
use DocumentTools in 
theta_d:=GetProperty('Dial9','value'); #Get value of theta from Dial
phi_s:=GetProperty('Slider9','value'); #Get value of phi from slider

expr:= GetProperty('MathContainer14','expression');
new_expr0:=algsubs('theta'=theta_d*Pi/180,expr);  # Put value of theta in expression
new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0);  # Put value of phi in expression
SetProperty('MathContainer15','expression',new_expr); # Update expression
end use:
end proc:

•	First click the button to rename it, you'll see the option in the context panel on the right. Then add the code above to the button in the same way as the Slider an Dial (Right click and select Edit Click Code).

Now it is easy to add new components

•	Now if we want to add new components we just have to change the one procedure. Let's add a Volume Gauge to change the value of A.

•	Click in the cell containing the Dial, the context panel will show the option to Insert a row below the Dial.

•	Now drag a Volume Gauge into the new cell.

•	Click in the cell and choose the alignment (from the context panel) that looks best to you. In this case I chose center:

Update the procedure code for the Gauge

•	Add two lines for the volume gauge to get the value and sub it into the expression

UpdateMath:=proc()

UpdateMath:=proc()
local theta_d, phi_s, expr, new_expr, new_expr0;
use DocumentTools in 
theta_d:=GetProperty('Dial11','value'); #Get value of theta from the Dial
phi_s:=GetProperty('Slider11','value'); #Get value of phi from the Slider
A_g:=GetProperty('VolumeGauge1','value'); #Get value of A from the Guage

expr:= GetProperty('MathContainer18','expression');
new_expr0:=algsubs('theta'=theta_d*Pi/180,expr);  # Put value of theta in expression
new_expr1:=algsubs('phi'=phi_s*Pi/180,new_expr0);  # Put value of phi in expression
new_expr:=algsubs('A'=A_g,new_expr1);  # Put value of A in expression

SetProperty('MathContainer19','expression',new_expr); # Update expression
end use:
end proc:

•

Now add

UpdateMath();

to the Gauge.

Plot the changing expression

•	Make a procedure to get the value in the second Math Container and plot it

PlotMath:=proc()

PlotMath:=proc()
	local expr, p;
	use DocumentTools in 

	expr:=GetProperty('MathContainer21','expression'); 

	p:=plot(expr,'t'=-Pi/2..Pi/2,'view'=[-Pi/2..Pi/2,-100..100]):

	SetProperty('Plot14','value',p)
	end use:
end proc:

•	Put this procedure in the Initialize button and the call to it in the components.

Tidy up the app

•	Now that we have an interactive app let's tidy it up a bit.

•	The first thing I'd recommend in your own app is moving the code from the initialize button to startup code. In this document we choose to use the button instead to preserve earlier versions.

•	You can also remove the borders around the components by clicking in the table and selecting "Interior Borders" > "None" and "Exterior Borders" > "None" from the context panel.

Now you have a Math App

•	You can upload your Math App to the Maple Cloud to share with others by going to "File" > "Save to Cloud".

•	I'd recommend also including a description of what your app does. You can do this nicely using another table and Text mode.

HowToMathApp.mw

A robust workaround for forget(thismodule)

by: Carl Love 28010 Product: Maple

May 13 2020

1 0

When re-initializing a module (for example, while executing its ModuleApply), you'd likely want to re-initialize all of the remember tables and caches of its procedures. Unfortunately, forget(thismodule) (a direct self reference) doesn't work (I wonder why not?), and it doesn't even tell you that it didn't work. You could do forget(external name of module) (an indirect self reference), but I consider that not robust because you'd need to change it if you change the external name. Even less robust is forget~([procedure 1, procedure 2, submodule 3, ...]). Here's something that works and is robust:

forget~([seq(x, x= thismodule)])[]

Or, using Maple 2019 or later syntax,

forget~([for local x in thismodule do x od])[]

Both of these handle submodules and ignore non-procedures. Both handle both locals and exports.

A Brief intro to Student Linear Algebra in Maple...

by: Valerie 132 Product: Maple 2020

April 29 2020

5 0

With the new features added to the Student[LinearAlgebra] package I wanted to go over some of the basics on how someone can do Linear Algebra in Maple without require them to do any programming. I was recently asked about this and thought that the information may be useful to others.

This post will be focussed towards new Maple users. I hope that this will be helpful to students using Maple for the first time and professors who want their students to use Maple without needing to spend time learning the language.

In addition to the following post you can find a detailed video on using Maple to do Linear Algebra without programming here. You can also find some of the tools that are new to Maple 2020 for Linear Algebra here.

The biggest tools you'll be using are the Matrix palette on the left of Maple, and the Context Panel on the right of Maple.

First you should load the Student[LinearAlgebra] package by entering:

with(Student[LinearAlgebra]);

at the beginning of your document. If you end it with a colon rather than a semi colon it won't display the commands in the package.

Use the Matrix Palette on the left to input Matrices:

Once you have a Matrix you can use the context panel on the right to apply a variety of operations to it:

The Student Linear Algebra Menu will give you many linear algebra commands.

You can also access Maple's Tutors from the Tools Menu

Tools > Tutors > Linear Algebra

If you're interested in using the commands for Student[LinearAlegbra] in Maple you can view the help pages here or by entering:

?Student[LinearAlegbra]

into Maple.

I hope that this helps you get started using Maple for Linear Algebra.

Maple for Beginners

by: ecterrab 14540 Product: Maple

April 15 2020

11 0

Maple_for_Beginners.mw

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

Vectors in Spherical Coordinates using Tensor...

by: ecterrab 14540 Product: Maple

April 07 2020

6 3

Vectors in Spherical Coordinates using Tensor Notation

Edgardo S. Cheb-Terrab¹ and Pascal Szriftgiser²

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

(1) Maplesoft

The following is a topic that appears frequently in formulations: given a 3D vector in spherical (or any curvilinear) coordinates, how do you represent and relate, in simple terms, the vector and the corresponding vectorial operations Gradient, Divergence, Curl and Laplacian using tensor notation?

The core of the answer is in the relation between the - say physical - vector components and the more abstract tensor covariant and contravariant components. Focusing the case of a transformation from Cartesian to spherical coordinates, the presentation below starts establishing that relationship between 3D vector and tensor components in Sec.I. In Sec.II, we verify the transformation formulas for covariant and contravariant components on the computer using TransformCoordinates. In Sec.III, those tensor transformation formulas are used to derive the vectorial form of the Gradient in spherical coordinates. In Sec.IV, we switch to using full tensor notation, a curvilinear metric and covariant derivatives to derive the 3D vector analysis traditional formulas in spherical coordinates for the Divergence, Curl, Gradient and Laplacian. On the way, some useful technics, like changing variables in 3D vectorial expressions, differential operators, using Jacobians, and shortcut notations are shown.

The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.640 or newer.

Start setting the spacetime to be 3-dimensional, Euclidean, and use Cartesian coordinates

>

$`Default differentiation variables for d_, D_ and dAlembertian are:`*{X = (x, y, z)}$

Physics:-g_[mu, nu] = Matrix(%id = 18446744078312229334)

(1)

I. The line element in spherical coordinates and the scale-factors

In vector calculus, at the root of everything there is the line element , which in Cartesian coordinates has the simple form

>

(1.1)

To compute the line element in spherical coordinates, the starting point is the transformation

>

(1.2)

>

(1.3)

Since in are just symbols with no relationship to start transforming these differentials using the chain rule, computing the Jacobian of the transformation (1.2). In this Jacobian J, the first line is , ,

>

So in matrix notation,

>

Vector[column](%id = 18446744078518652550) = Vector[column](%id = 18446744078518652790)

(1.4)

To complete the computation of in spherical coordinates we can now use ChangeBasis , provided that next we substitute (1.4) in the result, expressing the abstract objects in terms of .

In two steps:

>

(1.5)

The line element

>

(1.6)

This result is important: it gives us the so-called scale factors, the key that connect 3D vectors with the related covariant and contravariant tensors in curvilinear coordinates. The scale factors are computed from (1.6) by taking the scalar product with each of the unit vectors , then taking the coefficients of the differentials (just substitute them by the number 1)

>

(1.7)

The scale factors are relevant because the components of the 3D vector and the corresponding tensor are not the same in curvilinear coordinates. For instance, representing the differential of the coordinates as the tensor , we see that corresponding vector, the line element in spherical coordinates , is not constructed by directly equating its components to the components of , so

The vector is constructed multiplying these contravariant components by the scaling factors, as

This rule applies in general. The vectorial components of a 3D vector in an orthogonal system (curvilinear or not) are always expressed in terms of the contravariant components the same way we did in the line above with the line element, using the scale-factors , so that

$`#mover(mi("A"),mo("→"))` = Sum(h[j]*A^j*`#mover(mi("\`e__j\`"),mo("&circ;"))`, j = 1 .. 3)$

where on the right-hand side we see the contravariant components and the scale-factors . Because the system is orthogonal, each vector component satisfies

The scale-factors do not constitute a tensor, so on the right-hand side we do not sum over j. Also, from

it follows that,

A__j = A__j/h__j

where on the right-hand side we now have the covariant tensor components .

•	This relationship between the components of a 3D vector and the contravariant and covariant components of a tensor representing the vector is key to translate vector-component to corresponding tensor-component formulas.

II. Transformation of contravariant and covariant tensors

Define here two representations for one and the same tensor: will represent A in Cartesian coordinates, while will represent A in spherical coordinates.

>

${A__c[j], A__s[j], Physics:-Dgamma[a], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](S), Physics:-SpaceTimeVector[a](X)}$

(2.1)

Transformation rule for a contravariant tensor

We know, by definition, that the transformation rule for the components of a contravariant tensor is $`#mrow(msup(mi("A"),mi("μ",fontstyle = "normal")),mo("⁡"),mfenced(mi("y")),mo("="),mfrac(mrow(mo("&PartialD;"),msup(mi("y"),mi("μ",fontstyle = "normal"))),mrow(mo("&PartialD;"),msup(mi("x"),mi("ν",fontstyle = "normal"))),linethickness = "1"),mo("⁢"),mo("⁢"),msup(mi("A"),mi("ν",fontstyle = "normal")),mfenced(mi("x")))`$ , that is the same as the rule for the differential of the coordinates. Then, the transformation rule from to computed using TransformCoordinates should give the same relation (1.4). The application of the command, however, requires attention, because, as in (1.4), we want the Cartesian (not the spherical) components isolated. That is like performing a reversed transformation. So we will use

where on the left-hand side we get, isolated, the three components of A in Cartesian coordinates, and on the right-hand side we transform the spherical components , from spherical (4^th argument) to Cartesian (3^rd argument), which according to the 5^th bullet of TransformCoordinates will result in a transformation expressed in terms of the old coordinates (here the spherical ). Expand things to make the comparison with (1.4) possible by eye

>

Vector[column](%id = 18446744078459463070) = Vector[column](%id = 18446744078459463550)

(2.2)

We see that the transformation rule for a contravariant vector is, indeed, as the transformation (1.4) for the differential of the coordinates.

Transformation rule for a covariant tensor

For the transformation rule for the components of a covariant tensor , we know, by definition, that it is $`#mrow(msub(mi("A"),mi("μ",fontstyle = "normal")),mo("⁡"),mfenced(mi("y")),mo("="),mfrac(mrow(mo("&PartialD;"),msup(mi("x"),mi("ν",fontstyle = "normal"))),mrow(mo("&PartialD;"),msup(mi("y"),mi("μ",fontstyle = "normal"))),linethickness = "1"),mo("⁢"),mo("⁢"),msub(mi("A"),mi("ν",fontstyle = "normal")),mfenced(mi("x")))`$ , so the same transformation rule for the gradient , where and so on. We can experiment this by directly changing variables in the differential operators , for example

>

Physics:-d_[x] = (proc (u) options operator, arrow; ((-r*cos(theta)^2+r)*cos(phi)*(diff(u, r))+sin(theta)*cos(phi)*cos(theta)*(diff(u, theta))-(diff(u, phi))*sin(phi))/(r*sin(theta)) end proc)

(2.3)

This result, and the equivalent ones replacing x by y or z in the input above can be computed in one go, in matricial and simplified form, using the Jacobian of the transformation computed in . We need to take the transpose of the inverse of J (because now we are transforming the components of the gradient )

>

Vector[column](%id = 18446744078518933014) = Vector[column](%id = 18446744078518933254)

(2.4)

The corresponding transformation equations relating the tensors and in Cartesian and spherical coordinates is computed with TransformCoordinates as in (2.2), just lowering the indices on the left and right hand sides (i.e., remove the tilde ~)

>

Vector[column](%id = 18446744078557373854) = Vector[column](%id = 18446744078557374334)

(2.5)

We see that the transformation rule for a covariant vector is, indeed, as the transformation rule (2.4) for the gradient.

To the side: once it is understood how to compute these transformation rules, we can have the inverse of (2.5) as follows

>

Vector[column](%id = 18446744078557355894) = Vector[column](%id = 18446744078557348198)

(2.6)

III. Deriving the transformation rule for the Gradient using TransformCoordinates

Turn ON the CompactDisplay notation for derivatives, so that the differentiation variable is displayed as an index:

>

The gradient of a function f in Cartesian coordinates and spherical coordinates is respectively given by

>

(3.1)

>

(3.2)

What we want now is to depart from (3.1) in Cartesian coordinates and obtain (3.2) in spherical coordinates using the transformation rule for a covariant tensor computed with TransformCoordinates in (2.5). (An equivalent derivation, simpler and with less steps is done in Sec. IV.)

Start changing the vector basis in the gradient (3.1)

>

%Nabla(f(X)) = ((diff(f(X), x))*sin(theta)*cos(phi)+(diff(f(X), y))*sin(theta)*sin(phi)+(diff(f(X), z))*cos(theta))*_r+((diff(f(X), x))*cos(phi)*cos(theta)+(diff(f(X), y))*sin(phi)*cos(theta)-(diff(f(X), z))*sin(theta))*_theta+(-(diff(f(X), x))*sin(phi)+cos(phi)*(diff(f(X), y)))*_phi

(3.3)

By eye, we see that in this result the coefficients of are the three lines in the right-hand side of (2.6) after replacing the covariant components by the derivatives of f with respect to the j^th coordinate, here displayed using indexed notation due to using CompactDisplay

>

(3.4)

>

(3.5)

So since (2.5) is the inverse of (2.6), replace A by ∂ f in (2.5), the formula computed using TransformCoordinates, then insert the result in (3.3) to relate the gradient in Cartesian and spherical coordinates. We expect to arrive at the formula for the gradient in spherical coordinates (3.2) .

>

Vector[column](%id = 18446744078344866862) = Vector[column](%id = 18446744078344866742)

(3.6)

>

(3.7)

Simplifying, we arrive at (3.2)

>

(3.8)

>

(3.9)

IV. Deriving the transformation rule for the Divergence, Curl, Gradient and Laplacian, using TransformCoordinates and Covariant derivatives

•	The Divergence

Introducing the vector A in spherical coordinates, its Divergence is given by

>

(4.1)

>

(4.2)

>

%Divergence(%A__s_) = ((diff(A__r(S), r))*r+2*A__r(S))/r+((diff(`A__θ`(S), theta))*sin(theta)+`A__θ`(S)*cos(theta))/(r*sin(theta))+(diff(`A__φ`(S), phi))/(r*sin(theta))

(4.3)

We want to see how this result, (4.3), can be obtained using TransformCoordinates and departing from a tensorial representation of the object, this time the covariant derivative . For that purpose, we first transform the coordinates and the metric introducing nonzero Christoffel symbols

>

$`Default differentiation variables for d_, D_ and dAlembertian are:`*{S = (r, theta, phi)}$

Physics:-g_[a, b] = Matrix(%id = 18446744078312216446)

(4.4)

To the side: despite having nonzero Christoffel symbols, the space still has no curvature, all the components of the Riemann tensor are equal to zero

>

(4.5)

Consider now the divergence of the contravariant tensor, computed in tensor notation

>

(4.6)

>

(4.7)

To the side: the covariant derivative expressed using the D_ operator can be rewritten in terms of the non-covariant d_ and Christoffel symbols as follows

>

(4.8)

Summing over the repeated indices in (4.7), we have

>

%D_[j](%A__s[`~j`]) = diff(A__s[`~1`](S), r)+diff(A__s[`~2`](S), theta)+diff(A__s[`~3`](S), phi)+2*A__s[`~1`](S)/r+cos(theta)*A__s[`~2`](S)/sin(theta)

(4.9)

How is this related to the expression of the $VectorCalculus[Nabla].`#mover(mi("\`A__s\`"),mo("→"))`$ in (4.3) ? The answer is in the relationship established at the end of Sec I between the components of the tensor and the components of the vector $`#mover(mi("\`A__s\`"),mo("→"))`$ , namely that the vector components are obtained multiplying the contravariant tensor components by the scale-factors . So, in the above we need to substitute the contravariant by the vector components divided by the scale-factors

>

(4.10)

>

%D_[j](%A__s[`~j`]) = diff(A__r(S), r)+(diff(`A__θ`(S), theta))/r+(diff(`A__φ`(S), phi))/(r*sin(theta))+2*A__r(S)/r+cos(theta)*`A__θ`(S)/(sin(theta)*r)

(4.11)

Comparing with (4.3), we see these two expressions are the same:

>

%Divergence(%A__s_) = diff(A__r(S), r)+(diff(`A__θ`(S), theta))/r+(diff(`A__φ`(S), phi))/(r*sin(theta))+2*A__r(S)/r+cos(theta)*`A__θ`(S)/(sin(theta)*r)

(4.12)

•

The Curl

The Curl of the the vector $`#mover(mi("\`A__s\`"),mo("→"))`$ in spherical coordinates is given by

>

%Curl(%A__s_) = ((diff(`A__φ`(S), theta))*sin(theta)+`A__φ`(S)*cos(theta)-(diff(`A__θ`(S), phi)))*_r/(r*sin(theta))+(diff(A__r(S), phi)-(diff(`A__φ`(S), r))*r*sin(theta)-`A__φ`(S)*sin(theta))*_theta/(r*sin(theta))+((diff(`A__θ`(S), r))*r+`A__θ`(S)-(diff(A__r(S), theta)))*_phi/r

(4.13)

One could think that the expression for the Curl in tensor notation is as in a non-curvilinear system

But in a curvilinear system is not a tensor, we need to use the non-Galilean form , where is the determinant of the metric. Moreover, since the expression has one free covariant index (the first one), to compare with the vectorial formula (4.12) this index also needs to be rewritten as a vector component as discussed at the end of Sec. I, using

A__j = A__j/h__j

The formula (4.13) for the vectorial Curl is thus expressed using tensor notation as

>

(4.14)

>

%Curl(%A__s_) = Physics:-LeviCivita[i, j, k]*Physics:-D_[`~j`](A__s[`~k`](S), [S])/%h[i]

(4.15)

followed by replacing the contravariant tensor components by the vector components using (4.10). Proceeding the same way we did with the Divergence, expand this expression. We could use TensorArray , but Library:-TensorComponents places a comma between components making things more readable in this case

>

(4.16)

Replace now the components of the tensor by the components of the 3D vector $`#mover(mi("\`A__s\`"),mo("→"))`$ using (4.10)

>

(4.17)

>

%Curl(%A__s_) = [((diff(`A__φ`(S), theta))*sin(theta)+`A__φ`(S)*cos(theta)-(diff(`A__θ`(S), phi)))/(r*sin(theta)), (diff(A__r(S), phi)-(diff(`A__φ`(S), r))*r*sin(theta)-`A__φ`(S)*sin(theta))/(r*sin(theta)), ((diff(`A__θ`(S), r))*r+`A__θ`(S)-(diff(A__r(S), theta)))/r]

(4.18)

We see these are exactly the components of the Curl (4.13)

>

%Curl(%A__s_) = ((diff(`A__φ`(S), theta))*sin(theta)+`A__φ`(S)*cos(theta)-(diff(`A__θ`(S), phi)))*_r/(r*sin(theta))+(diff(A__r(S), phi)-(diff(`A__φ`(S), r))*r*sin(theta)-`A__φ`(S)*sin(theta))*_theta/(r*sin(theta))+((diff(`A__θ`(S), r))*r+`A__θ`(S)-(diff(A__r(S), theta)))*_phi/r

(4.19)

•	The Gradient

Once the problem is fully understood, it is easy to redo the computations of Sec.III for the Gradient, this time using tensor notation and the covariant derivative. In tensor notation, the components of the Gradient are given by the components of the right-hand side

>

%Nabla(f(S)) = `&dtri;`[j](f(S))/%h[j]

%Nabla(f(S)) = Physics:-d_[j](f(S), [S])/%h[j]

(4.20)

where on the left-hand side we have the vectorial Nabla differential operator and on the right-hand side, since is a scalar, the covariant derivative becomes the standard derivative .

>

(4.21)

The above is the expected result (3.2)

>

(4.22)

•	The Laplacian

Likewise we can compute the Laplacian directly as

>

(4.23)

In this case there are no free indices nor tensor components to be rewritten as vector components, so there is no need for scale-factors. Summing over the repeated indices,

>

%Laplacian(f(S)) = Physics:-dAlembertian(f(S), [S])+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2)

(4.24)

Evaluating the Vectors:-Laplacian on the left-hand side,

>

((diff(diff(f(S), r), r))*r+2*(diff(f(S), r)))/r+((diff(diff(f(S), theta), theta))*sin(theta)+cos(theta)*(diff(f(S), theta)))/(r^2*sin(theta))+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2) = Physics:-dAlembertian(f(S), [S])+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2)

(4.25)

On the right-hand side we see the dAlembertian , in curvilinear coordinates; rewrite it using standard diff derivatives and expand both sides of the equation for comparison

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diff(diff(f(S), r), r)+(diff(diff(f(S), theta), theta))/r^2+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2)+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2) = diff(diff(f(S), r), r)+(diff(diff(f(S), theta), theta))/r^2+(diff(diff(f(S), phi), phi))/(r^2*sin(theta)^2)+2*(diff(f(S), r))/r+cos(theta)*(diff(f(S), theta))/(sin(theta)*r^2)

(4.26)

This is an identity, the left and right hand sides are equal:

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(4.27)

Download Vectors_and_Spherical_coordinates_in_tensor_notation.mw

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

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