text:
School:Chemistry

Welcome to the School of Chemistry! FACULTY OF SCIENCES F ACULTY OF P HYSICAL S CIENCES  · F ACULTY OF L IFE S CIENCES  · F ACULTY OF E NGINEERING AND  T ECHNOLOGY  ·  F ACULTY OF M ATHEMATICS Biology · Chemistry · Computer Science ·Economics · Mathematics· Physics and Astronomy

Key Concepts

 * Fundamentals of Chemistry
 * Introduction to the Periodic Table, stoichiometry, chemical states, chemical equilibria, acid & base, oxidation & reduction reactions, chemical kinetics, inorganic nomenclature and chemical bonding.
 * Properties of Matter
 * Stoichiometry, Ideal gas law, electronic structure, chemical reactivity, inorganic and organic compounds.
 * Chemical Reactions
 * Liquids, solutions, principles of chemical equilibria, solubility, electrochemical processes, kinetics.

 * Concepts in Chemistry
 * Matter and Measurements
 * Atoms, Molecules, and Ions
 * Formulas, Equations, and Reactions
 * Mass Relations in Chemistry and Stoichiometry

 * Chemical Behavior
 * Electronic Structure
 * Ionic Bonding
 * Covalent Bonding
 * Thermochemistry
 * Quantum Theory of the Atom

 * States of Matter
 * Gases and Gas Laws
 * Solids
 * Liquids
 * Gases
 * Plasmas

 * Reactions and Equilibrium
 * Chemical kinetics
 * Chemical Equilibrium
 * Acids and Bases
 * Acid-Base Equilibrium
 * Solubility and Complex-Ion Equilibrium
 * Thermodynamics and Equilibrium

 * Other Essential Concepts in Chemistry
 * Fundamental Electrochemistry
 * Fundamental Nuclear Chemistry
 * Fundamental Organic Chemistry

Key Topics
General
 * Formulae, Equations, and Reactions

Inorganic Chemistry
 * Bonding and Chemical Structure
 * Symmetry Elements
 * Acid-Base Chemistry
 * Reduction and Oxidation Reactions
 * Organometallic Chemistry

Organic Chemistry
 * Nomenclature of Organic Chemistry
 * Molecular Recognition
 * Reactivity and Mechanism
 * Electronic Structure

Biological Chemistry
 * Biomolecular Equilibrium
 * Biochemical Kinetics
 * Ligand Binding and Multivalency
 * Biological Spectroscopy
 * Protein Stability
 * Biocalorimetry

Physical Chemistry
 * Fundamentals of Computational Chemistry
 * Thermodynamics
 * Quantum Mechanics
 * Chemical Kinetics

History of Chemistry

 * History of Chemistry
 * Basics of Chemistry

Chemistry Courses

 * Basic Chemistry
 * High School Chemistry
 * Advanced Chemistry
 * University Chemistry
 * Organic Chemistry

Departments

 * Analytical Chemistry
 * Atmospheric Chemistry
 * Biochemistry
 * Biophysical Chemistry
 * Electrochemistry
 * Food Chemistry

 * Green Chemistry
 * Inorganic Chemistry
 * Nanotechnology
 * Organic Chemistry
 * Pharmacology

 * Polymer Chemistry
 * Physical Chemistry

External Departments

 * Chemical Engineering

Youtube Videos
GENERAL CHEMISTRY explained in 19 Minutes - Wacky Science

 * This video offers an 18-minute and 49-second concise overview of General Chemistry, beneficial for review and serving as a brief introduction to the subject.

Wikipedia resources

 * Category:Chemistry journals
 * List of organic reactions
 * List of organic compounds
 * List of inorganic compounds
 * List of compounds
 * List of alloys
 * Inorganic compounds by element
 * List of minerals
 * List of biomolecules
 * List of basic chemistry topics
 * Table of standard reduction potentials for half-reactions important in biochemistry
 * List of phytochemicals and foods in which they are prominent
 * List of CAS numbers by chemical compound

 * List of chemistry topics
 * List of essential oils
 * List of basic biochemistry topics
 * List of biochemistry topics
 * Systematic name

Wikibooks

 * IB Chemistry
 * Mathematics for Chemistry
 * General Chemistry
 * Biochemistry
 * Biophysical Chemistry
 * Inorganic Chemistry
 * Organic Chemistry
 * Chemical Synthesis
 * Computational Chemistry
 * Crystallography
 * Analytical Forensic Pharmacology

Active participants
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

 * Sign your name here with four ~ keys sequentially.
 * Dominant group/Planetary science Cometary chemistry, Active since 11 September 2011. --Marshallsumter (discuss • contribs) 18:43, 22 January 2020 (UTC)

Things you can do!

 * Clean up Draft:Chemicals and move it to become a subpage of a supporting main page learning project.
 * Clean up Draft:Chemistry and move it to become a subpage of a supporting main page learning project.
 * Clean up Draft:Phosphate biochemistry and move it to become a subpage of a supporting main page learning project.

School news

 * 25 August 2006 - School founded!
 * September 2006 - Two journals started on the Academic Publishing Wiki [wiki] which are relevant to Chemistry: Interpretations in the Physical and Computational Sciences [] and Education in the Sciences [].
 * 10 October 2012 - Radiation astrochemistry announced on Main Page News!
 * 8 July 2013 - Renovation of the School begins!
 * 1 January 2015 - The full-semester course Principles of radiation astronomy is up and running, including the lecture Radiation astronomy/Chemistry.

Related news

 * April 18, 2007 - Researchers are studying converting carbon dioxide into fuel utilizing solar energy...

Humor
Take a break and prevent your head from exploding.

Rules of the lab
1. When you don't know what you're doing, do it neatly.

2. Experiments must be reproduceable, they should fail the same way each time.

3. First draw your curves, then plot your data.

4. Experience is directly proportional to equipment ruined.

5. A record of data is essential, it shows you were working.

6. To study a subject best, understand it thoroughly before you start.

7. To do a lab really well, have your report done well in advance.

8. If you can't get the answer in the usual manner, start at the answer and derive the question.

9. If that doesn't work, start at both ends and try to find a common middle.

10. In case of doubt, make it sound convincing.

11. Do not believe in miracles---rely on them.

12. Team work is essential. It allows you to blame someone else.

13. All unmarked beakers contain fast-acting, extremely toxic poisons.

14. Any delicate and expensive piece of glassware will break before any use can be made of it. (Law of Spontaneous Fission)

Brief guide to scientific literature
It has been long known = I haven't bothered to check the references

It is known = I believe

It is believed = I think

It is generally believed = My colleagues and I think

There has been some discussion = Nobody agrees with me

It can be shown = Take my word for it

It is proven = It agrees with something mathematical

Of great theoretical importance = I find it interesting

Of great practical importance = This justifies my employment

Of great historical importance = This ought to make me famous

Some samples were chosen for study = The others didn't make sense

Typical results are shown = The best results are shown

Correct within order of magnitude = Wrong

The values were obtained empirically= The values were obtained by accident

The results are inconclusive = The results seem to disprove my hypothesis

Additional work is required = Someone else can work out the details

It might be argued that = I have a good answer to this objection

The investigations proved rewarding = My grant has been renewed

Synthesised according to standard protocols = Purchased

Phrases
Remember, if you're not part of the solution, you're part of the precipitate!

Never replicate a successful experiment -Fett's law.

It takes alkynes to make a world.

A couple of months in the laboratory can frequently save a couple of hours in the library.

Make it myself? But I'm a physical organic chemist!

Old Chemists never die, they just fail to react.

First law of Laboratorics: Hot glass and cold glass look alike!

title: School:Chemistry


text:
School talk:Chemistry

Confusing
This structure seems confusing still, but in time, it should be alright. --HappyCamper 15:49, 27 August 2006 (UTC)

Would someone explain the difference between "Key Concepts" and "Key Topics"? Should these not be combined? 173.249.140.15 (discuss) 14:34, 6 August 2013 (UTC)

 * Hi, I know this is old but this is near the top and pretty relevant to the topic. But I will attempt to fix the structure of the school and yes, in time it will be alright :D Heartwarming (discuss • contribs) 01:21, 27 December 2015 (UTC)

Request for courses and information
Given a formula, how can an aspiring chemist make that substance or matter? What information is needed (if any, other than the formula) to produce it safely? (That reminds me: chemical and practical chemistry work safety).

Suggestions for practical work. Basic chemistry kit (advanced high school level and onward), things needed, how to use them, how to store chemicals and produced matter safely. Experiments to perform.

Suggested textbooks to get a student on-track to satisfying practical chemistry learning as soon as possible. How to discover new compounds. How to predict when experimenting whether it will blow up or be safe.

I have found most chemistry textbooks confusing and they almost never give any lab work to do. The best chapter on chemistry at the very introductory level I ever read was in Biology by Campbell and Reece (my edition is 6th).

--wanttoknow 18:27, 07 September 2006 (AEST)

Resource
This is a great resource. I am a chemistry student right now - but it needs a lot of work (although I realize it's all in Beta).

But, great idea nonetheless. I'll see what I can do as the semester goes on for the chem section. It could use some pictures.

128.255.166.141 22:28, 19 September 2006 (UTC) (Dan Cook)

Chemistry Userbox
Those that contribute to this school can put the following userbox on their userpage. It is currently on my userpage namespace, but I hesitate to move it for fear of deletion. Please let me know whether this is acceptable or not, any changes you might suggest, or anything of that nature. If it has adaquate support, I will move it to the Template namespace. -- Mathboy965 14:11, 27 September 2006 (UTC)

 * Hm...that looks like a pretty neat userbox. Personally, I think it's best to keep userboxes out of the template namespace. These things were quite polarizing on Wikipedia, and it's probably best not to import too much of the surrounding conflicts over to Wikiversity. --HappyCamper 14:40, 27 September 2006 (UTC)

 * Sounds good. --Mathboy965 15:40, 27 September 2006 (UTC)

 * Wikiversity currently has one set of userboxes that are approved to be located in the "Template:" namespace, the language boxes, see: Category:Users by language. Userboxes that say, "I participate in the Wikiversity school of Foo," can serve a useful function in helping build Wikiversity communities of users with common academic interests. Wikiversity schools and departments function as Wikiversity content creation/improvement projects. Is there really any reason to exclude "improvement project participant" templates from the "Template:" namespace? Wikipedia has w:Category:Wikipedians by collaboration and allows project participation userboxes in the "Template:" namespace. --JWSchmidt 16:27, 27 September 2006 (UTC)

 * Well, can we put those in the template namespace then? --HappyCamper 02:52, 28 September 2006 (UTC)
 * Unless someone can think of a reason not to. --JWSchmidt 02:54, 28 September 2006 (UTC)
 * Okay then, let's make it simple and say userboxes in the spirit of the above can be in the template namespace then. --HappyCamper 03:01, 28 September 2006 (UTC)
 * Moved to Template:User Chemcontrib. Can everyone put on their userpages instead? And Mathboy - we don't just delete pages because they're in the wrong location. Be bold and put it wherever you feel appropriate, and at the most someone will spend 10 seconds fixing it and telling you not to make the same mistake again. --Draicone (talk) 03:08, 28 September 2006 (UTC)

still some stuff on wikibooks
See b:Category:School of Chemistry

Any page that doesn't have the big banner on the top saying it was moved probably wasn't moved.

 * If you'd like them moved, please add this template to the wikibooks page: b:Template:wv-req
 * Otherwise maybe mark with b:Template:wv-delete - use this for stubs or for materials that have been replaced on wikiversity by better materials.

SB_Johnny | talk 23:19, 12 October 2006 (UTC)

Welcome from a new contributor
As it says - I intend to help out here - first with a section on 'How to do mole calculations' since experience tells me that this is an often-asked question (and major stumbling block for many!)

john courtneidge

John courtneidge 21:03, 14 October 2006 (UTC)
 * A tricky one indeed! I look forward to your contribution! Physchim62 13:56, 16 October 2006 (UTC)

Next steps?
I was thinking of creating flashcards for organic chemistry...other ideas? --HappyCamper 01:31, 21 February 2007 (UTC)

Department of the History of chemistry
Shouldn't all the history of chemistry be in one department? which could be separated into topics on different eras, or simply topics of different advances, like the history of the atom, which spans accross many many eras (since 512BC!!)

Where would oil refining fall?
I am a degreed chemist at an oil refinery. Even though most of my work is quality assurance and analytical chemistry, I have my fingers on the major issues of oil refining. Ethanol and Biofuels. I saw the need to better understand its social implications started an effort to study that on my home page. this then makes me a study of the history of chemistry as well. I'm not sure I fall neatly under the major divisions of chemistry.

 * Welcome to Wikiversity. Perhaps creating a new page to organize your ideas is best. How about something like...Topic:chemistry and its socioeconomic impact?--HappyCamper 05:00, 9 October 2008 (UTC)

Thank you--Jolie 11:50, 10 October 2008 (UTC)

Portal and School
Would anyone agree to having the portal more integrated with the school:chemistry as that section develops. It strikes me as confusing having so many links on the portal with equal significance to the school link. Shouldn't the school link be the focus rather than, say, outside links. --Bgreeson 22:03, 19 April 2010 (UTC)

Who's in charge here?
OR should I say who is active here? The most recent comments I see are from 2006-2007. I am about to begin an introductory chemistry class, and I would like to integrate my learning/notes as well as information from my text into an intro course. I feel like wikibooks should be the focus here, as any good class needs a text first. Come June I am just going to start changing stuff as far Chem 101 is concerned. Be warned, my grammar is not exceptional. I would like to hear from someone to discuss this with. Is it even worth the effort? It seems as if none of this is frequented, and it is terrible unorganized... grrrr --Bgreeson 22:08, 19 April 2010 (UTC)
 * - PlEASE Do!! unsigned edit by 91.65.132.76, 11:44, 22 August 2010 .--Abd 16:15, 22 August 2010 (UTC)

 * It is not very active, I agree. I have largely been away for a long time, but some general issues here, not Chemistry related, have drawn me back. There is an attempt at a text for Chem101 at wikibooks, but it is not very good. If you want to write one, I suggest your just modify and improve what is there. Then your learning notes and so on here would be most welcome. Let me know when you start and I'll keep an eye on the grammar and am happy to discuss it with. I taught 101 and introductory chemistry for many years before retiring. I will however be largely on a wikibreak from next week until early June as I am traveling. --Bduke 23:39, 19 April 2010 (UTC)

The course starts in early June, but I have already begun my studies for it. I will begin compiling information, and post it when it is substantial.--Bgreeson 19:40, 20 April 2010 (UTC)

 * Before I work too much on course materials here I feel I should compile a book that matches the course. I have started some light work on a book here at wikibooks.

--198.200.181.209 15:36, 28 April 2010 (UTC)

I already started Introduction to Chemistry but I didn't realize how far behind most of the topics were in the School:Chemistry. The organization and lack of courses are a big issue in the school of chemistry. Right now I am going to focus on Chemical Equilibrium though I am new to Wiki editing and not good at all, so I may need some clean up help on that. I am going to try and learn LaTeX so I can be of more use, but that would mean less information right away. Anyway I just finished my AP Chemistry course in High School so I hope to contribute a lot this summer (mainly from June 12th onwards). Charcoal (talk) 00:31, 25 May 2010 (UTC)

Renovation
As the School is in clear need for a redevelopment, I'm going to try to do this. I'll start by organising the topics and making them more consistent and coordinated. Then I'll work on developing some courses. However, I'm quite new to wiki editing and probably not very good, so any mistakes I make, please correct or contact me. Hopefully by the end, we'll have a glossy looking, structured, useful, informative School! Anyone wishing to discuss or find out more about my plans, don't hesistate to contact me. Mysterioso (discuss • contribs) 14:55, 12 July 2013 (UTC)

 * Hi! Glad to see some interest here. I moved the topics listed under courses into the Divisions and Departments column because a topic is considered a division or department rather than a course. Also, the courses I listed under courses are labeled as courses. Just FYI. Have fun renovating! --Marshallsumter (discuss • contribs) 14:07, 14 July 2013 (UTC)

 * Ohhh I see, I moved those courses into the relevant department page, as I was intending for any courses listed there to be multi-department. As for the courses that are there, I'll remove them from the topic namespace then! Mysterioso (discuss • contribs) 14:40, 15 July 2013 (UTC)

 * I just finished College Chemistry and will be taking Organic and Analytical Chemistry this upcoming school year. As a way of reinforcing my own learning and giving back to the educational community, I'd love to help!  Is there a structure or timescale to this renovation?  HallAtHarding (discuss • contribs) 15:11, 6 August 2013 (UTC)

 * Hi! My plan is to make all departments, courses and the school overall more consistent and structured. I'm trying to develop lessons for each department that are suitable for both generic students (i.e. ANY wikiversity participant) and also make them applicable for the courses I'm making (see Chemistry Courses on the school page) which are relevant to real life qualifications. This is what I've started doing now (in particular, Topic:Physical chemistry, Topic:Inorganic chemistry and Topic:Organic chemistry). As for timescale, this will probably take a while! Hopefully after this I thought about putting pages with more basic information in the "Fundamentals of chemistry" section on the school page, as most of the department/course lessons will be at a more advanced level. If you can add lessons and resources for organic and analytical chemistry, that would be great! Mysterioso (discuss • contribs) 12:51, 7 August 2013 (UTC)

Renovation 2015
Hi Wikiversity School of Chemistry, I am going to attempt to reorganize and complete the organization of the School of Chemistry. All that will hopefully be needed is information under a heading and its respective subsections. I don't know how long it will take, but I'm currently on break from University, so I will try my best! Heartwarming (discuss • contribs) 01:19, 27 December 2015 (UTC)

Areas for potential quizzes / testing
Hi Wikiversity Chemistry, I think it'd be a good idea to start adding some review quizzes / tests in a subsection of some of concepts and resources listed. I'll try to compile a list of possible ideas.
 * Nomenclature of inorganic chemistry

Fundamentals of Chemistry Needs to Move to a Portal: General Chemistry section
Fundementals of chemistry is an introductory chemistry section taught in general chemistry it is not its own section and if we are to manage this webpage efficiently it should be found with its appropriate subjects. This only makes sense since the whole thing needs order in which to function properly Saml214 (discuss • contribs) 16:24, 13 July 2023 (UTC)

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

title: School talk:Chemistry


text: School talk:Communication

title: School talk:Communication


text:
The Human Body

 * Circulatory system: Blood circulations with heart and blood vessels.
 * Digestive system: Processing food with mouth, stomach and intestines.
 * Endocrine system: Communicating within the body using hormones.
 * Immune system: Defending against disease-causing agents.
 * Integumentary system: Skin, hair and nails.
 * Lymphatic system: Structures involved in the transfer of lymph between tissues and the blood stream.
 * Muscular system: Moving the body.
 * Nervous system: Collecting, transferring and processing information with brain and nerves.
 * Reproductive system: The sex organs.
 * Respiratory system: The organs used for breathing, the lungs.
 * Skeletal system: Structural support and protection through bones.
 * Urinary system: The kidneys and associated structures involved in the production and excretion of urine.

Links to Related Fields of Study

 * Category:Biology
 * Category:Anatomy
 * Category:Physiology
 * Category:Anatomy stubs

Extraordinary care must be taken while studying human body for the purpose of treatment or education, etc. There are many extraordinarily yet importantly sophisticated natural systems which make up the whole human body. A few examples are the skeletal system, the muscular system, respiratory system, the reproductive system, the digestive system, the nervous system, the blood circulatory system, the urinary system, etc. The various systems consist of different organs and/or limbs which have specific functions. For example digestive system consists of hands, mouth, tongue, teeth, inner throat, stomach, small intestine, large intestine, etc. There are many parts of human body, namely: head, neck, shoulders, chest, arms, back, abdomin, buttock, legs, feet, etc. Health of every organ, limb or system is important. However, the health of some organs is vital for survival as that of heart, brain, liver, lungs, etc. Physical, psychological, mental and spiritual healths are inter-related and interdependant.

title: The Human Body


text:
Wikiversity talk:Naming conventions/Archive 1

Page Discussion Voting is below. Argh! I dropped in here to ask a specific question to find it subverted as a voting page. Maybe a good innovation for rapid creation of policy, if not we can take future voting back to the central policy voting page as various stuff is archived. Interesting experiment.! The intent of saying there will be no subpages confuses me:
 * 1. Are you saying the lesson space is reserved for main index pages of lesson plans and detailed treatment of specific material or issues (which can get lengthy) should be linked to in an alternate namespace?
 * 2. If I tackle creating detailed notes from an advanced physics books regarding the material which has lengthy multiple page examples (a typical problem set of 6 or 7 problems in physics at 400 level ran about 50 pages of hand algegra & calculus &esoteric assumptions) how do you want this handled?
 * 3. I hope this is not a reincarnation of Wikipedia's there can be only one article tendency/policy? A convenient way of keeping separate tracks of same topic is adding a suffix or prefix to each associated page. Surely we not trying to fan into single pages with links to only one from all lesson plans? Mirwin 00:06, 20 August 2006 (UTC)

If you think you have a better solution, feel free to replace the entire proposal. Messedrocker 23:54, 19 August 2006 (UTC) Yes, I didn't know where to put this... any ideas?--Rayc 01:22, 20 August 2006 (UTC)

=Voting= I guess this is as good as any place. I think there needs to be a vote on a couple of things.

Number of levels
You can express your position by voting, or  and signing. Please use the comments section for discussion about the proposal.

One
All school pages are considered equal, be it “engineering and technology” or “audio engineering”. No departments, no subschools. Just one level.

Two
Major school pages, like the ones listed on the browse page, will be given sort of a hub status. They will host any and all departments that belong to them and act as the coordination page for them. Whether or not they will host their own courses and/or research will be decided later.

Few
Major school page will be formed, and then the participants in each shall decided on how many levels each major school will have. For example Art might just have one (“Art” -> “Abstract Art” ), and engineering and technology might have three (“ engineering and technology” -> “engineering“ -> “electrical engineering” -> “Power  engineering”)

Many
Major school page will be formed, and when a new school is “created”, it will be placed by the creator into any school/department/subschool as they feel fit.

..(add others)

Major school name
You can express your position by voting, or  and signing. Please use the comments section for discussion about the proposal.

What should major schools be called?

Just schools
Put the major schools into the wikiversity schools category, and create subcats for there departments. All schools, major or minior, start with School:

Department
..(add others)

Minor school name
You can express your position by voting, or  and signing. Please use the comments section for discussion about the proposal.

What should minor schools be called?

Just schools
Put the minor schools into their subcats. All schools, major or minior, start with School:

Department
..(add others)

What should be names of the major schools
You can express your position by voting, or  and signing. Please use the comments section for discussion about the proposal.


text:
School:Engineering

كلية علوم الهندسة el:Σχολή:Πολυτεχνική Facultad de Ingeniería fr:Faculté:Sciences de l'ingénieur it:Facoltà:Ingegneria

title: School:Engineering


text:
School talk:Engineering

Topic talk:Engineering/Archive1 - from before the move to the beta site

Archive
I was going to archive everything on this page, but I'm going to ask here first, waite a day and then do it.--Rayc 23:51, 16 August 2006 (UTC)
 * Done--Rayc 01:37, 18 August 2006 (UTC)

Name
Shouldn't this be Portal:Engineering? Ah I see... why is there a Portal:Engineering and a School:Engineering? Shouln't we find a way to name this less confusingly? 71.77.207.50 06:48, 16 September 2006 (UTC)

Overlap
It seems that a lot of departments that are a specialization of another department exist. For example, the Electronic and Electrical department overlaps in terms of material covered with the "school of electronics" (?), computer science, physics. This results in a situation where topics are covered four or five times, and other topics are ignored because it is assumed they fall under the remit of another school. I have seen about 4 renditions of basic electronics, which is fine except that a large number of other topics on which time would be better spent are not covered. It might be a good idea to link to other schools content where appropriate, and also reduce the number of schools or at least agree what is covered between them. opinions? 88.212.136.185 14:38, 6 January 2007 (UTC)
 * Please create links between similar pages and encourage contributors to cooperate in avoiding duplications and covering needed topics. --JWSchmidt 01:36, 7 January 2007 (UTC)
 * I have restructured the Engineering Departments under established Major Branch Divisions and have implemented categorization using the new "Tree" system. Hopefully, others will follow suit and engineering will come back into the fold and take on the same adherence to the organizational standards that other schools have done recently. -Gustable 06:51, 14 July 2008 (UTC)
 * Along these lines, I would like to move Aerospace into the Interdisciplinary category. While mechanics are important to the aerospace field, equally-critical topics include materials, software, electronics, and systems/controls. Discussion is welcome, of course. TythosEternal (discuss • contribs) 00:11, 25 February 2013 (UTC)

Typical tempelates.
I would like to use most of the templates in this Wikiversity and translate them to be typical in Arabic (to be used by Arabic Wikiversity which is still under preparations) and hope this won't bother the original creators.--Email4mobile 13:26, 5 October 2009 (UTC)

title: School talk:Engineering


text:
User:Dvortygirl

If you need to contact me, it would be best to use Email this user on the left or leave a note at my Wiktionary page.

title: User:Dvortygirl


text:
School:Mathematics/Print Version

Welcome to the Wikiversity School of Mathematics, a collection of lesson plans and related materials to assist in the learning of mathematics.

Pre-university level course areas
See http://en.wikibooks.org/wiki/High_school_mathematics_extensions for more High School Mathematics info and http://en.wikibooks.org/wiki/Wikiversity:School_of_Mathematics#WIKIVERSITY_SCHOOL_OF_MATHEMATICS for more links and courses.

Print version created by: writerhawk 23:34, 30 May 2006 (UTC)

title: School:Mathematics/Print Version


text:
Portal:High school mathematics

Welcome to the Wikiversity Division of High School Mathematics, part of the School of Mathematics. This division focuses on high school mathematics. Progression should continue through each section. Each section may be instructed or learned thoroughly per year, or can be tailored to a custom study plan.

See also: School:Mathematics/Primary_School_Mathematics

Introduction
These are the topics which are normally taught in high school. It should tend to be inclusive. Topics which are only present in some curriculae and topics that are not normally taught at the high school level but which high school student are capable of learning should be put under supplementary materials.
 * Brief overview over each topic
 * Study plan

Building basic mathematical knowledge

 * Algebra 1 in Simple English (recent)
 * Function graphing
 * Systems of Equations
 * Solving equations
 * Solving linear inequalities
 * Functions
 * Sets and Boolean logic

Further algebra

 * Quadratic functions
 * Polynomials
 * Factorization of polynomials
 * Linear Algebra and Matrices

Functions

 * Trigonometry
 * Exponential and Logarithm
 * Complex numbers

Others

 * Sets
 * Series and Sequences
 * Permutation and Combination
 * Probability
 * Statistic
 * Vectors
 * Geometry

Calculus

 * Limits
 * Differentiation
 * Integration

Supplementary

 * Graph Theory
 * Group Theory
 * Logic
 * Number Theory
 * /Mathematical errors/

Country-specific courses

 * Advanced Level Mathematics - United Kingdom (Under Construction)

Extensions

 * High school mathematics extensions contains numerous topics not covered in the standard high school curriculum.

Links to other languages in other Wikimedia projects

 * 日本語高等学校数学 ＠Wikibooks - Japanese

title: Portal:High school mathematics


text:
Portal talk:High school mathematics

Thanks for adding Algebra 1 In Simple English to your project! Please visit http://www.hstutorials.net/olpcmath.htm to see my plans for the student-user interface. I think the design of the student-user interface is absolutely critical to the success of this project, so my focus will be split between sheparding the textbook and working on the interface.--HSTutorials 18:32, 17 July 2006 (UTC)

I got a question I'm not really good in mathématics, particularly in algebra... If I got X / Y = X + (-(Y-1))(X/Y) and I want to divide the equation by X. Do I divide all the three Xs, or just one on each side? I don't know because I got one X in a parenthese... FenixEden 08:36, 5 April 2007 (UTC)

 * We should add this as an example at Wikibooks. You could first factor "x" out of the two terms on the right side of the equation and then only have to divide by "x" twice. --JWSchmidt 13:53, 5 April 2007 (UTC)

Wikibooks
There is a well-designed book, i think, on Polish Wikibooks - If there's something intresting for you, maybe we (for now, I) could try to translate it into (Simple) English.. --
 * Hello, thx for the offer. Very kind of you. But why not ask at (Simple) English Wikibooks ? Here at Wikiversity we are more interested in learning projects. Perhaps some here want to join you in the effort to translate (you could start a learning project here to translate that book and then it could be moved to Wikibooks) ? See also for more info about Wikiversity here. Erkan Yilmaz Wikiversity:Chat  16:44, 22 February 2008 (UTC)

Nation specific maths.
I'm a new user here, and I plan on creating a fairly big, lesson by lesson maths course based on A level specification maths(16-18 years). I've already drawn up drafts, and am planning to make some videos with commentary explaining some of the more difficult questions. I find that some of the terms used here are ambiguous between countries, and it would make these pages a much more friendly place for would be students to see exactly what they're looking for.

For example, if I was studying Mechanics in the UK, I wouldn't know where to go, but if there was a link saying "A level Maths", things would be much more clear.

However, I see that this has not been done already, and I wouldn't want to be seen as a new user ruining the order of things. Would any one be in favour of this?

So far, I have enough to cover the following topics: Mechanics 1, Statistics 1, Core 1, Decisions 1. --Ohara 23:34, 14 November 2008 (UTC)

Grammatical Concerns
I was confused by this passage:

"These are the topics which are normally taught in high school. It should tend to be inclusive. Topics which only presents in some curriculum and topics that are not normally taught at high school level but in which high school student are in capacity of learning should be put under supplementary materials."

In English it doesn't really make sense. I would have corrected it, but I wasn't entirely sure what its intention was.

--Kierkk 05:49, 22 January 2011 (UTC)

title: Portal talk:High school mathematics


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Portal:Primary school mathematics

 * This section of the School of Mathematics focuses on providing teaching tools for parents and educators, whether in traditional or home-schooling environments. A parallel project is underway to teach students directly, whether children or adults, at this Wikibooks page.

Welcome to the Wikiversity Division of Primary School Mathematics, part of the School of Mathematics. The face of teaching mathematics is in constant flux. This course, therefore, is an attempt to stay in step with the current pedagogue(s) used to teach mathematics to the primary grades.

While educators should find this course helpful in learning or reacquainting themselves with some of the methodologies currently used, this course attempts to use as much lay language as possible to also be helpful to parents who, when looking over the shoulders of their children, are sometimes baffled by the "New Math". Those who wish to home-school their child(ren) may find the most value from this course; attempts will be made to cover a number of pedagogically related topics in depth; many core components (not necessarily math related) of learning will be discussed and links to more in-depth coverage of those topics will also be included. Also, if you have trouble with the material, links will be found on each page to other more basic and/or remedial sources in the Wiki community to help bring you up to speed. By providing links "in both directions" as described above, it is hoped that this course might serve as a starting point for those wishing to gain more insight into the teaching of mathematics to the primary grades.

What This Course is Not
This course assumes that the reader has reached a certain level of competency with the topics it covers. As such, this course is not designed to teach mathematics to the reader. For a very elementary introduction to mathematics, for the child or for the adult who considers themselves mathematically handicapped, you may want to read a book in development in Wikibooks, Primary Mathematics.

This course is meant to give the reader insight into how and why mathematical core understandings and skills are taught the way they are. A basic mathematics background should be all that is needed to find value in this course. Oftentimes parents (and even teachers) only understand a topic in the way they were taught it, but modern math curriculums are typically constructed such that topics are covered in many different ways in order to, among other things, accommodate the different learning styles of students.

There are many ways to teach any given mathematical understanding, and the depth of students' mathematical understandings are enhanced when they have explored it from multiple perspectives. For example, the Theory of Multiple Intelligences is but one perspective that teachers employ when leveraging student's individual strengths.

Although the idea that there are many learning styles is not well supported by the research in this area so far, this course recognizes that there are many teaching styles. As such, this course is not necessarily intended to be a math curriculum, although it could serve as part of the foundation of one. For example, this course will not teach the reader in step by step fashion how to multiply two large numbers, but the various building blocks and algorithms that students can be introduced to in the process of learning this skill will be explained.

There are many very good texts/curriculums/approaches out there; there are even "schools" of thought that suggest that math is best taught without textbooks if the teacher is skilled enough. This course could never pretend to cover the range of curricula out there. Emphasis will be on content knowledge and how it can be taught to the primary grade student. Nonetheless, activities will be suggested in this course that support the concept or skill being presented, which may serve to present the reader with ideas for their own course content.

Connections
One of the overarching ideas that should be highly leveraged in the teaching of mathematics in the primary grades involves the connections students make in their learning. All of our mathematical understandings are intertwined. For this reason, different skills such as multiplication and concepts like those learned from the study of geometry should not be explored by students in a vacuum. Rather, they should be taught together, in such a way that they reinforce each other. Internal links, when found in this course, will purposefully be placed there to emphasize the need for an ever-present awareness of the connectedness of mathematical understandings. Teachers should prefer not to "teach", they should prefer to "guide".

So how do teachers not teach? They do this by inspiring students to inspire their own learning. The most inspiring inquiries are often born of students' own interests. Primary and even upper school teachers recognize that understanding mathematics in the abstract is not the goal of most students. They need to see connections to the real world to inspire their learning. Teachers should prefer to use real world problems that require the need for mathematical models (see below). Students should then be encouraged to make connections by looking for patterns, exploring extremes, and forming and testing conjectures. At the same time teachers should be aware that all mathematics, from basic numeracy upwards is an abstraction and that some ideas are better explained as abstractions or generalizations from what is already known, rather than by constructing tedious and implausible 'real world' examples. After all, how often do carpenters really use trigonometry to measure the height of a flagpole?

Modern educators realize that students gain true "ownership" of their understandings through, inasmuch as it is possible, making connections on their own - by way of their own work and explorations. "Telling them how to do it" does not respect or celebrate their abilities. On the other hand, letting students explore concepts, learning from their mistakes as they go, ultimately leads to much stronger mathematical understandings. It also leads them to form learning habits that make their future explorations more efficient and successful. Connecting with a child's personal interests is also important. However, it is unrealistic to expect students to construct their own understandings of many areas in Mathematics. For example, calculus eluded mathematicians for centuries before it was created by Sir Isaac Newton. More appropriately for the primary school level, the long division algorithm is not something that students can be expected to figure out for themselves.

Manipulatives and Models
One very important component of the contemporary methods used to teach math is the use of manipulatives (such as toys) and models (visual representations) to give added dimension to students' understandings. In each chapter of this course, the reader will find various examples of models used to teach different mathematical understandings. Often, they will find that these models serve to make connections to material covered in other sections. Keep in mind that teachers in the classroom tend to be very creative and resourceful. Often, the models found in this course have many possible permutations, and can come in various shapes, sizes, and guises. They are purposely presented here in simple forms to facilitate, for the reader, their identification.

In this course, the word model is occasionally used in a subtly different context as well - one that is commonly used by teachers of any discipline. We like to model good mathematical habits just as we like to see any kind of good behavior modeled. Occasionally, teachers should model problems by doing them as if they are a student.

Course Contents
Note: a copy of some of the following pages initially was created from Wikibooks, but the content will start to diverge to take the two different audiences into account. See Import.

Numbers and Operations

 * Numbers
 * Adding numbers
 * Subtracting numbers
 * Multiplying numbers
 * Dividing numbers

Pre-Algebra

 * Negative numbers
 * Powers, roots, and exponents
 * Fractions
 * Working with fractions
 * Percentages
 * Factors and Primes
 * Method for Factoring
 * Proportions
 * Introduction to variables

Measurement and Geometry

 * Polygons and Area
 * Angles, see also: Angles
 * Solids

Advanced topics

 * Probability
 * Measures of Central Tendency: Mean, Median, and Mode
 * Sets and Boolean logic

Links commonly found in this course

 * The Use of Calculators

Move on to High School Mathematics

title: Portal:Primary school mathematics


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School:Mathematics/Course guide

 * Algebra I
 * Algebra II
 * Precalculus
 * Calculus (introductory)
 * Cartesian Geometry (introductory)
 * Elementary Logic
 * Elementary Set Theory
 * Real analysis
 * Abstract algebra
 * Geometry (Introductory Euclidean)
 * Infinitesimals Introduction
 * Limits Introduction
 * Mathematical Statistics (Topics)
 * Non-Euclidian Geometry Introduction
 * Probability Introduction
 * Statistics Introduction

title: School:Mathematics/Course guide


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School talk:Mathematics/Course guide

What is this supposed to be? --Fephisto 19:32, 19 April 2006 (UTC)

 * The list of suggested courses? I presume. -- Taku 06:58, 11 August 2006 (UTC)

 * Wikiversity is still very young - you can shape however the project evolves into - what is written is not set in stone, many things can change - and you can do just that with the Wiki! --HappyCamper 07:08, 21 August 2006 (UTC)

Do the Introdutory Statistics course fits in here? I believe Mathematical Statistics would be more adequated. --Lucas Gallindo 03:25, 11 September 2006 (UTC)

Spelling in Geometry
The correct is Euclidean Geometry (usage on latin languages) or Euclidian Geometry? --Lucas Gallindo 22:43, 2 October 2006 (UTC)

List of courses order
Should we rearrange the order of the courses so that it will be listed in alphabetical order? --Zoo 19:53, 14 November 2006 (UTC)

title: School talk:Mathematics/Course guide


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School:Mathematics/Facultative courses

On this page will be listed additional math courses for Wikiversity School of Mathematics, which are not part of the obligatory University program but may be either reasonably added to University program in the future either be taught in a University as useful facultative courses, as well as special courses for particular specializations.

Category theory
Additional courses in the field of category theory:

Category of Endomorphisms (Currently not a part of University program but is thought to be recommended to be added even as an obligatory course for certain specializations.)

Category of Endomorphisms. Endomorphisms of any category are objects of certain categories. This produces a rather elaborate theory, especially in the case if the morphisms of the given category are partially ordered.

Study of these categories is important for specialists in mathematical logic, computer science, abstract algebra, and some other fields of mathematics.

title: School:Mathematics/Facultative courses


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School talk:Mathematics/Facultative courses

The link to wikipedia turns up nothing for facultative. If you are going to use a term like facultative, you should probably ensure that wikipedia and wikionary both have entries for it. A quick google define: search yields mostly biology terms, but a few things about "optional" at the bottom. I would not presume to provide a definition for the wikis, however, as this is the first time I've ever heard the term.


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Actuarial science

Actuarial science applies mathematical and statistical methods to finance and insurance, particularly to risk assessment. Actuaries are professionals who are qualified in this field through actuarial examinations or university-based education (but the credentialing usually requires passing a series of examinations), and work experience.

Actuarial mathematics deal with the mathematics of uncertainty and risk. Some of the key areas where actuarial mathematics are principally applied are mortality studies, financial risk, risk and ruin theory, credibility theory, demography, reliability study and graduation of statistical data.

Actuarial science includes a number of interrelating disciplines, including probability and statistics, finance, and economics. Historically, actuarial science used deterministic models in the construction of tables and premiums. Modern approaches incorporate the stochastic simulation of random variables, such as interest rates.

Studying Actuarial Science
Studying Actuarial Science can be a task that is both challenging and rewarding. It requires a very good basis in mathematics as it requires heavy use of statistics. In the U.K., few universities offer an actuarial related degree. Among those, the Queen's University of Belfast, Heriot-Watt, City and Kent universities are some of them that can offer full exemptions from the CT series of examinations of the Institute or Faculty of Actuaries.

Actuaries are involved in the following areas :
 * Life insurance
 * Short term or general insurance
 * Pensions
 * Institutional investments, etc.

Study materials for actuarial examinations
Society of Actuaries (SOA):

 * Exam P
 * Wikibooks: Probability

 * Exam FM
 * Wikibooks: Financial Math FM

title: Actuarial science


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Talk:Actuarial science

So, is there going to be any actual 'stuff' about actuarial science in here? Fephisto 22:38, 8 July 2006 (UTC)

I'm starting to wonder how stating that actuarial sciences is hard would aid in learning. I have several friends who've (unofficially) taken the actuarial tests, and most have stated that they're not hard until the fifth one. Mo Anabre 20:17, 9 November 2006 (UTC)

Intersection with the department of statistics
When the subject is mathematical foundation, 90% of the topics studied wil be both in the Actuaries Course and in the Statistics Course. How we are going to deal with that?

Ah, forget it, this thing looks dead. --Lucas Gallindo 16:37, 9 October 2006 (UTC)

well... we can try to add what know so far.....

title: Talk:Actuarial science


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School:Mathematics/Undergraduate/Pure Mathematics

Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. From the eighteenth century onwards, this was a recognised category of mathematical activity, sometimes characterised as speculative mathematics, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering and so on.

Courses
more needs to be added here
 * Foundations of Pure Mathematics
 * School of Mathematics:Introduction to Proofs
 * School:Mathematics/Foundation of mathematical concepts
 * School of Mathematics:Introductory Real Analysis
 * School:Mathematics/Introduction to Abstract Algebra
 * School of Mathematics:Introduction to Graph Theory
 * School:Mathematics/Calculus
 * Introduction to Set Theory

Algebra

 * Proofs
 * Complex numbers
 * Vectors in two dimensions
 * Matrices
 * Vector spaces
 * Linear transformations
 * Eigenvalues and eigenvectors

Abstract algebra

 * Groups
 * Rings

 * Matrices
 * Eigenvalues and eigenvectors
 * Vector spaces
 * Linear transformations
 * Inner product spaces
 * Proofs

Analysis

 * Complex Analysis
 * Real Analysis
 * Topology

Calculus

 * Functions
 * Limits
 * Differentiation
 * Applications of Derivatives
 * Higher Order derivatives
 * More differentation rules
 * Summation notation
 * Integration
 * Vectors
 * Applications

Discrete mathematics

 * Set theory
 * Functions and relations
 * Number theory
 * Logic
 * Enumeration
 * Graph theory
 * Recursion
 * Number representations
 * Modular arithmetic
 * Polynomials and number theory
 * Finite fields

Differential geometry

 * Lie Algebras
 * Information geometry

Foundations

 * Set Theory


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School talk:Mathematics/Undergraduate/Pure Mathematics

Q: Why is graph theory under discrete math instead of topology? A: Although some areas of Graph Theory (for example Kuratowski's Theorem) deal with Topological concepts, other areas deal with Probability, and yet others with algebraic concepts such as eigenvalues, the common ground between all of the areas is the graph itself, which is a purely discrete combinatorial concept. It can be depicted in a topological space, but it is not a topological concept.

Modular functions - help needed
I need help with Modular functions at Why 10 dimensions. --JWSchmidt 01:08, 23 August 2006 (UTC)
 * Do you want a definition?--Hillgentleman|User talk:hillgentleman 05:29, 17 November 2006 (UTC)

Undergraduate
The term Undergraduate mathematics means differently in different countries, and within each country, different universities. The content in the School: page is the syllibus for a common university in the united states of America. In an English university, an undergraduate curriculum often also includes a selection from functional analysis, algebraic topology, Lie group, dynamical systems, commutative algebra, algebraic geometry, etc.

 * Agreed. Further, I find the distinction between undergraduate and graduate-level courses unhelpful to begin with.  It's not like Wikiversity is going to refuse undergrads access to advanced courses.  --Jorend 16:34, 5 January 2007 (UTC)


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Foundations of mathematical concepts

{| An introductory course from the School of Mathematics
 * bgcolor="#E6CFDD" style="border:1px solid #cfcfcf;padding:1em;padding-top:0.5em;padding-bottom:0em;"| Foundation of mathematical concepts
 * bgcolor="#E6CFDD" style="border:1px solid #cfcfcf;padding:1em;padding-top:0.5em;padding-bottom:0em;"| Foundation of mathematical concepts

 * bgcolor="#F4E3F3" style="border:1px solid #cfcfcf;padding:1em;padding-top:0.5em;padding-bottom:0em;"|
 * bgcolor="#F4E3F3" style="border:1px solid #cfcfcf;padding:1em;padding-top:0.5em;padding-bottom:0em;"|

This course aims to ensure that the student will have a practical and theoretical basis for the application of mathematical knowledge in fields that require it or in future mathematics courses.

Course requirements
To start this course, you should have:
 * knowledge of basic calculus, how to differentiate, integrate, evaluate limits, etc.
 * knowledge of basic algebra, manipulation of equations, solving quadratic equations, etc.
 * knowledge of basic geometry, trigonometry, trigonometric identities, geometric proofs, etc.

Lecture series
not available yet

Examinations
not available yet
 * }

title: Foundations of mathematical concepts


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Talk:Foundations of mathematical concepts

May I help? --Krishnaaz 16:33, 30 January 2006 (UTC)
 * Absolutely :-) --HappyCamper 07:29, 21 August 2006 (UTC)


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Introduction to Real Analysis

Course outline
This course aims to provide a thorough introduction to the subject of real analysis. The work done corresponds to the period of Fall, 2008.

Course requirements
The following knowledge is required or desirable on commencement of study of this course:
 * Knowledge of basic methods of proof
 * Calculus of One Variable
 * Differential Equations
 * Basics of Set Theory

Syllabus
The course will follow Rudin's Principles of Mathematical Analysis scheme. Course material can be obtained from Wikibooks, MIT's OCW and other free online resources, including the work done by participants during this and other semesters (cycles).

Lecture series

 * 1) The Real and Complex Number System
 * 2) Basic Topology
 * 3) Numerical Sequences and Series
 * 4) Continuity
 * 5) Differentiation
 * 6) The Riemann-Stieltjes Integral
 * 7) Sequences and Series of Functions
 * 8) Some Special Functions
 * 9) Functions of Several Variables
 * 10) Integration of Differential Forms
 * 11) The Lebesgue Theory

Assignments
Will be posted, with suggestions of solutions, after each lecture is finished

Examinations

 * Examination #1 should be taken after all the material from lectures 1-3 is studied.
 * Examination #2 should be taken after all the material from lectures 4-6 is studied.
 * Examination #3 should be taken after all the material from lectures 7-9 is studied.
 * Examination #4 should be taken after all the material from lectures 10-11 is studied.
 * The Final Examination should be taken after all the material from all the lectures is studied.

Recommended student evaluation scheme
Group work is encouraged: discussions, corrections and observations are greatly welcome. Questions can be posted in the discussion pages for each topic and answers can be attempted by all members of the class.

Evaluation is self-assesed. Each assignment should be done after the material from the lectures is understood, and should be used as preparation for the examinations. Solutions will be posted with each assignment and examination.

A success percentage above 75% indicates dominion over the topics.

Sign Up List
If you are interested in taking (or helping with) this course, please indicate so here. I'm as serious as you are:

 * --Cœlispex 06:17, 2 September 2008 (UTC)
 * Love to take it. W3asal 23:50, 20 September 2008 (UTC)
 * I would be happy to help give it. Thenub314 15:11, 20 February 2009 (UTC)
 * I would also love to take this course. Sorry, I don't have a login name.  96.28.49.232 09:33, 2 April 2009 (UTC)
 * Would also love to take this. Sorry I don't have a login name either
 * I plan to take this course in my university next semester. I am willing to help whenever needed.--Fj217 23:32, 7 November 2009 (UTC)

title: Introduction to Real Analysis


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School:Mathematics/Courses by level

This is a list of the courses provided by the Wikiversity School of Mathematics, sorted by course level

Introductory - Level I

 * School:Mathematics/Foundation of mathematical concepts
 * Synthetic division

Level II

 * School of Mathematics:Introductory Real Analysis
 * School:Mathematics/Calculus

title: School:Mathematics/Courses by level


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Introduction to graph theory

Introduction to Graph Theory
An introductory course from the School of Mathematics

This course aims to provide a thorough introduction to the subject of graph theory.

Course requirements
The following knowledge is required or desirable on commencement of study of this course:

 * knowledge of basic methods of proof
 * knowledge of basic probability

Course outline
This is an approximate depiction of the course:

 * Definitions
 * Bipartite Graphs
 * Hamilton Cycles and Eulerian Circuits
 * Planar Graphs
 * Statement of Kuratowski's Theorem
 * Matchings in Bipartite Graphs
 * Hall's Theorem
 * Connectivity
 * Menger's Theorem
 * Extremal Graph Theory
 * Hamilton and other cycles
 * Turan's Theorem
 * Ramsey's Theorem
 * Graph Colourings
 * Chromatic Polynomial
 * Vizing's Theorem
 * Four-Colour and Five-Colour Theorems
 * Extensions to other surfaces
 * Eigenvalues
 * Applications to Strongly Regular Graphs
 * The Probabilistic Method
 * Lower bounds for Ramsey numbers
 * Graphs with large girth and chromatic number

Lecture series

 * /Lecture 1/ Introduction and Definitions
 * /Lecture 2/ Bipartite Graphs and Trees
 * /Lecture 3/ Hamilton Cycles and Eulerian Circuits
 * /Lecture 4/ Graph Traversal
 * /Lecture 5/ Flows and Cuts
 * /Lecture 6/ Planar Graphs

List of Definitions

Assignments

 * /Problems 1/ (on Lectures 1-5)
 * /Problems 2/ (on Lectures 6-10)

title: Introduction to graph theory


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Introduction to graph theory/Lecture 1

Introduction
Although Graph Theory, and Combinatorics in general, has very few prerequisites, an introductory course must unfortunately start with many definitions. This lecture may therefore be a little dry, but it will provide the necessary backbone for the rest of the course.

What is a graph?
A lot of situations from the real world can usefully be described by means of a diagram consisting of a set of points together with various lines joining certain pairs of these points. For example, if you are planning a wedding, the points could represent your guests, with the lines representing pairs of people who it would be bearable to have sitting together. Another example is the Kevin Bacon Graph, in which points represent Hollywood actors, and lines represent collaborations between actors. Yet another example is a computer network, in which the points could represent the computers and printers, with lines representing physical and wireless connections.

A graph is a mathematical abstration of the above. Formally, a graph $$G$$ consists of a set of vertices $$V(G)$$, and a set of edges $$E(G)$$, where an edge $$E(G)$$ is a 2-element subset of $$V(G)$$. A graph is usually represented by a picture, in which the vertices are represented by points, and the edges by lines or curves joining  two elements.

Examples
Let $$V(G)=\{a,b,c,d,e\}$$, and let $$E(G)=\{ab,bc,cd,de,ae\}$$. (The edges are usually written as $$ab$$ rather than $$\{a,b\}$$ for ease of notation.) This can be depicted by the following picture:

For another example, let $$V(G)=\{1,2,3,4,5\}$$, and let $$E(G)=\{12,13,14,25,34,45\}$$. This can be depicted by the following picture:

Isomorphism
Given a particular graph, it can be depicted in countless ways, and there is usually no "best" way of depicting a graph. For example the second graph could also be depicted in the following way:. Note that this is still the same graph - the location of the vertices and edges in the diagram makes no difference to the graph.

This rearrangement of the vertices has shown a connection between our two graph examples. If we relabel the vertices of this new drawing in such a way that $$a=1$$, $$b=2$$, $$c=5$$, $$d=4$$ and $$e=3$$, we get the first diagram. In the vast majority of Graph Theory examples and results, the choice of labels for the vertices are pretty much irrelevant, and most graph theorists would see these two graphs as being the same. The relationship between these two graphs is an isomorphism, and they are said to be isomorphic.

Formally, an isomorphism from graph $$G$$ to graph $$H$$ is a mapping $$\phi:V(G)\to V(H)$$ which is one-to-one ($$\phi(u)=\phi(v)\iff u=v$$) , onto (for all $$h\in V(H)$$, there exists $$g\in V(G)$$ such that $$\phi(g)=h$$), and such that for any vertices $$u,v\in V(G)$$, the edge $$uv$$ is contained in $$E(G)$$ if and only if the edge $$\phi(u)\phi(v)$$ is contained in $$E(H)$$. Graphs $$G$$ and $$H$$ are said to be isomorphic if there is an isomorphism from $$G$$ to $$H$$. Unless specified, we will not distinguish between isomorphic graphs.

Note that it is by no means easy to tell if two graphs are isomorphic. See Problem 1 on the problem sheet for examples.

Necessary Definitions
There are various terms relating to the vertices and edges of a graph which, although obvious, need to be defined. Given vertices $$u,v$$ and edge $$uv$$ of graph $$G$$, edge $$uv$$ is said to join the vertices $$u$$ and $$v$$, which are called the endvertices of the edge. $$u$$ and $$v$$ are neighbouring or adjacent vertices of $$G$$, and are incident with edge $$uv$$.

The order of $$G$$ is the number of vertices, whereas the size is the number of edges.

The Degree Sequence
Given a vertex $$v\in V(G)$$, the degree of $$v$$ (denoted by $$d_G(v)$$) is the number of edges incident with $$v$$. The smallest degree of the vertices of $$G$$ is called the minimum degree of $$G$$ and is denoted by $$\delta(G)$$. The maximum degree is denoted by $$\Delta(G)$$.

In the examples above, each vertex has degree 2, so the minimum and maximum degree are 2. A graph in which every vertex is of degree $$k$$ is called $$k$$-regular, and a graph is called regular if it is $$k$$-regular for some $$k$$.

We now have our first theorem of the course. It is the belief of many proponents of Combinatorics that the best way to teach it is through problems. To that end, the theorems are not provided with full proofs in the lectures, but instead with a roadmap with which the student can find the proof themselves. At the end of each theorem a link is provided to a proof page.

Theorem 1: The sum of the degrees of the vertices of a graph $$G$$ is precisely twice the size of $$G$$.

Proof Roadmap: Let $$S=\{(v,e):v\in V(G), e\in E(G), v$$ incident with $$e\}\!\,$$. By considering how many elements of $$S$$ each vertex $$v$$ is part of, count the number of elements of $$S$$. Count the number of elements of $$S$$ again, this time by considering how many elements each edge $$e$$ is part of.

QED (Full Proof)

If the graph $$G$$ has vertices $$v_1,v_2,\ldots,v_n$$, then the degree sequence of $$G$$ is the sequence $$(d_G(v_1), \ldots, d_G(v_n))$$. For ease of notation, this sequence is often written in increasing order.

Common graphs
In the same way that the alternating, cyclic, dihedral and symmetric groups occur so often that it is useful to have fixed notation for them, similarly there are various graphs which it is useful to be able to describe easily.

Empty Graphs
The empty graph $$E_n$$ has $$V(E_n)=\{1,\ldots,n\}$$ and $$E(E_n)=\emptyset$$. That is to say that is has $$n$$ vertices and 0 edges. It is called the empty graph on $$n$$ vertices.

Complete Graphs
In contrast to the empty graph, the complete graph $$K_n$$ has every edge. That is to say that $$V(K_n)=\{1,\ldots,n\}$$ and $$E(K_n)=\{ij: 1\le i\le j\le n\}$$. It has $$n$$ vertices and $$\frac{n(n-1)}2$$ edges. It is called the complete graph on $$n$$ vertices.

Paths
The path graph $$P_n$$ has $$V(P_n)=\{1,\ldots,n\}$$ and $$E(P_n)=\{12,23,34,\ldots,(n-1)n\}$$. It has $$n$$ vertices and $$n-1$$ edges. It is called the path on $$n$$ vertices or simply the n-path.

Cycles
For $$n\ge 3$$, the cycle graph $$C_n$$ has $$V(C_n)=\{1,\ldots,n\}$$ and $$E(C_n)=\{12,23,\ldots,(n-1)n,1n\}$$. It has $$n$$ vertices and $$n$$ edges It is called the cycle on $$n$$ vertices or simply the n-cycle.

Subgraphs
A question of common importance in Graph Theory is to tell, given a complicated graph $$G$$, whether we can, by removing various edges and vertices, show the presence of a certain other graph $$H$$. In this case $$H$$ is said to be a subgraph of $$G$$. Formally, given graphs $$G, H$$ with $$V(G)\subseteq V(H)$$, $$G$$ is a subgraph of $$H$$ if $$E(G)\subseteq E(H)$$. The subgraph is said to be spanning if $$V(G)=V(H)$$, and induced if $$E(H)$$ consists of every edge of $$G$$ with both vertices in $$V(H)$$.

Cliques, Cycles and Paths
There are a few examples of subgraphs which are particularly important. These correspond to the special graphs mentioned above. Examples in this section will be subgraphs of the Kevin Bacon graph mentioned above.

Cliques
If subgraph $$H$$ of $$G$$ is isomorphic to a complete graph $$K_n$$ for some $$n$$, then $$H$$ is said to be a clique of $$G$$. Note that all cliques are induced subgraphs. A trivial example in the Kevin Bacon graph would be a subgraph H consisting of the entire cast of some film, with all edges present. Clique 1 is such an example with the film Who's Afraid of Virginia Woolf? (1966). Cliques can also arise in other ways. For example, in Clique 2 each edge corresponds to a different film.

The size of the largest clique in graph $$G$$ is the clique number of $$G$$. The largest clique in the Kevin Bacon Graph probably consists of the largest cast of any film.

Independent Sets
If induced subgraph $$H$$ of $$G$$ is isomorphic to an empty graph $$E_n$$ for some $$n$$, $$H$$ is said to be an independent set of $$G$$. In other words, an independent set is a set of vertices between which no edges lie. For example, Angelina Jolie, Catherine Zeta-Jones, Drew Barrymore, Dennis Hopper, Eddie Murphy, Kevin Bacon, Kevin Spacey, Madonna, Mel Gibson and Shirley Maclaine form an independent set in the Kevin Bacon graph. The size of the largest independent set in graph $$G$$ is the independence number of $$G$$. In the case of the Kevin Bacon graph this probably consists of a collection of extras who have each appeared in one movie, these movies being different.

Cycles
If subgraph $$H$$ is isomorphic to a cycle $$C_n$$ for some $$n$$, then $$H$$ is a cycle of $$G$$. Here is an example of a cycle in the Kevin Bacon Graph which is currently an induced subgraph.

The size of the smallest cycle in a graph $$G$$ is the girth of $$G$$. The size of the largest cycle is the circumference of $$G$$. For the Kevin Bacon graph, the girth is 3, as Robert De Niro, Jodie Foster and Joe Pesci can attest. The circumference is a much harder question, but probably will consist of most of the vertices on the graph of degree at least 2.

Paths
If subgraph $$H$$ is isomorphic to a path $$P_n$$ for some $$n$$, then $$H$$ is a path of $$G$$. Here is an example of a path in the Kevin Bacon Graph which is currently an induced subgraph.

If the vertices at the end of the path are $$u,v$$, it is called a $$u,v$$-path. If there exists a path between $$u$$ and $$v$$, the distance between $$u$$ and $$v$$ is the length (number of edges) of the shortest $$u,v$$-path. The diameter of a graph is the longest distance between two points. If, as is commonly suggested, every actor can be connected to Kevin Bacon with at most 6 links, the diameter must be at most 12.

r-partite Graphs
A graph $$G$$ is said to be bipartite with vertex classes $$V_1$$ and $$V_2$$ if the vertex set of $$G$$ is the disjoint union of $$V_1$$ and $$V_2$$, and each edge joins a vertex of $$V_1$$ to a vertex of $$V_2$$. There is no edge joining two elements in $$V_1$$ (or in $$V_2$$). Similarly $$G$$ is r-partite with vertex classes $$V_1, V_2, \ldots, V_r$$ iff the vertex set is the disjoint union of the $$V_i$$, and no edge goes between vertices in the same class.

Connected Components
If graph $$G$$ has the property that for each pair of vertices $$u, v$$, there is some $$u,v$$-path, then $$G$$ is said to be connected. If $$G$$ is not connected, then it breaks down uniquely into several connected subgraphs. A maximal connected induced subgraph of $$G$$ is called a component.

A vertex whose deletion increases the number of components is called a cutvertex, whilst an edge with the same property is called a bridge.

Conclusion
This lecture contained a necessary initial set of definitions and concepts, along with some examples to simplify understanding. Next time we will ground these by investigating properties of bipartite graphs, and of a particular kind of graph called a tree.

title: Introduction to graph theory/Lecture 1


text:
User talk:WAvegetarian

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title: User talk:WAvegetarian


text:
Introduction to graph theory/Lecture 2

Introduction
It is useful, and indeed vital, to have the collection of definitions that were doled out in the last lecture. Nevertheless, in order to get a feel for what these definitions mean, one must actually get one's hands dirty. Today we start investigating the consequences of these definitions. We start by investigating graphs with no cycles.

What is a forest?
A graph is said to be a forest if it contains no cycles (this property is also called being acyclic). A sample forest from the Kevin Bacon Graph is given below. The vertices have been placed in order to give some idea as to the reason for the name forest.

It is time for the second theorem of the course. To reiterate what was said in the first lecture, proofs to the theorems will not be given in the main body of the lecture. Instead, there will be a "roadmap" of the proof, for the student to fill in himself. At the end of each theorem there is a link to a proof page.

Theorem 2: A graph is a forest if and only if for every pair of distinct vertices $$u,v$$, there is at most one $$u,v$$-path.

Proof Roadmap: QED (Full Proof)
 * Suppose you have a graph $$G$$ which is not a forest.
 * What must G contain?
 * Take an example of the thing that is in all non-forest graphs.
 * Choose two vertices from your example, and try to find two paths between them.
 * Suppose you have a graph $$G$$ with two vertices $$u,v$$ with two $$u,v$$-paths.
 * By drawing a picture, allowing your paths to cross over each other many times, attempt to see why this graph cannot be a forest.
 * Now formalise this argument, making certain that what you construct uses no vertex more than once.

If a forest is connected, it is called a tree. This naming may seem a little counterintuitive, but note that forests consist of a collection of trees. If deforestation were to continue to the extent that Sherwood Forest was reduced to having only one plant in it, then the forest may well be renamed Sherwood Tree. The previous theorem has the following easy corollary.

Corollary 3: A graph is a tree if and only if for every pair of distinct vertices $$u,v$$, there is exactly one $$u,v$$-path.

Proof Roadmap: QED (Full Proof)
 * Suppose you have a graph $$G$$ which is not a forest.
 * Use Theorem 2 to show that there is some pair of vertices $$u,v$$ for which there is not exactly one $$u,v$$-path.
 * Suppose you have a graph $$G$$ which is a forest, but is not a tree.
 * What property must $$G$$ fail to have?
 * Looking back at the definition in the previous lecture, if necessary, show that this means that there is some pair of vertices $$u,v$$ for which there is not exactly one $$u,v$$-path.
 * Now suppose you have a graph $$G$$ which is a tree.
 * Use Theorem 2 and the same definition to show that for every pair of vertices $$u,v$$, there is exactly one $$u,v$$-path.

As a tree is such a useful concept, we will give (in the following theorem) two alternative characterisations of trees.

Theorem 4: For a graph $$G$$, the following are equivalent:
 * $$G$$ is a tree
 * $$G$$ is a minimal connected graph, in the sense that the removal of any edge will leave a disconnected graph.
 * $$G$$ is a maximal acyclic graph, in the sense that the addition of any missing edge will create a cycle.

Proof Roadmap QED (Full Proof)
 * Suppose $$G$$ is a tree, and let $$uv$$ be an edge not in $$G$$.
 * Use Corollary 3 to show that adding $$uv$$ to $$G$$ will create a cycle.
 * Suppose $$G$$ is a tree, and let $$uv$$ be an edge of $$G$$.
 * Find two vertices such that the graph $$G-uv$$ can have no path between them.
 * Suppose $$G$$ is a minimal connected graph, but is not a tree.
 * From the definition of a tree, what else does this mean that G is not?
 * Therefore, what must $$G$$ contain?
 * Find an edge you can remove from $$G$$ without disconnecting the graph.
 * Suppose $$G$$ is a maximal acyclic graph, but is not a tree.
 * Since acyclic graphs and forests are the same thing, what property does $$G$$ not have?
 * Using this, show that there are two non-adjacent vertices which can be joined without creating a cycle.

We will need one more theorem about trees here. It is not immediately obvious that all trees have one fewer edge than vertex. The proof, as you shall see, is easy.

Theorem 5: A tree of order $$n$$ has $$n-1$$ edges.

Proof Roadmap QED (Full Proof)
 * Prove this by induction on $$n$$.
 * Prove this first (by experimenting) for $$n=1,2,3$$. Find all trees for these values of $$n$$.
 * Now let $$T$$ be a tree of k vertices, and suppose it is true for all $$n<k$$. Remove an edge from T.
 * By Theorem 4, the new graph cannot be connected. Can it have three or more components?
 * What must each component be?
 * How many vertices are there in total in the components?
 * Therefore, using the induction statement, how many edges must there be?

title: Introduction to graph theory/Lecture 2


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Introduction to graph theory/Glossary

]] ]]
 * Forest : [[School of Mathematics:Introduction to Graph Theory:Lecture 2#What is a forest?|A graph with no cycles.
 * Tree : [[School of Mathematics:Introduction to Graph Theory:Lecture 2#What is a forest?|A connected graph with no cycles.

title: Introduction to graph theory/Glossary


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Talk:Introduction to Graph Theory/Lecture 1

Too many references to the "Kevin Bacon" graph. Get more mathematical--if this is to be a course, make it look like one.

There is nothing wrong with the refrences p.s when are the problem sets coming in


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School:Mathematics/Calculus

{|
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 * bgcolor="#E6CFDD" style="border:1px solid #cfcfcf;padding:1em;padding-top:0.5em;padding-bottom:0em;"| Calculus An introduction

from the School of Mathematics

This course aims to provide an introduction to Differential and Integral Calculus for year one students.
 * bgcolor="#F4E3F3" style="border:1px solid #cfcfcf;padding:1em;padding-top:0.5em;padding-bottom:0em;"|
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Course requirements
The following is required or desirable before commencement of study of this course:
 * High School Algebra and Trigonometry
 * For proofs you will need some basic Set Theory

Instructions
Please note that there will be no deadlines in this course. You will have to go through the syllabus at your own pace. Questions and Memos will be posted regularly, and since you will be required to grade your own work (at times) please do it as honestly as possible.

Prescribed textbook: Calculus

Recommended books: https://en.m.wikibooks.org/wiki/High_School_Trigonometry

COURSE OUTLINE

Course Outlineine
will use the calculus book often, or other sources. As you learn these things by yourself you become accostumed to the book that taught you, so I may show some pages from one of my calculus textbooks.

How I'm thinking about doing this is once you contact me (send me a pretest and any questions/comments on my discussion page), I'll make lecture pages, assignments, exams, etc. for you exclusively and you can come here to answer/ask questions directly on these pages.

Looking forward to this experiment. Fephisto 03:41, 9 July 2006 (UTC)

Update
I've ended this class now, since it looks like it has ended, hope the very very few that were in this program found it helpful, I may do it again next year, for now I'll leave it here as an open program for those interested, I won't be adding new stuff (probably), but if you need help, want to ask something, or have something checked, leave it on my discussion page. Chau! Fephisto 23:49, 19 August 2006 (UTC)

People Signed Up
Vitalij_zad

Superwiki

Lectures
Building up to the Riemann-Darboux Definition

The Riemann-Darboux Integral, Integrability criterion, and monotone/Lipschitz function

Integrability of monotone/Lipshitz functions, Linearity of the Integral, and The Dirichlet Everywhere Non-integrable Function

Assignments
They're on the lecture pages, but if you want to keep track, feedbacked is pretty much the same thing as done, I'm not going to stop you if you've got what ideas you've wanted to take from the exercise.:

Vitalij
Pre-test (Ok)

In Building up to the Riemann-Darboux Definition:
 * A Small Exercise (feedbacked) (Added stuff about the Archimedean Property to the module)
 * Summing Exercise (feedbacked)
 * Exercise (feedbacked)
 * Approxiamation Property Proof exercise
 * Return of the summing exercise

In The Riemann-Darboux Integral, Integrability criterion, and monotone/Lipschitz function:
 * Yet ANOTHER return of that summing exercise
 * Integrability criterion proof

Super-Wiki
Pre-test (Waiting)

Examinations
Calculus pre-test

Recommended student evaluation scheme
not available yet

 * }

title: School:Mathematics/Calculus


text:
Calculus pre-test

To take this test, click [ here] to put the test on your userpage. Click the link and save. Now you have your test, take it!

General Questions
What are you looking to get out of this course?

What would you say is your general knowledge of math?

Would you like a formal treatment of Calculus (very heavy, requires set theory), or an intuitive treatment of Calculus?

What times do you have available/would prefer for lectures?

Any other comments?

Pre-Calculus

 * 1)  What is a function?
 * 2)  What is an inverse function?
 * 3)  What is a logarithm?
 * 4)  What is a radian?
 * 5)  What is a trigonometric function?
 * 6)  Name a trigonometric identity.
 * 7)  What is a parametric equation?
 * 8)  What is a slope-intercept equation?
 * 9)  What does it mean for something to be proportional?
 * 10)  What does injective, surjective, and bijective mean?
 * 11) What are logarithmic equations?

Calculus I

 * 1)  I will accept an intuitive or formal answer for this, although a formal answer is obviously preferred: what is a limit?
 * 2)  What is a one-sided limit?
 * 3)  What does it mean for a function to be continuous?  Is it possible for a function to be discontinuous everywhere?
 * 4)  What is a derivative?
 * 5)  What is the product rule?
 * 6)  What is the relation between differentiability and continuity?
 * 7)  What is the chain rule?
 * 8)  What is the relation between a local extrema and a derivative of a function?
 * 9)  What is the mean value theorem?
 * 10)  What is L'Hôpital's rule?

Calculus II

 * 1)  What is an integral?
 * 2)  What is a Riemann Sum?
 * 3)  What is the Fundamental Theorem of Calculus?
 * 4)  What is the substitution method?
 * 5)  What is the difference between a definite and indefinite integral?

title: Calculus pre-test


text:
School talk:Mathematics/Calculus

I would like to take this class. --Superwiki 03:33, 23 July 2006 (UTC)

Is someone going to teach this class? If so, I'd like to be a TA- i've taken 3 semesters of college calculus, and differential equations. DettoAltrimenti 07:47, 24 October 2005 (UTC)

I think we should discuss here the topics we want to tackle in the course. Have you seen the Calculs book? Is it good enough for you or you need to go deeper in some topic? What have you answered to the questions in the Calculus pre-test? --Vitalij_zad 14:30, 29 July 2006

A merger with Precalculus is a great idea Graemeb1967 21:29, 25 November 2009 (UTC)

I think we should have a pretest with actual problems to solve, to complement the one with just definitions (Ex. If sin(a) = pi/2 and 0<a<2pi, solve for a). Sometimes the best way to learn is by doing. Crusader8389

title: School talk:Mathematics/Calculus


text:
Introduction to graph theory/Proof of Theorem 1

Statement
The sum of the degrees of the vertices of a graph $$G$$ is precisely twice the size of $$G$$.

Proof
Let $$S=\{(v,e):v\in V(G), e\in E(G), v$$ incident with $$e\}\!\,$$. We will count the number of elements of $$S$$ in two different ways, using the method of double counting.

The degree of a vertex is defined to be the number of edges incident with that vertex. Therefore vertex $$v$$ is the vertex-part of precisely $$d(v)$$ elements of $$S$$. Thus the number of elements of $$S$$ is the sum of the degrees of the vertices.

The other obvious way to count the number of elements of $$S$$ is in terms of the edges. Clearly each edge is incident with precisely two vertices. Therefore edge $$e$$ is the edge-part of precisely 2 elements of $$S$$. Thus the number of elements of $$S$$ is twice the number of edges.

Since a set has the same number of elements, regardless of how you count them (unless, of course, you miscount them), it follows that the sum of the degrees of the vertices is twice the number of edges.

Alternate Proof
In combinatorics, you can prove most results a number of different ways. Here is an alternate proof by induction on the number of edges $$m$$. If $$m=0$$, there are no edges and hence each degree is 0, so the total of the degrees is indeed twice the number of edges.

Now suppose that we have proved the theorem for all graphs with $$m-1$$ edges, and that graph $$G$$ has $$m$$ edges. Let $$uv$$ be an edge of $$G$$. Denote by $$G-uv$$ the graph identical to $$G$$ in all respects, save the omission of the edge $$uv$$. Then clearly $$G-uv$$ has $$m-1$$ edges, and so its degrees sum to $$2m-2$$. $$G$$ has identical degrees to $$G-uv$$, except for an increase by one at the vertices $$u$$ and $$v$$. Thus the sum of the degrees of $$G$$ is $$(2m-2)+2=2m$$, proving the theorem by induction.

Comments
The method of double counting is often used in Combinatorics, and is worth getting to know well.


text:
Introduction to graph theory/Proof of Theorem 2

Statement
A graph is a forest if and only if for every pair of distinct vertices $$u,v$$, there is at most one $$u,v$$-path.

Proof
Suppose you have a graph $$G$$ which is not a forest. Since a forest is defined to be a graph containing no cycles, it is clear that $$G$$ must have a cycle $$C$$. All cycles have at least three vertices. Let $$v_1, v_2, \ldots, v_n$$ (with $$n\ge 3$$) be the vertices on $$C$$ in order. Then there are two distinct $$v_1,v_2$$-paths, namely $$v_1v_2$$ and $$v_1v_nv_{n-1}\ldots v_3v_2$$.

Now suppose that $$G$$ is a forest, but has two distinct $$u,v$$-paths, namely $$ua_1a_2\ldots a_nv$$ and $$ub_1b_2\ldots b_mv$$. We would like to say that these create a cycle $$ua_1\ldots a_nvb_mb_{m-1}\ldots b_1u$$. However, a cycle can only include each vertex once, and it is possible that one of the $$b_i$$ is equal to one of the $$a_j$$. To that end, write $$a_0=b_0=u$$ and $$a_{n+1}=b_{m+1}=v$$. Let $$i$$ be the largest value of $$i<=n$$ such that $$a_i=b_j$$ for some $$j$$. Then we find a cycle, namely $$a_ia_{i+1}\ldots a_nvb_mb_{m-1}\ldots b_j$$. By our choice of $$i$$, clearly no vertex appears more than once.


text:
Introduction to graph theory/Proof of Corollary 3

Statement
A graph is a tree if and only if for every pair of distinct vertices $$u,v$$, there is exactly one $$u,v$$-path.

Corollary of
Theorem 2: A graph is a forest if and only if for every pair of distinct vertices $$u,v$$, there is at most one $$u,v$$-path.

Proof
A tree is defined to be a connected forest. Furthermore, a graph is defined to be connected if and only if for every pair of distinct vertices $$u,v$$, there is at least one $$u,v$$-path. The proof should now be evident, but for completeness:

If graph $$G$$ is not a tree, it is either not a forest, or not connected. If $$G$$ is not a forest, by Theorem 2, there exists a pair of vertices $$u,v$$ with more than one $$u,v$$-path. If $$G$$ is not connected, there exists a pair of vertices $$u,v$$ with no $$u,v$$-path. Thus if $$G$$ is not a tree, it is not the case that each pair of vertices has precisely one $$u,v$$-path.

If graph $$G$$ is a tree, fix a pair of vertices $$u,v$$. As G is a forest, by Theorem 2, there exists at most one $$u,v$$-path. Since $$G$$ is connected, there is at least one $$u,v$$-path. Thus there is exactly one $$u,v$$-path. Since we have proved this for any fixed pair of vertices $$u,v$$, it follows that if G is a tree, each pair of vertices has precisely one path.


text:
Introduction to graph theory/Proof of Theorem 4

Statement
For a graph $$G$$, the following are equivalent:
 * $$G$$ is a tree
 * $$G$$ is a minimal connected graph, in the sense that the removal of any edge will leave a disconnected graph.
 * $$G$$ is a maximal acyclic graph, in the sense that the addition of any missing edge will create a cycle.

Corollary of
Corollary 3: A graph is a tree if and only if for every pair of distinct vertices $$u,v$$, there is exactly one $$u,v$$-path.

Proof
Suppose $$G$$ is a tree, and let $$uv$$ be an edge not in $$G$$. By Corollary 3, there is exactly one $$u,v$$-path. If this path is $$ux_1\ldots x_kv$$, then adding the edge $$uv$$ will create the cycle $$ux_1\ldots x_kvu$$. As a tree is defined to be a connected forest and a forest is defined to be acyclic. A tree is therefore a maximal acyclic graph.

Now suppose $$G$$ is a tree, and let $$uv$$ be an edge of $$G$$. By Corollary 3, there is exactly one $$u,v$$-path. This path is clearly just the edge $$uv$$. Thus, in the graph $$G-uv$$, there is no $$u,v$$-path, and hence $$G-uv$$ is disconnected. Since a tree is defined to be a connected forest, a tree is therefore a minimal connected graph.

Now suppose $$G$$ is a maximal acyclic graph, but is not a tree. Since an acyclic graph is called a forest, and a tree is defined to be a connected forest, $$G$$ must be disconnected. Thus there exists vertices $$u,v$$ such that there is no $$u,v$$-path in $$G$$. In particular $$uv$$ is not an edge of $$G$$. As $$G$$ is a maximal acyclic graph, $$G+uv$$ must have a cycle. Since $$G$$ has no cycle, this cycle must include the edge $$uv$$. Let this cycle be $$ux_1\ldots x_kvu$$. Then $$ux_1\ldots x_kv$$ is a $$u,v$$-path in $$G$$. This contradiction shows that maximal acyclic graphs are trees.

Finally, suppose $$G$$ is a minimal connected graph, but is not a tree. As a tree is defined to be a connected forest, it follows that $$G$$ is not a forest, and hence (by definition), $$G$$ must have a cycle. Let $$u,v$$ be adjacent vertices on the cycle, and let the cycle be $$ux_1\ldots x_kvu$$. As $$G$$ is a minimal connected graph, $$G-uv$$ must be disconnected. Consider any vertex $$x$$. In $$G$$, there was a path from $$x$$ to $$v$$. If the path went through the edge $$uv$$, then in $$G-uv$$, there is clearly an $$x,u$$-path. Otherwise, there is clearly an $$x,v$$-path in $$G-uv$$. Either way, the connected component of $$G-uv$$ contains either $$u$$ or $$v$$. Since $$ux_1\ldots x_kv$$ is a $$u,v$$-path in $$G-uv$$, the connected component containing $$u$$ also contains $$v$$. Thus every vertex in $$G-uv$$ is contained in the same connected component, meaning that $$G-uv$$ is connected. This contradiction shows that minimal connected graphs are trees.


text:
Introduction to graph theory/Proof of Theorem 5

Statement
A tree of order $$n$$ has $$n-1$$ edges.

Corollary of
Theorem 4:For a graph $$G$$, the following are equivalent:
 * $$G$$ is a tree
 * $$G$$ is a minimal connected graph, in the sense that the removal of any edge will leave a disconnected graph.
 * $$G$$ is a maximal acyclic graph, in the sense that the addition of any missing edge will create a cycle.

Proof
For $$n=1$$, all graphs have $$0=n-1$$ edges. For $$n=2$$, there are two graphs, one with an edge and one without. The graph without an edge is disconnected, and therefore is not a tree, whereas the one with an edge is connected and has no cycles and therefore is a tree.

For $$n=3$$, there are 4 graphs, one each with 0, 1, 2 or 3 edges. The graphs with 0 and 1 edges are disconnected, while the graph with 3 edges has a cycle. The graph with 2 edges is connected and has no cycle, so the theorem is proved for $$n\le 3$$.

Now suppose we have proved this theorem for all $$n<k$$, and let $$T$$ be a tree of order $$k$$. Remove an edge $$e$$ from $$T$$ to form a new graph $$T-e$$. By Theorem 4, $$T$$ is a minimal connected graph, and hence $$T-e$$ is disconnected. Any connected component of $$T-e$$ (by the connectedness of $$T$$) must have one of the vertices of $$e$$ in it, and hence $$T-e$$ has exactly two components.

The two components are connected, and have no cycles, and are therefore trees. Let there be $$n_1, n_2$$ vertices in the two components of $$T-e$$. Then $$n_1+n_2=k$$. Further, there are $$n_1-1, n_2-1$$ edges in the components, so $$T-e$$ has $$k-2$$ edges in total. Therefore, putting edge $$e$$ back into the graph, $$T$$ has $$k-1$$ edges.


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School:Mathematics/Undergraduate/Applied Mathematics

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WIKIVERSITY SCHOOL OF MATHEMATICS - APPLIED MATHEMATICS
Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations research, continuous modelling, mathematical biology and bioinformatics, and many others.
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Structured courses
more needs to be added here
 * School:Mathematics/Foundation of mathematical concepts

Texts for reference

 * }


text:
Portal:Thermodynamics

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Laws of Thermodynamics (Simplified)
1. Energy cannot be created or destroyed. In other words, in a closed system, the total amount of energy that can be taken out of the system will be equal to the total amount of energy that was put into the system.

2. In any given exchange of energy, there will always be energy lost. This is referred to as entropy. This basically means that in any system, energy will always be lost in some means, be it friction, or some random quantum effect.

This also implies that there can be no such thing as a perpetual motion machine as energy will always be lost in some form.

 * Note that this lost isn't mean to say "destroyed". It rather means unusable. That lost energy is transferred to the microscopical degrees of freedom of the system, as are molecular vibrations, place in which we call it "heat".

3. No system can reach absolute zero temperature. This is due to the fact that at absolute zero, a system has no energy, and thus does not move. Although this does not cause any problems in the sense of classical mechanics; it does cause problems on the quantum level. If a particle had no movement at all, its speed would be exactly known (zero, exactly), which is forbidden by Heisenberg's uncertainty principle.

 * }

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Introduction
Engineering is an area in which mathematics is applied to actual real world problems. The basic fundamentals of Engineering all require Calculus to understand. The more advanced topics require knowledge of both ordinary and partial differential equations. Actual applications require the usage of multiple forms of mathematics in tandem with each other. Frequently, multiple partial differential equations are solved at the same time. To solve these equations effectively, a combination of linear algebra and differential equations is used.

One example is in solving the gas equation in chemistry. Standard equations of the "ideal" gas law is PV = nRT. Engineers understand that the "ideal" gas does not exist in reality, and a compression ratio must usually be used. As such the equation is often used as " PV = znRT " where z is used as a compression ratio.

The compression ratio though is a mathematically derived constant which is based on experimental data. The data is taken and has curvefitting applied to it. Basic algebraic curve fitting does not do an adequate job, so a differential approach is applied to it. In order to do this Maxwell's equations must be applied to get the ideal Gas law into a partial differential format.

One of Maxwell's equations is:

 * $$ T = \left( \frac{\partial H}{\partial S} \right)_P $$

where T is temperature, H is enthalphy, S is entropy, and p is pressure.

So, in essence, the temperature is equal to the change in enthalpy with respect to the change in entropy, assuming that pressure is constant. The $$\partial$$ operator states that the enthalpy (H) is dependent on multiple variables (S,p) and that S is just one of them. By making pressure a constant, the derivative of H just depends on S, and not on p.

By making substitutions like this into the ideal gas equation, the compression factor "z" can be determined for each gas.

 * }

title: Portal:Thermodynamics


text:
Philosophy of mathematics

This is intended to become the script for a course on the Philosophy of Mathematics. It may then serve as basis for books and other materials as well as discussions and research. Also see the discussion page of this module.

About Philosophy of Mathematics
This is an introductory meta chapter on the topic and the course of Philosophy of Mathematics (PhilM). It might be helpful for deciding on taking this course as well as for understanding its approach.

The Scope

 * The Topic PhilM: What is Mathematics? How is Philosophy seen here? Who is PhilM for? What is PhilM about?
 * The Course PhilM: didactics, boundaries, ...

The Structure

 * The Topic PhilM: What are the structuring principles? What does the structure look like?
 * The Course PhilM: didactics, approach, ...

The Basics

 * /Defining Mathematics/ - The subject usually called Mathematics.
 * Other Basics used within the course - misc. basics can be added here ...

The History
Understanding the historical development of PhilM ...

Philosophy of Supplying Mathematics
A Mathematician is thinking about the degrees of freedom he or she has while doing his or her job, finally starting to question the foundations of a mathematician's job.

Questioning the Method
Judging mathematical work beyond right and wrong: How to do the Mathematician's every day job? How to exploit the degrees of freedom a Mathematician has in doing a Proof, judging the importance of Sentences and overthrowing Theories?

 * Questioning the Instance of Proof - The Exam Situation. Why do I do my Proof the way I do? What is a good or bad proof? Is a proof an explanation? ...
 * Questioning the Sentence - Normal Science. What Sentence to prove? What is an important Sentence? ...
 * Questioning the Theory - Revolutionary Science. How do Theories interrelate? What is an important Theory? ...

Questioning the Foundations
How well-founded is Mathematics? Is there a better Mathematics than the one we employ today?

 * Questioning the Concept of Proof - Metamathematics. Is there a mathematical definition of a mathematical proof?
 * Questioning the Language - Metamathematics. What can and what can't be expressed with the usual mathematical Language? Are there more expressive alternatives?  Does increased expressiveness come for free? ...

Philosophy of Applying Mathematics
Or "Philosophy of Logic"

Questions concerning justification and use:

 * How can a Theorem/Theory be of any use in general?

All theorems are either (a) an assumption agreed upon by a posteriori knowledge (something we intuitively observe), or (b) a priori knowledge. (something we assume is true within a rational language by strict rules of inference).

In other words, theorems are either facts we can generally agree on about the way the universe works around us (If you put one apple in a box with one other apple, the box will have 2 apples), or they are concepts which we base a formal language on (1+1=2).

Deductive reasoning (A priori): By collecting a vast amount of theorems, we can use these theorems together and derive / dedeuce other assumptions about the world around us. Ie: We can take different things that we all assume, and use them together to assume more things. Eg: "If a+b=c, and c-b=a, then c-a=b"

Inductive reasoning (A posteriori): By collecting a vast amount of theorems, we can connect some of our assumptions and approximate highly probable theories about how they function together. Eg: "Out of every of the thousands of DNA strands we've examined on Earth, every one of them contains the base-pairs of Adenine, Guanine, Thymine, Cytosine. So it is fair to assume that with possible rare exceptions- all DNA strands on Earth have the base pairs of Adenine, Guanine, Thymine, Cytosine"

 * How beneficiary is the Theorem/Theory, esp. to science and engineering?
 * Where and how may one apply this Theorem/Theory?

Further Questions:

...

title: Philosophy of mathematics