text:
Technical writing/Tools/Wiki Tables

Back to Tools

Why make tables?
Tables are a very easy way of organising information.

Basic markup
All tables begin with {| and end with |}.

The most common markup used is known as the "pipe" - which looks like this |.

Building a basic table
The following code will build a basic table.

In the first line we set the widths of the borders and the space between cells.

{| border="1" cellpadding="1" cellspacing="1"

On the second line we enter the title of the table.

|+ 2006/07 Premier League Table

The following lines create the column headings, when creating the column headings we use a "!" instead of a pipe.

|-    !Position !Club Name !Points

Now we create the rows and the information that goes in them. Each line starts with a pipe and each row of information is seperated by a |-.

|-  |1  ||  Manchester United  ||  89 |-  |2  ||  Chelsea  ||  83 |-  |3  ||  Liverpool  ||  68 |-  |4  ||  Arsenal  ||  68 |-  |5  ||  Tottenham  ||  60 |-  |6  ||  Bolton  ||  59

Notice that each cell is split by a double pipe ||. Usually there would be no space between the pipes, here the space is there to make reading the code easier.

This will create a basic table, which looks like this.

Useful extras
Table width:

In the first line of the table it is possible to define the table's width, for example:

{| width=1000pt border="1" cellpadding="1" cellspacing="1"

Column width:

To define the column width just add a percentage of the overall width to the column heading lines, as follows:

! width=25% | Position ! width=50% | Club Name ! width=25% | Points

The table now looks like this:

Common mistakes
One of the most common mistakes people make with tables is forgetting to put pipes in the code.

Also common is forgetting to finish the table with |}

title: Technical writing/Tools/Wiki Tables


text:
Technical writing/Tools/Wiki Tables/Demo

Back to Wiki Tables

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
User talk:Iamsaa

Hello Iamsaa, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Riaz Ahmed Gohar Shahi
Hi Iamsaa. Thank you for your message on Requests for Deletion. To find out what kind of thing you can put on Wikiversity, I suggest you look around the place a lot - start with some of the links HappyCamper gave you just above. Basically Wikiversity is about education and creating teaching materials. Biographies of notable living people should go on Wikipedia instead. If you wish, you can request the deletion of the Riaz Ahmed Gohar Shahi page. Everyone makes mistakes - so don't worry about it! People often request deletion of their own pages when they make their first edits. McCormack 09:47, 10 August 2007 (UTC)

title: User talk:Iamsaa


text: Technical writing/Tools/Making Wiki Tables/Technical writing alistairtable


text: Technical writing/Dima Screengrab2

title: Technical writing/Dima Screengrab2


text: User:Foxlangserv/Technical writing Adam Screengrab


text: Technical Writing/Vlodek Screengrab

title: Technical Writing/Vlodek Screengrab


text: Technical writing/Tools/Making Wiki Tables/Technical writing adamtable


text: Technical writing/Tools/Making Wiki Tables/Technical writing Martinstable


text: Technical writing/Tools/Making Wiki Tables/Technical writing sasha


text: Technical writing/Tools/Making Wiki Tables/Technical writing Jana


text: Technical writing/Tools/Making Wiki Tables/Technical writing Zdenkatable


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User:Tanya/Technical writing tanyahanatable

=Tanya's and Hana's Table= {|width=500pt border="1" cellpadding="1" cellspacing="1" !width=33%|Friend's Name !width=67%|Favourite Drink
 * + Our Friends' Favourite Drinks
 * Hana||Kubik
 * Tanya||Belarusian balsam
 * Sasha||Klubnichnyj Kefir
 * Dima||style=background:red|Vodka Stolichnaya
 * Fred||style=background:red|Fresh blood of little old lady
 * Johny||style=background:orange|Any gold tequila
 * -}
 * Dima||style=background:red|Vodka Stolichnaya
 * Fred||style=background:red|Fresh blood of little old lady
 * Johny||style=background:orange|Any gold tequila
 * -}
 * Johny||style=background:orange|Any gold tequila
 * -}
 * -}


text: Technical writing/Tools/Making Wiki Tables/Technical writing tapootable


text: Technical writing/Tools/Making Wiki Tables/Technical writing Dmitrytable


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User talk:Dmitry Bogatyrev

Hello Dmitry Bogatyrev, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Page redirects and titles
Hi Dmitry. Some of your new pages and redirects seem to be getting a bit mixed up. Do you need a custodian to help sort things out? McCormack 07:58, 23 August 2007 (UTC)

Image copyright problem with Image:CIMG3404.JPG
Thanks for uploading Image:CIMG3404.JPG. However, the image may soon be deleted unless we can determine the copyright holder and copyright status. The Wikimedia Foundation is very careful about the images included in Wikiversity because of copyright law (see Wikiversity' Copyright policy).

The copyright holder is usually the creator, the creator's employer, or the last person who was transferred ownership rights. Copyright information on images is signified using copyright templates. The three basic license types on Wikibooks are open content, public domain, and fair use. Find the appropriate template in Image copyright tags and place it on the image page like this:.

Please signify the copyright information on any other images you have uploaded or will upload. Remember that images without this important information or giving false information can be deleted by an administrator. If you have any questions, feel free to contact me, or ask them at Colloquim. Thank you. – Mike.lifeguard | talk 23:30, 28 September 2007 (UTC)

Image copyright problems

 * Your currently affected uploads are listed below. For each, please provide a license template and source. Thanks. – Mike.lifeguard | @en.wb 23:54, 13 February 2008 (UTC)

 * 2007-08-09 15:27, Dmitry Bogatyrev: Dima.jpg: untagged! ; 1 (1)
 * 2007-08-23 07:17, Dmitry Bogatyrev: Dmitry B.JPG: untagged! ; 0 (0)

title: User talk:Dmitry Bogatyrev


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User talk:Alexandra~enwikiversity

Hello Alexandra, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Pictures, and pages
Hi there - I made a few bold edits...could you take a look at Http://en.wikiversity.org/wiki/Technical writing Sasha Sladeckova and see if it is the effect you wanted? Also, would you prefer your article to be in another location? --HappyCamper 19:48, 22 August 2007 (UTC)

Sasha Hi, thank you for helping me :) I'm a wiki-novice, so it takes me time to make it look like it has to be. What location do you mean?
 * Well, how about putting it at Technical writing/Sasha Sladeckova instead? That way, others won't have to type in the http:// stuff in order to find the page. --HappyCamper 16:03, 23 August 2007 (UTC)

Invitation
Dear user of Wikiversity,

Do you like to join a reading club, where you can read and discuss literature? Reading clubs could have active and inactive participants. Active participants could be required to edit on a regularly basis untill a deadline is reached and another topic can be chosen for reading and discussing. Hopefully, reading clubs can ensure a social network on Wikiversity of people cooperating with eachother and becoming friends while discussing knowledge.--Daanschr 09:42, 1 November 2007 (UTC)

Image copyright problems

 * Your currently affected uploads are listed below. For each, please provide a source and license information. Thanks .– Mike.lifeguard | @en.wb 23:40, 13 February 2008 (UTC)

 * 2007-08-09 15:26, Alexandra: SunsetSasha.jpg: untagged! ; 1 (1)
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Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Alexandra. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Alexandra~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:27, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:02, 19 April 2015 (UTC)

title: User talk:Alexandra~enwikiversity


text:
User talk:Foxlangserv

Hello Foxlangserv, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Image copyright problems

 * This is the image in question. – Mike.lifeguard | @en.wb 21:54, 13 February 2008 (UTC)

title: User talk:Foxlangserv


text:
User talk:Pospisilik

Hello Pospisilik, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 17:10, 9 August 2007 (UTC)

Image:Martinsscreengrab.jpg
Please provide a license template and source for this image. Thanks. – Mike.lifeguard | @en.wb 22:08, 13 February 2008 (UTC)

title: User talk:Pospisilik


text:
User talk:El-vis

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title: User talk:El-vis


text:
User talk:JanaS

Hello JanaS, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Image copyright problems

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title: User talk:JanaS


text: File:OdysseyConsole.jpg

title: File:OdysseyConsole.jpg


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C Programming/Basic Output/Example 1

This example will show you how to use escape sequences and do text formatting.

Here is the source code:

If you will notice, at the end of every string in the  functions there is a \n, this is so a new line starts for every line.

In the first  call there is a \" escape sequence. This is so the double quote will be printed to the console and not be seen by the compiler as something terminating or initiating a string. Same goes for the single quote in the third call of.

If you intend to print unformatted strings, you can use the function  to supply the same output:

always appends a newline to whatever it writes, so in this example there is no need for the newline escape sequences, except for the very last one where we require two (one supplied by, one supplied by our string).

title: C Programming/Basic Output/Example 1


text:
Help:Abaqus

Welcome to this help desk on !

Abaqus is a commercial software package for finite element analysis.

Help Desk Summary

 * Project code:
 * Portal:Engineering and technology
 * School:Engineering
 * Department:Mechanical engineering . Civil engineering
 * Stream: Applied mechanics . Solid mechanics
 * Level: Undergraduate . Graduate . Professional

Content summary
The help desk seeks to collect pointers to solving problems with Abaqus. The content is in the form of questions followed by answers.

Learning materials

 * Introduction to Abaqus
 * Advanced simulation with Abaqus

Resources

 * Abaqus
 * User's Manuals
 * ABAQUS/CAE User's Manual
 * ABAQUS Analysis User's Manual
 * ABAQUS Example Problems Manual
 * ABAQUS Benchmarks Manual
 * Tutorials
 * Getting started with ABAQUS
 * Getting started with ABAQUS/Standard (Keywords version)
 * Getting started with ABAQUS/Explicit (Keywords version)
 * Reference and Theory Manuals
 * ABAQUS Keywords Reference Manual
 * ABAQUS Theory Manual
 * ABAQUS User SUbroutines Manual

Questions and Answers
Each of these pages contains questions and answers on the topics suggested below. If your topic does not fall into any of these categories please add another category to the list. Please post your questions and answers in the appropriate category page.

 * Getting started
 * Graphical User Interface
 * Scripting
 * User subroutines
 * Explicit dynamics
 * Vibrations
 * Linear Elasticity
 * Nonlinear Elasticity
 * Viscoelasticity
 * Plasticity
 * Viscoplasticity
 * Damage mechanics
 * Cohesive zone modeling
 * Fracture
 * Fatigue
 * Deformable Rolls
 * Heat Transfer

Active participants
Active participants in this Learning Group

{| width="100%" style="background:#FFFCCF; border:1px solid #DAA520"

Learning materials
Learning materials and learning projects are located in the main Wikiversity namespace. Simply make a link to the name of the lesson (lessons are independent pages in the main namespace) and start writing!

You should also read about the Wikiversity:Learning model. Lessons should center on learning activities for Wikiversity participants. Learning materials and learning projects can be used by multiple projects - and you are encouraged to cooperate with other departments that use the same learning resource.
 * }

title: Help:Abaqus


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User:Zath42

Hello, I'm a Civil Engineering student in Minnesota.

I currently attend The University of Minnesota Institute of Technology. I transfered to the U of M after completing an AAs degree in Civil Engineering Technology, from Lake Superior College.

I'm not involved in any research at this time, though research opportunity's are extremely prevalent here are the U, it is only a matter of time, and I will be involved...

I work as a superintendent for a highway/heavy construction company, working manly on earthen damns in northeastern Minnesota.Hoover Construction is the company that I work for, we do all forms of major Civil construction, major site work projects, highways, drilling and blasting, and earthen damns.

Help:Formula

title: User:Zath42


text:
User talk:Zath42

Hello Zath42, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --HappyCamper 01:24, 10 August 2007 (UTC)

title: User talk:Zath42


text:
User talk:EmiLiuS

Hello EmiLiuS, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 02:19, 10 August 2007 (UTC)

title: User talk:EmiLiuS


text:
Category:Computational mechanics

Computational mechanics deals with numerical methods of solving problems in continuum mechanics and molecular dynamics. Materials can be solid, fluids, or plasmas.

title: Category:Computational mechanics


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Category:Computational solid mechanics

Computational solid mechanics deals with the numerical solution of problem in the continuum mechanics of solids.

title: Category:Computational solid mechanics


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Portal:Finite element analysis/Subtopics

 * Linear finite element analysis
 * Nonlinear finite element analysis
 * Analysis of plates and shells

 * Fluid structure interaction
 * Thermal analysis
 * Mesh generation


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Portal:Finite element analysis/Featured learning resource

Nonlinear finite element analysis


text:
Portal:Finite element analysis/Participants

Please add your name to the list:
 * User:Banerjee


text:
Portal:Finite element analysis/Discussion forum

This forum is for the initial discussions on finite element analysis. If there is a significant amount of interest, the discussion will be moved under the discussion tab above.


text:
Portal:Finite element analysis/Research projects

Add you research questions/ideas here.


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Portal:Finite element analysis/Things you can do

 * Add a course on introductory finite elements


text:
Nonlinear finite elements

Welcome to this learning project about nonlinear finite elements!

Introduction
Finite element analysis (FEA) is a computer simulation technique used in engineering analysis. It uses a numerical technique called the finite element method (FEM) to solve partial differential equations. There are many finite element software packages, both free and proprietary. Development of the finite element method in structural mechanics is usually based on an energy principle such as the virtual work principle or the minimum total potential energy principle.

Project metadata

 * Suggested Prerequisites:
 * Linear algebra
 * Partial differential equations
 * Introduction to finite elements
 * Continuum mechanics
 * Time investment: 6 months
 * Portal:Engineering and Technology
 * School:Engineering
 * Department:Mechanical engineering
 * Level: First year graduate

Content summary
This is an introductory course on nonlinear finite element analysis of solid mechanics and heat transfer problems. Nonlinearities can be caused by changes in geometry or be due to nonlinear material behavior. Both types of nonlinearities are covered in this course.

Goals
This learning project aims to.
 * provide the mathematical foundations of the finite element formulation for engineering applications (solids, heat, fluids).
 * expose students to some of the recent trends and research areas in finite elements.

Here's a short quiz to help you find out what you need to brush up on before you dig into the course:
 * Assessment Quiz

Contents
{|width=100%
 * valign=top width=50%|

Syllabus and Learning Materials

 * 1) Mathematical Preliminaries
 * 2) Set notation
 * 3) Functions
 * 4) Vectors
 * 5) Matrices
 * 6) Tensors
 * 7) Partial differential equations
 * 8) Variational calculus
 * 9) Linear finite element basics
 * 10) An example: Axially loaded bar
 * 11) * Strong form: governing differential equation
 * 12) * Weak form: integral equation
 * 13) * Approximate solution: Galerkin method
 * 14) * Approximate solution: Finite element method
 * 15) More examples: Some model problems
 * 16) * Weak form: Weighted residual methods
 * 17) * Choosing a weight function
 * 18) * Bubnov Galerkin methods
 * 19) * Finite element basis functions
 * 20) * Example of a finite element approximation
 * 21) A time-dependent problem: the heat equation
 * 22) * Steady state heat conduction
 * 23) * Weak form of the steady state heat equation
 * 24) * Weak form of the time-dependent heat equation
 * 25) * Finite element approximation of Poisson equation
 * 26) * Finite element approximation of the heat equation
 * 27) * Time integration of the heat equation.
 * 28) Nonlinear finite element basics
 * 29) Nonlinearities in solid mechanics
 * 30) Nonlinear deformation of an axial bar
 * 31) * Weak form and finite element approximation
 * 32) * The Newton-Raphson method
 * 33) * The Newton method applied to finite elements
 * 34) * The Newton method applied to the axially loaded bar

 * 1) Lagrangian and Eulerian descriptions of motion
 * 2) Lagrangian finite elements
 * 3) * Motion from the Lagrangian point of view
 * 4) * Total Lagrangian approach
 * 5) * Updated Lagrangian approach
 * 6) * An example: Effect of mesh distortion
 * 7) Solution procedure
 * 8) Special case: Natural vibrations
 * 9) Nonlinear deformation of beams
 * 10) Euler-Bernoulli beams
 * 11)  Timoshenko beams
 * 12)  Buckling of beams
 * 13) Nonlinear deformation of plates and shells
 * 14) Basic linear plate and shell elements
 * 15) Nonlinear plates and shells
 * 16) Time-dependent deformation of shells
 * 17) Basic continuum mechanics
 * 18) Kinematics
 * 19) Motion, displacement, velocity, acceleration
 * 20) Stresses and strains in one and two dimensions
 * 21) Strains and deformations in three-dimensions
 * 22) Polar decomposition
 * 23) Spectral decompositions of kinematic quantities
 * 24) Volume change and area change
 * 25) Time derivatives and rate quantities
 * 26) Objectivity of kinematic quantities
 * 27) Stress measures and stress rates
 * 28) Stress measures
 * 29) Deviatoric and volumetric stress
 * 30) Objective stress rates
 * 31) Balance laws
 * 32) Overview of balance laws
 * 33) Balance of mass
 * 34) Balance of linear momentum
 * 35) Balance of angular momentum
 * 36) Balance of energy
 * 37) Entropy inequality
 * 38) Constitutive models
 * 39) Objectivity of hyperelastic relations
 * 40) Rate form of hyperelastic relations
 * 41) Nonlinear elasticity
 * 42) Plasticity
 * 43) Viscoplasticity
 * 44) Viscoelasticity.
 * 45) Finite element formulation in three dimensions.
 * 46) Updated Lagrangian formulation
 * 47) Verification and Validation.

 * valign=top width=50%|

Assignments

 * Homework 1 Problem set
 * Solutions
 * Homework 2 Problem set
 * Solutions
 * Homework 3 Problem set
 * Solutions
 * Homework 4 Problem set
 * Hints
 * Solutions
 * Homework 5 Problem set
 * Solutions
 * Homework 6 Problem set
 * Hints
 * Solutions
 * Homework 7 Problem set
 * Hints
 * Solutions
 * Homework 8 Problem set
 * Solutions
 * Homework 9 Problem set
 * Solutions
 * Homework 10 Problem set
 * Solutions
 * Homework 11 Problem set
 * Solutions

Tests and Quizzes

 * Quiz 1
 * Solutions

 * }

Textbooks

 * An Introduction to Nonlinear Finite Element Analysis by J. N. Reddy, Oxford University Press, 2004, ISBN 019852529X.
 * Nonlinear Finite Elements for Continua and Structures by T. Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000.
 * Computational Inelasticity by J. C. Simo and T. J. R. Hughes, Springer, 1998.
 * The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by T. J. R. Hughes, Dover Publications, 2000.

Reading List

 * Taylor, R.L., Simo, J.C., Zienkiewicz, O.C., and Chan, A.C.H, 1986, The patch test - a condition for assessing FEM convergence,  International Journal for Numerical Methods in Engineering, 22, pp. 39-62.
 * Simo, J.C. and Vu-Quoc, L., 1986, A three-dimensional finite strain rod model. Part II: Computational Aspects, Computer Methods in Applied Mechanics and Engineering,  58, pp. 79-116.
 * Ibrahimbegovic, A., 1995, On finite element implementation of geometrically nonlinear Reissner's beam theory: Three-dimensional curved beam elements, Computer Methods in Applied Mechanics and Engineering,  122, pp. 11-26.
 * Buchter, N., Ramm, E., and Roehl, D., 1994, Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept,  Int. J. Numer. Meth. Engng.,  37, pp. 2551-2568.
 * Rouainia, M. and Peric, D., 1998, A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications,  Int. J. Numer. Meth. Engng.,  42, pp. 289-311.

title: Nonlinear finite elements


text:
Category:Finite element analysis

Finite element analysis is a numerical method that is widely used in computational mechanics and other field of engineering and physics.

title: Category:Finite element analysis


text:
User:Snooze~enwikiversity

My name is Bob

title: User:Snooze~enwikiversity


text:
Donations

If you are interested in donating time and knowledge, money, land, or other resources to Wikiversity, this is the place to be.

Would you have land somewhere on Earth you might be interested in donating to Wikiversity?

All donations are appreciated. Aside from one's time, one way to contribute to Wikiversity is through monetary support. The easiest and most sure-fire way to give Wikiversity monetary support would be to donate directly to the Wikimedia foundation. Currently, systems are not setup within Wikiversity to allow for people to sponsor projects or directly support the endeavors of Wikiversity itself.

One novel situation that could potentially become reality would be for Wikiversity to reach brick-and-mortar status. It seems the fastest way for this to occur would be for a group or individual to donate some land to Wikiversity. Perhaps, if this were to occur, then participants could maintain the land in a wiki fashion...

title: Donations


text:
Talk:Donations

See this, which is linked from the sidebar. Thunderhead 09:43, 10 August 2007 (UTC)

title: Talk:Donations


text:
Function (mathematics)

One of the most neglected topics in high school is the study of functions. In this lesson there are three rather lengthy chapters directly related to functions and several others that are indirectly related. There are two reasons for this: functions are important, and most calculus courses assume you know this topic almost perfectly, an unrealistic assumption. So let's get started at the beginning.

Introduction
We will start with an excruciatingly theoretical and general definition of a function in mathematics, and then look at the topic in a more down-to-Earth way.

Function: Given a set D. To each element in D, we assign one and only one element.

Example 1

The table below represents a function. 1 maps to a, 2 maps to 3, 3 maps to 3, and 4 maps to "pig". Each element in D is assigned one and only one element.

 * $$D\ \ \ R\,$$
 * $$1  \to   a$$
 * $$2  \to   3$$
 * $$3  \to   3$$
 * $$4  \to   $$ "pig"

The set D is called the domain. There is a second set that arises, called the range. Notice that the domain and the range can contain the same thing (the number 3) or vastly different things (3 and "pig"). The domain and range are arbitrary mathematical sets. In this example, the domain is the set of integers 1 to 4, and the range might be the union of the set of integers, the set of letters, and the set of farm animals.
 * $$D = \{\text{integers from 1 to 4}\}\,$$
 * $$R = \{\text{integers}\} \cup \{\text{letters}\} \cup \{\text{farm animals}\}$$

The rule (the arrows in the table above) is called the map or mapping. 1 is mapped into a; 2 is mapped into 3; 3 is mapped into 3; and 4 is mapped into "pig".

The true theoretical definition of a function is: A function is a domain, a range, and set of ordered pairs such that
 * The first item in each pair is an element of the domain.
 * The second item is an element of the range.
 * No two pairs have the same first element. This property is sometimes expressed by saying that a function must be single valued.
 * Every element in the domain appears as the first item of one (and only one) of the pairs.

In this example, the set of ordered pairs would be
 * $$\{ (1, a), (2, 3), (3, 3), (4, \text{pig}) \}\,$$

A few things to note:
 * It is perfectly legitimate for an element of the range to appear as the second item in more than one pair. That is, multiple domain elements can map to the same range element.
 * Not every element of the range needs to appear as the second item in some pair. That is, there can be elements of the range that nothing maps to.  In this example, no number maps to "cow", even though the range includes all farm animals.  The subset of the range that consists only of those elements that do get mapped to is called the image.  We will say more about that later.

Example 2

This does not represent a function, since 1 would map to both a and d. It is not single-valued.
 * $$\{ (1, a), (1, d), (2, e), (3, f), (4, g) \}\,$$

Function's Notation

 * [[Image:functionEx.png|right|300 px|Visualizing functions.]]

The first depicts a function as sort of a "machine" that receives inputs and emits outputs. This is sometimes useful when dealing with composition of functions, as discussed below. The second depicts a function in terms of arrows going from points in the domain to points in the range. The diagram shows just 3 points in some very abstract representations of the domain and range.

If the function defined earlier is denoted f, we say: f(1) = a (read "f of 1 equals a"); f(2) = 3; f(3) = 3; and f(4) = "pig".

Mathematical notation
 * $$f(x)=y$$

Where
 * $$f(x)$$ - Function of variable x
 * $$x$$ - Function's variable
 * $$y$$ - Function's value

Function Definition by a Rule
The definition of functions in terms of sets of ordered pairs is suitable only for abstract discussions of set theory. In practice, functions are usually defined by giving a rule, and having the domain and range be numbers. For example, here is a definition of a function whose domain and range are the real numbers:
 * $$f(x) = x^2+4x+7 \,$$

If x is 2 then
 * $$f(x) = 2^2+4 \times 2+7 \ = 4 + 8 + 7 = 19$$

Functions as Graphs
A way of visualizing functions that is extremely useful in practical situations, is as a graph. It only works for functions from real numbers to real numbers, but that is an extremely common case. The x-axis of the graph shows the domain, and the y-axis shows the range. A line of the required shape shows the function values (that is, y-values) for any given x-value.

The diagram below shows this (simultaneously) for two well-known trigonometric functions. The sine function is shown in red, and the cosine function in blue. Horizontal and vertical divisions are each 0.5.
 * $$f(x)=Sin x$$ . In red
 * $$f(x)=Cos x$$ . In blue
 * [[Image:Sincosgraph.png|400 px|Graphs of the sine (red) and cosine (blue) functions.]]

Looking at the graph of a function makes it very easy to see a number of interesting characteristics, such as continuity. One can also estimate the derivative. For example, the graph on the right makes it quite plausible (and it is in fact true) that the derivative of the sine is the cosine, and the derivative of the cosine is the negative of the sine.

Inspection of a graph can also make it clear when a purported function does not satisfy the requirement of single-valuedness. A vertical line should cut the graph in only one place. The graph labeled "No" clearly doesn't satisfy that. One can see that it attempts to have f(p) = q and f(p) = r, which is not allowed.
 * [[Image:vtlinetest1.jpg|200px]]   [[Image:vtlinetest2.jpg|200px]]

Example 3

Let $$y = f(x) = x^2+4x+7 \,$$. The possible values of $$x\,$$ form the domain.

Let us assume that there are only three possible values of $$x\,$$, that is, $$ \mathcal{D} = \{1,-3,10\}\,$$.

Then, the function $$f(x)\,$$ converts these three values of $$x\,$$ into three different values of $$y\,$$ which form the set $$\mathcal{R}$$. The set $$\mathcal{R}$$ is called the range of the function.

In this example, the three values that form the set $$\mathcal{R}$$ are found to be

\begin{align} f(1) & = (1)^2+4(1)+7 = 12 \\ f(-3) & = (-3)^2+4(-3)+7 = 4 \\ f(10) & = (10)^2+4(10)+7 = 147 \end{align} $$

Therefore the range is $$ \mathcal{R} \equiv \{4,12,147\}$$. If we were to draw the points $$[x, f(x)]\,$$ on a graph, we would have to graph (1,12), (-3,4) and (10,147).

NOTE

Instead of graphing points (x,y), we are graphing points (x,f(x)). For our purposes, the notation is different, but the meanings are the same.

Example 4

Let g(x) = x²-5x-9. D = {4,0,-3,$$a^4$$,x+h}. Find the elements in the range.

This is a pretty crazy example, but there are reasons to do it.

g(4) = (4)²-5(4)-9 = -13 g(-3) = (-3)²-5(-3)-9 = 15

g(0) = 0²-5(0)-9 = -9

g($$a^4$$) = ($$a^4$$)²-$$5a^4$$-9 = $$a^8$$-$$5a^4$$-9

g(x+h) = (x+h)²-5(x+h)-9 = x² + 2xh +h²-5x-5h-9                                               Wherever there is an x,                                                                                you replace it by x + h!

The range is {-13,-9,15,$$a^8$$ -$$5a^4$$-9,x² +2xh +h²- 5x -5x -9}.

Example 5

If $$f(x)=\frac{x}{x+5}$$, find $$\frac{f(x+h)-f(x)}{h}$$:

 * $$\frac{f(x+h)-f(x)}{h} = \frac{x+h} {x+h+5}-\frac{x}{x+5}/{h}$$    (Add the fractions. Two tricks: a/b-c/d=(ad-bc)/bd;(e/f)/h=e/fh.)

 * $$\frac{f(x+h)-f(x)}{h} = \frac{(x+h)(x+5)-x(x+h+5)}{(x+h+5)(x+5)h}$$    (Multiply out the top; never multiply out the bottom.)

 * $$\frac{f(x+h)-f(x)}{h} = \frac{5h}{(x+h+5)(x+5)h} = \frac{5}{(x+h+5)(x+5)}$$    (Cancel the h's.)

This kind of problem occurs in almost every precalc book. What you should ask is why the heck it is here. I will tell you. This is very close to the topic you deal with in calculus. Here is a preview.

We have learned that the slope of a straight line is always the same. However, if we draw any curve and draw all its tangent lines, the slope changes. We would like to study this and algebraize it.

Given the point $$P(x,f(x))\,$$. A little bit away from P is point Q. Its x value is x + h, where x + h is an x value a little bit away from x. If the first coordinate is x + h, the second coordinate is f(x + h). Draw PQ, PR (horizontal line), and QR (vertical line). On any horizontal line all y values are the same. P and R have the same y values. Q and R have the same x values.

Since Q and R have the same x values, the length of QR, which is the change in y or Δy, is $$f(x + h) - f(x)\,$$. Since P and R have the same y values, the length of PR, which is the change in x or Δx, is $$(x + h) - x - = h\,$$.

The slope of the secant line l2 joining the points P and Q is

 * $$\frac{\Delta y}{\Delta x} = \frac{f(x+h)-f(x)}{h}$$

which is why we study this expression. But here's the conclusion. If we let h go to 0, graphically it means the point [x+h,f(x+h)] gets closer and closer to [x,f(x)]. If we do this process to the left of P as well as here to the right of P, and if they both approach the line l_1, then what we have calculated is the slope of the tangent line l1 at the point [x,f(x)]!!!!! You have taken your first step into calculus!!!!!

1-1 function. 1-1 is a property we need occasionally.

DEFINITION

If f(a) = f(b), then a = b.

Examples 6 and 7

f(x) = 2x is 1-1, but g(x) = x² is not 1-1.

If f(a) = f(b), then 2a = 2b and a = b.

If g(a) = g(b), then a² = b². But a could equal b or -b; therefore, not 1-1.

One-to-One, Onto, One-to-One Correspondence, and Inverse Function
If every point in the range of a function is actually mapped to (that is, the range is the same as the image), the function is said to be onto. A fancy term for this is that the function is surjective or is a surjection.

If no two points in the domain map to the same value, the function is said to be one-to-one. A fancy term for this is that the function is injective or is an injection.

If a function is both one-to-one and onto, it is said to be a one-to-one correspondence. A fancy term for this is that the function is bijective or is a bijection.

When a function is a one-to-one correspondence, something interesting happens. Refer back to the four fundamental principles given earlier. The third principle is that no two pairs have the same first element. But if the function is one-to-one, no two pairs have the same second element either. The fourth principle is that every element in the domain appears as the first item of one of the pairs. But if the function is onto, every element of the range appears as the second item of one of the pairs. This means that, if the function is both one-to-one and onto, we can swap the domain and range, and swap the first and second items of each pair, and still get a function. That function is the inverse of the original function. A bijection is invertible. The inverse of the function f is written f-1. We have
 * f ( f-1 ( x ) ) = x for all x
 * f-1 ( f ( y ) ) = y for all y

Image and Inverse Image
It is sometimes useful to speak of all of the function values for an entire set of points in the domain. Normally, when we say y=f(x), we mean that x is a specific element of the domain (for example, a specific number), and y is the single element that x maps to.

But sometimes we want to write something like Y=f(X), where X is a whole set. In this case, Y is the whole set of function values for all elements of X.
 * $$f(X) = \{\ y\ |\ y=f(x)\ \text{for}\ \text{some}\ x\ \in\ X\ \}$$

Y=f(X) is called the image of X. It is illustrated in the diagram to the right. The horizontal black line along the bottom edge depicts an open interval that we wish to find the image of. The vertical black line along the right edge depicts the image. The part of the function graph forming the image is shown in blue; the rest of the graph is in red. The grey lines are visual guides.

Also, X is called the inverse image of Y, denoted X=f-1(Y). That is, f-1(Y) is the set of points that map into Y.
 * $$f^{-1}(Y)\ =\ \{\ x\ |\ f(x)\ \in\ Y\ \}$$

This is illustrated in the diagram to the right. The vertical black line along the right edge depicts an open interval that we wish to find the inverse image of. The horizontal black lines along the bottom edge depict the two open intervals comprising the inverse image. The part of the function graph forming the inverse image is shown in blue; the rest of the graph is in red. The grey lines are visual guides.

title: Function (mathematics)


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User talk:25member~enwikiversity

Hello 25member, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 13:50, 10 August 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called 25member. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name 25member~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:26, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:01, 19 April 2015 (UTC)

title: User talk:25member~enwikiversity


text:
Template:Free learning

  Free Learning
 * Participating at Wikiversity
 * Wikimedia projects
 * Internet resources
 * World 1.0

title: Template:Free learning


text:
How to use wiki technology as a free learner

This learning project is for participants who create learning resources about how to use wiki technology as a tool to support communities of online learners. The goal is to learn about wiki and other computer technologies by creating digital media files that illustrate strategies for online learning that fit well with the Wikiversity "learn by doing" model for online learning.

Collaborative video creation

 * Experimental attempt to create learning resources that combine conventional screencasting and 3D animation. The basic idea is to create tutorials about using wiki and other computer technology. The basic structure of these tutorials will be the screencast format in which images of webpages are used to provide "walk throughs" of how to use computer software. The screencast format will include elements of video podcasts.

A basic goal is to replace two conventional elements of screencasts with elements of 3D animation.
 * 1) the cursor - rather than just using an arrow to point at items shown on the screen image, use an animated character
 * 2) disembodied voice - rather than just use a disembodied voice to narrate the screencast, use an animated character that is physically interacting with the web page content.

Editing tutorial
The editing tutorial video used at Introduction has been in need of an update. The old video uses the old style edit tab that just says "edit", but Wikiversity switched to "edit this page". Also, the old version was done in the Wikiversity Reports format which is only a distraction for a simple tutorial on a wiki page.

Figure 1 shows a quick "proof of concept" for placing a 3D animated human figure on a wiki page. The idea was to "pop" a human figure out of the image on the main page and then animate that human figure to draw attention to a link on the main page. The 3D character is acting like a fancy mouse cursor. Figure 1 (to the right) is an animated gif that indicates what the 3D human figure looks like. The ogg format video makes use of narration from the original tutorial and should probably be modified so as to allow the 3D character to appear to say things during the tutorial. An important change from the original editing tutorial's screencast was to set the minimum font size for the browser to 18 points. One goal is to see if useful Wikiversity tutorials can be uploaded to YouTube which has a 320 x 240 pixel video format. The Original editing tutorial was done at 640 x 480 and has smaller font sizes.

A "bluescreen" technique was used to layer the animated 3D figure onto the screencast video (see Figure 2 for an example). The approach used to create a human figure for "bluescreen" overlay of a webpage was to create a DAZ Studio "primitive", a flat plane, and assign to that plane a saved image file of the wiki page to be shown behind the human figure. In the DAZ Studio scene, the human figure was placed between that plane and the position of the camera. This image of the webpage was used as a guide for the 3D character animation so that the character would correctly point to features of the page being described in the narration of the tutorial. If you look closely at the ogg format video for the "bluescreen" example, you can see some green tint on the human figure, particularly the hands. Green was used for this example rather than blue because of the blue clothing for the human figure. This example was made using the "eZeScreen plugin for iMovie.

The new "3D" version of the editing tutorial movie has been uploaded to YouTube (see). The version at YouTube is not too bad for 320 x 240 video of a webpage. Almost all of the webpage text shown in the video is readable; the "inset" "edit this page" button that is inside an image of the webpage is not really readable at 320 x 240. It looks good in the 640 x 480 OGG video format version (9.2 MB, 1 minute 17 seconds).

A modified version needs to be made for use at Introduction. All mentions of "visit Wikiversity" need to be changed for use of the tutorial inside Wikiversity.

Related reading

 * 3D animated movie project
 * Examples of collaborative video creation


text: File:FunctionEx.png

title: File:FunctionEx.png


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User:Crochet.david.bot

Flag de Bot

 * Actuellement en cours sur : JA et PT
 * Validé sur CS, ES, FR, EN ,DE, EL et IT

Script
The script running can be see there.

ar:مستخدم:Crochet.david.bot cs:Uživatel:Crochet.david.bot de:Benutzer:Crochet.david.bot el:Χρήστης:Crochet.david.bot es:Usuario:Crochet.david.bot fi:Käyttäjä:Crochet.david.bot fr:Utilisateur:Crochet.david.bot it:Utente:Crochet.david.bot ja:利用者:Crochet.david.bot ko:사용자:Crochet.david.bot pt:Utilizador:Crochet.david.bot ru:Участник:Crochet.david.bot sl:Uporabnik:Crochet.david.bot sv:Användare:Crochet.david.bot

title: User:Crochet.david.bot


text:
User:Crochet.david.bot/Navigation

cs:Uživatel:Crochet.david.bot/Navigation de:Benutzer:Crochet.david.bot/Navigation el:Χρήστης:Crochet.david.bot/Navigation es:Usuario:Crochet.david.bot/Navigation fi:Käyttäjä:Crochet.david.bot/Navigation fr:Utilisateur:Crochet.david.bot/Navigation it:Utente:Crochet.david.bot/Navigation ja:利用者:Crochet.david.bot/Navigation pt:Usuário:Crochet.david.bot/Navigation sv:Användare:Crochet.david.bot/Navigation

title: User:Crochet.david.bot/Navigation


text: Category:Complex analysis

title: Category:Complex analysis


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Biographical science journalism

This is a "learn by doing" project in which participants read and discuss news articles that focus on the scientists who are involved in scientific discoveries.

Introduction
In some cases, reporting about some science-related stories becomes heavily involved in details about specific scientists. Why do some scientists become news-worthy celebrities while other scientists are almost invisible? Do news reports about famous scientists change in time, particularly after their death? What processes determine if a story about a scientist is "newsworthy"?

Carl Sagan

 * 1,950,000 internet hits (Google)
 * 12,700 news hits (Google)

Carl Sagan and Immanuel Velikovsky. Carl Sagan was heavily involved in Planetary science, particularly the use space craft to collect data from planets such as Venus and Mars. Sagan worked to educate non-scientists about how science is done and why everyone should be skeptical about extraordinary claims such as those made by Velikovsky in his 1950 book "Worlds in Collision". After its publication, the New York Times listed "Worlds in Collision" for many weeks as a non-fiction bestseller. Astronomers such as Sagan dismissed Velikovsky's work as non-scientific guess work and an attempt to construct an astronomical basis for ancient myths.

By the time of the Mariner 2 mission (1962), attempts were being made by scientists such as Sagan to use observations from space craft to study the atmosphere of Venus and understand the basis of the greenhouse effect on Venus. Science Journalists misunderstood one of the Mariner mission scientists and published stories saying that the atmosphere of Venus contains hydrocarbons that contribute to the greenhouse effect. Some followers of Velikovsky's "Worlds in Collision" felt that NASA's scientific observations of Venus had found supporting evidence for one of Velikovsky's claims about the planet Venus, which Velikovsky claimed originated as a comet ejected from Jupiter several thousand years ago. Scientists such as Sagan had to speak up in order to correct the record, as described in the later editions of Sagan's book, "Broca's Brain".

Exercises and discussion

 * Find, read and discuss some of the news reports that were written following the A.A.A.S. Symposium on "Velikovsky’s Challenge to Science" held on February 25, 1974.
 * Which book continues to sell more copies, "Worlds in Collision" or "Broca's Brain"?

Usenet and other forum discussions have noted the claim that in several instances, raw data involved in phenomena bearing on Velikovsky's theories actually support Velikovsky but that by the time the stories are published, there are invariably explanations of how the experiments in question must have failed, and the data published is that which would have coincided with standard theories since those are always assumed to be correct. The most major such case is the question of albedo (reflectivity) values for Venus as described in an article by F.W. Taylor of the Clarendon Laboratory at Oxford in an article on "VENUS", Hunton, Colin, Donahue, Moroz, Univ. of Ariz. Press, 1983, ISBN 0-8165-0788-0. Taylor notes that the observed albedo value of .080 would require the planet to be massively out of thermal balance (as Velikovsky predicted) and that, therefore the value .076 which would produce thermal balance, required by the conventional theory for explaining the surface temperature of the planet, is the "most probable value"

Other such phenomena include ancient motion charts for the planet Venus, and infrared flux measurements associated with the Pioneer Venus mission.

 * Was the 1983 book talking about Bond albedo as shown in this table? Are you saying that a difference between 0.08 and 0.076 supports the claim that Venus is of recent origin? --JWSchmidt 04:08, 13 December 2007 (UTC)

Linus Pauling

 * 944,000 (Google)
 * 7,430 (Google)

Francis Crick

 * 602,000 (Google)
 * 4,370 (Google)

Stephen Hawking

 * 1,950,000 (Google)
 * 14,200 (Google)

Francis Collins

 * 282,000 (Google)
 * 5,360 (Google)

E. O. Wilson

 * 544,000 (Google)
 * 2,170 (Google)

title: Biographical science journalism


text:
File:Refardeon Seduced by the Dark Side.pdf

Used with the WikiU Film School Lesson on how to format a movie script


text:
User talk:BomberJoe

Bomber Joe is a retired CFI and propellerhead with 30 years in the right seat

title: User talk:BomberJoe


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History of Vietnam (1945-2013 History)

Summary
The Vietnam War occurred in present-day Vietnam in Southeast Asia. It showed a successful attempt on the part of the Democratic Republic of Vietnam (North Vietnam, DRV) and the National Front for the Liberation of Vietnam (Viet Cong) to unite and impose a communist system over the entire nation.

Opposing the DRV was the Republic of Vietnam (South Vietnam, RVN), backed by the United States. The war in Vietnam occurred during the Cold War, and is generally viewed as a bitter relationship between the United States and Soviet Union, with each nation and its allies supporting one side. Sources: Military History

title: History of Vietnam (1945-2013 History)


text:
User talk:Gatesofeden

Hello Gatesofeden, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 03:35, 11 August 2007 (UTC)

title: User talk:Gatesofeden


text:
Nonlinear finite elements/Quiz 0

Please answer the following questions. Answers such as "don't know", "knew once but have forgotten", and "never seen that before" are acceptable.

 { +Linear algebra (Matrices, Vector algebra). { +Vector calculus. { +Tensor algebra and calculus. { +Partial Differential Equations. { +Variational calculus. { +Numerical analysis. { +Finite element analysis. { +Elasticity. { +Plasticity. { +Viscoelasticity. { +Continuum Mechanics. { +Heat Transfer. { +Fluid Dynamics. { +Electromagnetism.
 * Mark the topics on the following list that you have studied in the past (in class or on your own):
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * What do the three terms in the following equation mean?

\mathbf{K} \mathbf{d} = \mathbf{f} $$
 * What is the physical meaning of the equation (think Newton's laws)?

 * What physical process does the following equation model?

\boldsymbol{\nabla} \bullet \boldsymbol{\sigma} = 0 $$
 * What do the symbols mean?

 * What physical process is modeled by the following equation?

\frac{\partial T}{\partial t} - \frac{1}{C_v~\rho} \boldsymbol{\nabla} \bullet (\boldsymbol{\kappa}\bullet\boldsymbol{\nabla} T) = Q       $$
 * What do the symbols mean?

 { +Lagrange multipliers. { +Euler-Lagrange equations. { +Weak and strong forms. { +Newton's method for finding roots.
 * Have you ever encountered:
 * coef="0"}
 * coef="0"}
 * coef="0"}
 * coef="0"}

title: Nonlinear finite elements/Quiz 0


text:
File:Film School Movie Script Romeo Juliet et al.pdf

Used with the WikiU Film School course on filmmaking for high school drama departements


text: File:Film school script page Romeo Juliet el al.png


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Nonlinear finite elements/Sets

A familiarity with the notation of sets is essential for the student who wants to read modern literature on finite elements. This handout gives you a brief review of set notation. More details can be found in books on advanced calculus.

Sets
A set is a well-defined collection of objects. As far as we are concerned, these objects are mainly numbers, vectors, or functions.

If an object $$x$$ is a member of a set $$A$$, we write

x \in A (x~\text{belongs to}~A)~. $$

If $$x$$ is not a member of $$A$$, we write

x \notin A (x~\text{does not belong to}~A)~. $$

An example of a  finite set (of functions) is

S = \{x, \sin(x), \cos(x), \exp(x), \log(x)\}~. $$

Another example is the set of integers greater than 5 and less than 12

B = \{6, 7, 8, 9, 10, 11\}~. $$

If we denote the set of all integers by $$\mathbb{Z}$$, then we can alternatively write

B = \{n ~|~ n \in \mathbb{Z}, 5 < n < 12\}~. $$

The set $$\mathbb{Z}^{+}$$ of positive integers is an  infinite set and is written as

\mathbb{Z}^{+} = \{n ~|~ n \in \mathbb{Z}, n > 0\}~. $$

An  empty (or  null) set is a set with no elements. It is denoted by $$\varnothing$$. An example is

\varnothing = \{n ~|~ n \in \mathbb{Z}^{+}, n < 0\}~ = \{\}~. $$

Subsets
If $$A$$ and $$B$$ are two sets, then we say that $$A$$ is a  subset of $$B$$ if each element of $$A$$ is an element of $$B$$.

For example, if the two sets are

A = \{\sin(x), \cos(x)\}~ B = \{x, \sin(x), \cos(x), \exp(x), \log(x)\}\text{and} $$ we write

A \subset B (A~\text{is a proper subset of}~B)~. $$

On the other hand, if $$A$$ is a subset of $$B$$ which may be the set $$B$$ itself we write

A \subseteq B (A~\text{is a subset of}~B)~. $$

If $$A$$ is not a subset of $$B$$, we write

A \not\subset B (A~\text{is not a subset of}~B)~. $$

Equality of sets
Two sets $$A$$ and $$B$$ are equal if they contain exactly the same elements. Thus,

A = B \iff A \subseteq B ~\text{and}~ B \subseteq A ~. $$ The symbol $$\iff$$ means  if and only if.

For example, if

A = \{x ~|~ x^2 = 4\} ~\text{and}~ B = \{2, -2\} $$ then $$A = B$$.

Union, Intersection, Difference of Sets
The  union of two sets $$A$$ and $$B$$ is the set of all elements that are in $$A$$ or $$B$$.

C = A \cup B = \{x ~|~ x \in A ~\text{or}~ x \in B\}~. $$

The  intersection of two sets $$A$$ and $$B$$ is the set of all elements that are both in $$A$$ and in $$B$$.

D = A \cap B = \{x ~|~ x \in A ~\text{and}~ x \in B\}~. $$

The  difference of two sets $$A$$ and $$B$$ is the set of all elements that are in $$A$$ but not in $$B$$.

E = A - B = \{x ~|~ x \in A ~\text{and}~ x \not\in B\}~. $$

The  complement of a set $$A$$ (denoted by $$A^{'}$$) is the set of all elements that are not in $$A$$ but belong to a larger universal set $$U$$.

A^{'} = U - A = \{x ~|~ x \not\in A\}~. $$

Countable Sets
Suppose we have a set $$A$$. Such a set is called  countable if each of its members can be labeled with an integer subscript of the form

A = \{a_1, a_2, a_3, a_4, \dots\}~. $$

Obviously, each finite set is countable. Some infinite sets are also countable. For instance, the set of integers is countable because you can label each integer with an subscript that is also an integer. However, you cannot do that with the real numbers which are  uncountable.

The set of functions

P = \{f_k ~|~ f_k(x) = x^k, k=1,2,\dots\} $$ is countable.

The set of points on the real line

A = \{x ~|~ 0 \le x \le 1\} $$ is not countable because the points cannot be labeled $$a_1$$, $$a_2$$, $$\dots$$.

Cartesian Product
The Cartesian product of two sets $$A$$ and $$B$$ is the set of all  ordered pairs $$(a,b)$$, such that

A \times B = \{(a,b) ~|~ a \in A, b \in B \}~. $$ In general, $$A \times B \neq B \times A$$.

For example, if

A = \{1, 5, 7\} ~\text{and}~ B = \{3, 6\} $$ then

A \times B = \{(1,3),(1,6),(5,3),(5,6),(7,3),(7,6)\} $$ and

B \times A = \{(3,1),(3,5),(3,7),(6,1),(6,5),(6,7)\} \neq A \times B~. $$

The Set of Real Numbers
The set of real numbers ($$\mathbb{R}^{}$$) can be visualized as an infinitely long line with each real number being represented as a point on this line. We usually deal with subsets of $$\mathbb{R}^{}$$, called  intervals.

Let $$a$$ and $$b$$ be two points on $$\mathbb{R}^{}$$ such that $$a \le b$$. Then,

 * The  open interval $$(a,b)$$ is defined as

(a,b) = \{x ~|~ x \in \mathbb{R}, a < x < b\}~. $$
 * The  closed interval $$[a,b]$$ is defined as

[a,b] = \{x ~|~ x \in \mathbb{R}, a \le x \le b\}~. $$
 * The  half-open intervals $$(a,b]$$ and $$[a,b)$$ are defined as

(a,b] = \{x ~|~ x \in \mathbb{R}, a < x \le b\} ~\text{and}~       [a,b) = \{x ~|~ x \in \mathbb{R}, a \le x < b\}~. $$

Let $$p \in \mathbb{R}^{}$$ and $$\epsilon > 0$$. Then the  neighborhood of $$p$$ is defined as the  open interval

\text{nbd}(p;\epsilon) := (p - \epsilon, p+\epsilon) = \{ x ~|~ p - \epsilon < x < p + \epsilon \} $$

Let $$X \subset \mathbb{R}^{}$$. Then $$p$$ is an  interior point of $$X$$ if we can find a nbd($$p$$) all of whose points belong to $$X$$.

If every point of $$X$$ is an interior point, then $$X$$ is called an  open set. For example, the interval $$(a,b)$$ is an open set. So is the real line $$\mathbb{R}^{}$$.

A set $$X \subset \mathbb{R}^{}$$ is called  closed if its complement $$X^{'} = \mathbb{R}^{} - X$$ is open.

The  closure $$\bar{X}$$ of a set $$X \subset \mathbb{R}^{}$$ is the union of the set and its boundary points (a rigorous definition of closed sets can be made using the concept of points of accumulation).

Open and Closed Sets in Rn
The concept of the real line can be extended to higher dimensions. In two dimensions, we have $$\mathbb{R}^{2}$$ which is defined as

\mathbb{R}^{2} = \mathbb{R}^{}\times\mathbb{R}^{} = \{(x,y) ~|~ x,y \in \mathbb{R}^{}\}~. $$ $$\mathbb{R}^{2}$$ can be thought of as a two-dimensional plane and each member of the set $$\mathbf{x} = (x,y)$$ represents a point on the plane.

In three dimensions, we have

\mathbb{R}^{3} = \mathbb{R}^{}\times\mathbb{R}^{}\times\mathbb{R}^{} = \{(x,y,z) ~|~ x,y,z \in \mathbb{R}^{}\}~. $$

In $$n$$ dimensions, the concept is extended to mean

\mathbb{R}^{n} = \mathbb{R}^{}\times\mathbb{R}^{}\times\mathbb{R}^{}\dots\times\mathbb{R}^{} = \{(x_1,x_2,x_3,\dots,x_n) ~|~ x_1,x_2,x_3,\dots,x_n \in \mathbb{R}^{}\}~. $$

In the case of sets in $$\mathbb{R}^{n}$$ the concept of distance in $$\mathbb{R}^{}$$ is extended so that

\text{nbd}(\mathbf{p};\epsilon) := \{ \mathbf{x} ~|~ \mathbf{p} \in \mathbb{R}^{n}, |\mathbf{x} - \mathbf{p}| < \epsilon \} $$ where

|\mathbf{x} - \mathbf{p}| = \sqrt{(x_1 - p_1)^2 + (x_2 - p_2)^2 + \dots + (x_n - p_n)^2}~. $$

The definition of  interior point also follows from the definition in $$\mathbb{R}^{}$$. Thus if $$\Omega \subset \mathbb{R}^{n}$$, then $$p \in \mathbb{R}^{n}$$ is an interior point if we can always find a nbd$$(p;\epsilon)$$, all of whose points belong to $$\Omega$$. If every point on $$\Omega$$ is an interior point, then $$\Omega$$ is an  open set. As in the real number line, a  closed set is the complement of an open set. One way of creating a closed set is by taking an open set $$\Omega$$ and its boundary $$\Gamma$$. This particular closed set is called the  closure $$\bar\Omega$$ of $$\Omega$$. A rigorous definition can once again be obtained using the concept of points of accumulation.

title: Nonlinear finite elements/Sets


text:
Nonlinear finite elements/Functions

-- Back to Nonlinear finite elements --

There are certain terms involving relationships between functions that you will often encounter in papers dealing with finite elements and continuum mechanics. We list some of the basic terms that you will see. More details can be found in books on advanced calculus and functional analysis.

Functions
Let $$A$$ and $$B$$ be two sets. A function is a rule that assigns to each $$a \in A$$ an element of $$B$$. A function is usually denoted by

f : A \rightarrow B ~(f~\text{takes}~A~\text{to}~B)~. $$ Sometimes, one also writes

a \mapsto f(a) ~(a~\text{is mapped to the element}~f(a))~. $$ For example, if the function is $$f(x) = x^2$$, then we may write $$x \mapsto x^2$$.

Domain and Range
For a function $$f : A \rightarrow B$$, the set $$A$$ is called the domain of $$f$$. See Figure 1.

The range of $$f$$ is the set

R(A) = \{ f(x) \in B ~|~ x \in A \} $$ Therefore, $$R(A) \subset B$$.

One-to-one mapping (injection)
A function $$f : A \rightarrow B$$ is called one-to-one (or an injection) if no two distinct elements of $$A$$ are mapped to the same element of $$B$$.

Onto mapping (surjection)
A function $$f : A \rightarrow B$$ is called onto if for every $$b \in B$$ there is an $$a \in A$$ such that $$f(a) = b$$.

If that case, $$R(A) = B$$.

One-to-one and onto mapping (bijection)
When a mapping is both one-to-one and onto it is called a bijection. For example, if $$f : A \rightarrow B$$ and $$f(x) = x^2$$ with

A = \{x \in \mathbb{R}^{} ~|~ x \ge 0\} ~; B = \{x \in \mathbb{R}^{} ~|~ x \ge 0\} $$ the map is one-to-one and onto.

On the other hand, if

A = \{x \in \mathbb{R}^{} ~|~ x \ge 0\} ~; B = \{x \in \mathbb{R}^{}\} $$ the map is one-to-one but not onto.

If we choose

A = \{x \in \mathbb{R}^{}\} ~; B = \{x \in \mathbb{R}^{}\} $$ the map is neither one-to-one nor onto.

Image, pre-image, and inverse functions
Suppose we have a function $$f : A \rightarrow B$$. Let $$D$$ be a subset of $$A$$ (see Figure~1). Let us define

f(D) := \{f(d) \in B ~|~ d \in D\}~. $$ Then, $$f(D)$$ is called the image of $$D$$.

On the other hand, if $$C$$ is a subset of $$B$$ and we define

f^{-1}(C) := \{a \in A ~|~ f(a) \in C\}~. $$ Then, $$f^{-1}(C)$$ is called the inverse image or pre-image of $$C$$.

If $$f : A \rightarrow B$$ is one-to-one and onto, then there is a unique function $$f^{-1} : B \rightarrow A$$ such that

f(f^{-1}(b)) = b ~\text{for all}~ b\in B \text{and}~ f^{-1}(f(a)) = a ~\text{for all}~ a\in A ~. $$ The function $$f^{-1}$$ is called an inverse function of $$f$$.

Identity map
The map $$f : A \rightarrow A$$ such that $$f(x) = x$$ for all $$x \in A$$ is called the identity map. This map is one-to-one and onto.

Composition
A notation that you will commonly see in papers on nonlinear solid mechanics is the composition of two functions. See Figure 2.

Let $$f$$ and $$g$$ be two functions such that $$f : A \rightarrow B$$ and $$g : B \rightarrow C$$. The composition $$g \circ f : A \rightarrow C$$ is defined as

g \circ f(a) := g(f(a)) $$

Let us consider a stretch ($$f$$) followed by a translation ($$g$$). Then we can write $$f : x \mapsto \lambda x$$ and $$g : y \mapsto y + c$$.

The composition $$g \circ f$$ is given by

g \circ f : x \mapsto \lambda x + c $$

The inverse composition $$f \circ g$$ is given by

f \circ g : x \mapsto \lambda(x + c) $$

Isomorphism and Homeomorphism
You will also come across the terms isomorphism and homeomorphism in the literature on nonlinear solid mechanics.

 Isomorphism is a very general concept that appears in several areas of mathematics. The word means, roughly, "equal shapes". It usually refers to one-to-one and onto maps that preserve relations among elements.

A homeomorphism is a continuous transformation between two geometric figures that is continuous in both directions. The map has to be one-to-one to be homeomorphic. It also has to satisfy the requirements on an equivalence relation.

Continuously differentiable functions
A function

f : \Omega \rightarrow \mathbb{R}^{} $$ is said to be  $$k$$-times continuous differentiable or of class $$C^k$$ if its derivatives of order $$j$$ (where $$0 \le j \le k$$) exist and are continuous functions.

Figure 3 shows three functions ($$f(x)$$, $$g(x)$$, $$h(x)$$) and their derivatives.

C-1 Functions
The function $$f(x)$$ is called the Heaviside step function (usually written $$H(x)$$) which is defined as

H(x) = \begin{cases} 0 & ~\rm{if}~ x < 0 \\ 1 & ~\rm{if}~ x \ge 0 \end{cases} $$ The derivative of the Heaviside function is the Dirac delta function (written $$\delta(x)$$) which has the defining property that

\int_{-\infty}^{\infty} F(x)\delta(x-c)~dx = F(c) $$ for any function $$F(x)$$ and any constant $$c$$. The delta function is singular and discontinuous. Hence, the Heaviside function is not continuously differentiable. Sometimes the Heaviside function is said to belong to the class of $$C^{-1}$$ functions.

C0 Functions
The function $$g(x)$$ in Figure 3 (also called a hat function) is continuous but has discontinuous derivatives. In this particular case, the function has the form

g(x) = \begin{cases} \cfrac{x}{a} +1 & ~\text{if}~ -a \le x \le 0 \\ 1 - \cfrac{x}{a} & ~\text{if} 0 \le x \le a           \end{cases} $$ Such functions that are differentiable only once are called $$C^0$$ functions.

C∞ Functions
The function $$h(x)$$ in Figure 3 is infinitely differentiable and has continuous derivatives every time it is differentiable. Such functions are called $$C^{\infty}$$ functions. Since the function can be differentiated once to give a continuous derivative, it also falls into the category of $$C^1$$ functions.

Sobolev spaces of functions
You will find Sobolev spaces being mentioned when you read the finite element literature. A clear understanding of these function spaces needs a knowledge of functional analysis. The book Introduction to Functional Analysis with Applications to Boundary Value Problems and Finite Elements by B. Daya Reddy is a good starting point that is just right for engineers. We will not get into the details here.

Of particular interest in finite element analysis are Sobolev spaces of functions such as

\mathcal{H}^k = \{w ~|~ w \in \mathcal{L}_2, \frac{\partial w}{\partial x} \in \mathcal{L}_2, \dots, \cfrac{\partial^k w}{\partial x^k} \in \mathcal{L}_2\} $$ where

\mathcal{L}_2 = \{w ~|~ \int_0^1 w^2 ~dx < \infty \} ~. $$

The function space $$\mathcal{L}_2$$ is the space of square integrable functions.

Of interest to us is an outcome of Sobolev's theorem which says that if a function is of class $$\mathcal{H}^{k+1}$$ then it is actually a bounded $$C^k$$ function. If we choose our basis functions from the set of square integrable functions with continuous derivatives, certain singularities are automatically precluded.

title: Nonlinear finite elements/Functions


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Urantia Book/Philosophy of the Urantia Book

From an introductory philosophy course, Plato asks if our senses and thoughts reveal reality or just a shadow of reality. The Urantia Book repeatedly refers to our material senses and teaches us how to hone our spiritual senses. Various passages assert that finite, material reality is like a shadow of infinite, spiritual reality. By choosing to uphold spiritual values, the Urantia Book teaches that our minds, in liaison with the inner, spiritual fragment of the Eternal Father, gives birth to our infant soul that has the potential to live on in eternity. This soul is our inheritance in the afterlife.

((primary citations needed))


text:
User:Prof. Vivek Sharma

About Vivek Sharma
Teacher, author and consultant with over 20 years of experience in a really wide range of fields including And so on to a wide variety of audience from under matriculate EDP tranees, school kids, under and post graduate students to retired & sr. Officers.
 * Entrepreneurship
 * Economics
 * Small and Medium business management
 * Marketing
 * Strategic management
 * Environmental issues
 * Soft Skill development
 * Statistics
 * Mentoring budding entrepreneurs

Skill Sets

 * Earned a rich practical knowledge working on a large number of projects including on *Entrepreneurship Development
 * Identifying market requirements
 * developing and running of innovative, multi-disciplinary and new courses based upon those emerging trends
 * contributed in quality control and teacher education in Higher Education
 * conducted, coordinated, contributed in or guided various surveys and studies, EDPs, EEDPs, SME development programs
 * cluster development, training and development of managers, stakeholder management in SME,
 * Project design, coordination and implementation in business, marketing, education, public policy
 * HR in education, SME and not for profit organizations
 * trained and coached budding entrepreneurs for change management.

Currently working on
A teacher with commitment to subject

Key Competencies
Ability to communicate and relate well with diverse groups - with a layman as well as a 'techi', from schoolgirls to retired officers and managers. Teaching, researching and developing new and interdisciplinary areas. Ability to bridge theoretical and practical aspects to facilitate learning Ability to identify areas, facilitate and develop collaborations for cooperation. Design and develop new, innovative, path breaking courses in higher education and development and dissemination of e-learning materials, Ability to conceive, develop, organize workshops, seminars, management game sessions, training sessions, manage speakers and participant concerns. Continuous learning – studied over 60 subjects at the university level, but learned more including Hinduism, Photography, ICT, Web content development, engaging people in fruitful activities.

Interests

 * Entrepreneurship Development using inputs from culture and religion.
 * Developing n – ach and other competencies using inputs from Gita the holy book of The Hindus.
 * SME, rural, micro and green entrepreneurship.
 * Collaboration development online and in-person.
 * Innovation, creativity and Intrapreneurship training
 * Change management related to Small & Medium scale Industry, mergers & acquisitions
 * Change management related to business process improvement
 * Developing and conducting personal mentoring programs for individuals in family business.
 * Developing succession management plan for family business
 * Marketing, communications and event management for 'Educational projects and initiatives'.
 * Identification of staff training needs and developing programs for SME.
 * Creating strategies for engaging, empowering and coaching various groups for development.
 * Business process improvement in SME.

Recent
I have been working upon varoius projects: E learning model and portal development, training of entrepreneurs in India using the inputs from Indian culture. Has anybody tried it in other regions? Need for achievement can be incorporated in one's personality by using concepts from 'Gita'- a holy book of Hindus, work ethics by using the concepts of 'kirat karo' from Sikhism, 'Halaal' from Islam

I am looking forward to similar work being done by others. Please leave me a massage or mail me. Vivek.

title: User:Prof. Vivek Sharma


text:
User talk:Prof. Vivek Sharma

I have been working upon the training of entrepreneurs in India using the inputs from Indian culture. Has anybody tried it in other regions? Need for achievement can be incorporated in one's personality by using concepts from 'Gita'- a holy book of Hindus, work ethics by using the concepts of 'kirat karo' from Sikhism, 'Halaal' from Islam

I am looking forward to similar work being done by others. Vivek.

Welcome
Visitors are welcome to leave massages or to contact me at profvs AT THE RARTE gmail dot com

About Vivek Sharma
Teacher, author and consultant with over 20 years of experience in a really wide range of fields including Entrepreneurship, Economics, Small and Medium business management,  Marketing, Strategic management, Environmental issues, Soft Skill development, Statistics and so on to  a wide variety of audience from under matriculate EDP tranees, school kids, under and post graduate students to retired sr. officers. Earned a rich practical knowledge working on a large number of projects including on Entrepreneurship Development, Identifying market requirements, developing and running of innovative, multi-disciplinary and new courses based upon those emerging trends, contributed in quality control and teacher education in Higher Education, conducted, coordinated, contributed in or guided various surveys and studies, EDPs, EEDPs, SME development programs, cluster development, training and development of managers, stakeholder management in SME, project design, coordination and implementation in business, marketing, education, public policy, HR in education, SME and not for profit organizations, trained and coached budding entrepreneurs for change management.

Currently working in SDCC - a college designated as a college with a potential for excellence by UGC, (Ministry of HRD, Govt. of India). H. Professor and coordinator EDPs, Abza.Org

Hony. Professor at Abza Universe and Abza Institute for Entrepreneurship.

Visiting prof. at various places.  Key Competencies
 * Ability to communicate and relate well with diverse groups - with a layman as well as a 'techi', from schoolgirls to retired officers and managers.

 * Teaching, researching and developing new and interdisciplinary areas.

 * Ability to bridge theoretical and practical aspects to facilitate learning

 * Ability to identify areas, facilitate and develop collaborations for cooperation.

 * Design and develop new, innovative, path breaking courses in higher education and development and dissemination of e-learning materials,

 * Ability to conceive, develop, organize workshops, seminars, management game sessions, training sessions, manage speakers and participant concerns.

 * Continuous learning – studied over 60 subjects at the university level, but learned more including Hinduism, Photography, ICT, Web content development,

Interests

 * Entrepreneurship Development using inputs from culture and religion.

 * Developing n – ach and other competencies using inputs from Gita the holy book of The Hindus.

 * SME, rural, micro and green entrepreneurship.

 * Collaboration development online and in-person.
 * Innovation, creativity and Intrapreneurship training

 * Change management related to Small & Medium scale Industry, mergers & acquisitions

 * Change management related to business process improvement

 * Developing and conducting personal mentoring programs for individuals in family business.

 * Developing succession management plan for family business

 * Marketing, communications and event management for 'Educational projects and initiatives'.

 * Identification of staff training needs and developing programs for SME.

 * Creating strategies for engaging, empowering and coaching various groups for development.

 * Business process improvement in SME.

 * Socio Economic projects for development and environment.

Services Offered

 * Development of Study Material on Entrepreneurship Development under creative commons license or OCW.

--Prof. Vivek Sharma 05:41, 16 January 2008 (UTC) Hello Prof. Vivek Sharma, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity!

P.S. Do you want live feedback ? You can find us in #wikiversity-en (see Chat) Erkan Yilmaz ( evaluate me!, discussion ) 14:19, 9 September 2007 (UTC)

P.S. Why do you have the same text on both user and discussion page ?

Professor Sharma, welcome to Wikiversity. May I also suggest that you have a look at WikiEducator as well; they have a very strong group of Indian educators collaborating on projects. more at: http://www.wikieducator.org/India You can contact me on that wiki at http://www.wikieducator.org/User:BrentSimpson Countrymike 07:11, 10 January 2008 (UTC)

Comments
Hi Professor Sharma, there have been some updates to the Social Entrepreneurship portal which might interest you, and you might like to join the mailing list. - Kim 22:23, 22 August 2008 (UTC).

What about this page ?
Hi, you once created this page: User:Prof.Vivek Sharma. There are still some pages linking to it. What shall we do with that page ? Erkan Yilmaz uses the Chat (try) 13:48, 23 August 2008 (UTC)

Can it be merged with this page, if not, Please delete it

Vivek --Prof. Vivek Sharma 06:31, 29 August 2008 (UTC)
 * There is now a redirect. What about the talk page ? You can also create a redirect if you like, Erkan Yilmaz uses the Chat (try) 14:37, 6 September 2008 (UTC)

title: User talk:Prof. Vivek Sharma