text:
User talk:Robert Huber~enwikibooks

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
 * You should sign your name on Talk and vote pages using four tildes, like this: ~~~~.
 * Remember to conduct any editing experiments in the sandbox.
 * If you're a Wikipedian, see Wikibooks for Wikipedians for a primer on how things work here (it's a little different).
 * If you want to base your work here on materials from Wikipedia, please use WB:RFI to bring the material over in compliance with the GFDL.
 * If you're an instructor and plan on using wikibooks for a class project, see Guidelines for class projects
 * Please say hello at the Staff Lounge with any questions or ideas.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Please take a look at Naming policy before starting a new book.
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck!--Sundance Raphael 21:21, 10 January 2007 (UTC)

Dupe pages?
Hi I see you have created two Animal Behavior/Cover pages (the other one has different capitalisation). Do you require them both or should one be deleted, thanks -- Herby talk thyme 12:12, 11 January 2007 (UTC)

Equally you have two pages Animal Behavior/Contributors/ (one with a trailing slash the other without)? -- Herby talk thyme 12:20, 11 January 2007 (UTC)

re: Dupe pages?
hi herby, what a reat. i was just looking to see whether i could scrounch up an admin to delete the two unused files - one cover and on for contributors. i am completely new and am trying to figure still everything out. i am not even sure whether this is the right way to respond to your message. i think i have just managed to grasp file names and conventions, so i hope i wont create many more duplicates :-).

thanks again and please delete away, much obliged, ciao, 129.1.84.29

i guesss this is what the tildes do when you are logged in :-) Robert Huber

 * No problem at all Robert - always ask - there are folk around who will help if they can. You'll soon get the hang of it! I'll delete them anyway - regards -- Herby  talk thyme 14:34, 11 January 2007 (UTC)

Another thought!
Hi Robert - I've piped a couple of links using "|", you can see what I have done here. I won't do anymore as you may not feel it improves things but if you do you'll learn by doing it yourself (much of what I know I have learnt from other people!). If I can help let me know - regards -- Herby talk thyme 18:36, 11 January 2007 (UTC)

 * Great, thanks, that does look mighty fine. Your changes are greatly appreciated and make my life for adding in the material that much easier.

 * I have also just run into a different problem and was wondering what the proper solution would be. Another wikibook has a section that would be very applicable to this book.

 * Biological basis of behavior

 * i can link it into my table of context but then the navigation items are to the original table of contents and not to this book. Alternatively, if i copy the text, then there are two places that need to be updated in identical ways. it would really be best if the section could be used for both but the navigation would be book specific - how do i handle this all best? thanks, ciao,

 * oops Robert Huber

 * End of dayish in my time zone Robert so I'll take a look at this tomorrow and see if I can help (I'd like to say "I'm sure I can") then. Hope that is ok, regards -- Herby  talk thyme 19:28, 11 January 2007 (UTC)

Hi Robert - ok - copying text is a no no - licensing issues means that you are cutting out the "contributors". Apart from anything else I would consider a message on User talk:ArrowStomper the contributor's talk page for that page. I guess I would go for a "see also" approach in both articles rather than a navigation between them. the "See also" idea is not used here as much as Wikipedia but I do think it is relevant in this case.
 * reset

The book looks promising - the nature/nuture debate has always fascinated me (won't say more - don't want to fall out!!). I'll certainly keep watching but let me know if I can help - regards -- Herby talk thyme 13:07, 12 January 2007 (UTC)

 * hi herby, thanks for your help. i contacted arrowstomper to see what s/he has to say. more soon, ciao, Robert Huber

Importing
also, it is cool to import material via WB:RFI - right? could i do that with the materials from the psych page or does that simply apply to wikipedia entries?, ciao, Robert Huber

 * not sure I understand this. To me this is for importing from external Wikis (such as WP).  You may be better posting this one in the Staff lounge where more folks (& more experienced folks) will see it.  I see you have linked the Psych page anyway -- Herby  talk thyme 15:52, 12 January 2007 (UTC)

 * i had played with this yesterday.i take if off for now and see what arrowstomp says, thanks, ciao, Robert Huber

Links to Wikipedia
sorry, one more question ... i am trying to link to a subsection of a wikipedia page using Mind/body dualism, but that does not seem to work as i would have hoped. thanks for any suggestions, ciao, Robert Huber

 * Again not sure - to me what you have done works just fine. The link takes you to the relevant spot in WP.  If you want the WP content that would be a WB:RFI issue (and can be done).  However bear in mind that way you get a snapshot of the WP page, edits after the tranwiki'ing will not be there?  Cheers -- Herby  talk thyme 15:52, 12 January 2007 (UTC)

 * i get to the overal entry for descartes, but it fails to jump to the #dualism subsection. does this actually work for you, maybe it is my browser, thanks, ciao Robert Huber

 * Interesting - goes to Dualism for me - using Firefox, works in Opera to (please apologise first if you are using IE !) -- Herby talk thyme 16:53, 12 January 2007 (UTC)

 * thanks. great to know. nah, definitely no IE here, i m using safari on the mac and i am proud to offer a microsoft free environment for the kids in my classes ;-) ciao and thanks again Robert Huber

Animal behaviour patterns
Is it just me (or Euro based view) Konrad Lorenz (sp?) springs to mind? Bird behaviour I think -- Herby talk thyme 12:15, 13 January 2007 (UTC)

 * Konrad_Lorenz?? -- Herby talk thyme 12:18, 13 January 2007 (UTC)

 * indeed, this chapter is next :-) bird behavior it is. this is the part where i have quite a bit of material as i was one of his last graduate students - i even have pictures which i will be happy to bring to the public domain, more soon, Robert Huber
 * Excellent - as you will have guessed it does interest me. And must have been interesting to have worked with someone like that - Thanks and regards -- Herby  talk thyme 12:42, 13 January 2007 (UTC)

Deletions
No problem - will do. In case I am not around take a look at WB:TM & particularly Template_messages/Deletion as this will allow you to "tag" such pages as you require deleted yourself and any admin will get them as they will be placed in that category which is checked regularly. Regards -- Herby talk thyme 13:19, 20 January 2007 (UTC)
 * OK - done some - however there are pages on that list that have contributions (in some cases substantial) from IP addresses (Animal Behavior/History of Neuroscience for example). As such I am not comfortable with deleting the page from a licensing and approval standpoint.  If you "retain" information that others have placed it would be far better to move or merge content.  Thanks -- Herby  talk thyme 13:42, 20 January 2007 (UTC)
 * hi herby, the tcpip address was also me, i had just forgotten to log in :-) Robert Huber 14:00, 21 January 2007 (UTC)
 * Ok there are just three left that have more that one IP address contributing to them (widely different ones)? Thanks -- Herby talk thyme 15:25, 21 January 2007 (UTC)
 * thanks, the other ips should be mine as well, 129.1.84.29 is me at work and the other looks like me at the coffee shop :-) thanks,Robert Huber 12:08, 22 January 2007 (UTC)

templates for a animal behavior stub
hi, i would like to create a book specific stub, i found where the existing ones are listed but not on how to create one. any advice greatly appreciated, thanks, Robert Huber 16:18, 25 January 2007 (UTC)

Copyright problem with Image:LobsterFight.ogg

 * Just go to the image description page, click the edit tab at the top. Replace what's there with plus:

 * – Mike.lifeguard  | talk 02:21, 29 November 2007 (UTC)

 * Thanks, Mike, i did that, can you please double check? on previous uploads i got a box to select the type of permission to grant, did i miss it this time or is it gone? what is the type of permission that wikibooks is favoring? GNU? how does wikimedia fit in? is there a page on this somewhere? Robert Huber 12:10, 29 November 2007 (UTC)

Animal Behaviour book in catalan
Hello Robert,

First of all, congratulations for your book. I will read it entirely when I can. For now I've read the first pages and I like it very much.

I've just start "my" own wikibook about Animal Behaviour. It is in catalan, but if you are interested in a particular page I can bring you a quick translation in my poor english (or very quick in spanish). It will be structured in my own point of view, heavily influenciated for my Animal Behaviour teacher, but your text will be very useful to me.

I whish the help will be reciprocal. Ask me all that you want. Could I ask you questions if I have any? For now, what pages on your book do you think I can improve?

Thank you --Loquetudigas (awnser here please!) 00:33, 19 October 2008 (UTC)

Your account will be renamed
Hello,

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

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

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

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:38, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:25, 19 April 2015 (UTC)

title: User talk:Robert Huber~enwikibooks


text:
Cookbook:Raisin Crumb Pie

| American cuisine | Vegetarian Cuisine | Dessert

Ingredients

 * 1 pound raisins
 * 1½ cups cold water
 * 3 cups white granulated sugar, divided
 * 2 cups flour
 * ¼ cup shortening
 * 1 cup milk
 * 2 eggs, well-beaten
 * 2 teaspoons baking powder

Procedure

 * 1) Wash and clean raisins and add the cold water and 1½ cups of sugar.
 * 2) Cook until the raisins are tender and plump.
 * 3) Set aside to cool.
 * 4) Preheat oven to 440°F.
 * 5) Mix the flour, 1½ cups of sugar, and the shortening, crumbling well with the fingers. Take out 1 cup of these crumbs and set aside.
 * 6) To the remaining crumb mixture, add the eggs and milk; mix well.
 * 7) Stir in the baking powder.
 * 8) Line 2 large pie tins with pie crust and fill with cooked raisins. Pour over this the batter and sprinkle the top with the cup of reserved crumbs.
 * 9) Bake in a hot oven at 440°F for 10 minutes, then reduce heat to moderate oven (350°F) and finish baking, about 35 minutes.

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

title: Cookbook:Raisin Crumb Pie


text:
User talk:68.237.11.102

Regarding edits made during January 10 2007
Please stop adding nonsense to Wikibooks. It is considered vandalism. If you would like to experiment, use the sandbox. Thank you. If this is an IP address, and it is shared by multiple users, ignore this warning if you did not make any unconstructive edits. Az1568(Talk) 21:33, 10 January 2007 (UTC)

title: User talk:68.237.11.102


text:
User talk:82.15.61.141

Regarding edits made during January 10 2007
Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. If this is an IP address, and it is shared by multiple users, ignore this warning if you did not make any unconstructive edits. Az1568(Talk) 21:33, 10 January 2007 (UTC)

title: User talk:82.15.61.141


text:
User talk:75.68.100.2

Regarding edits made during January 10 2007
Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. If this is an IP address, and it is shared by multiple users, ignore this warning if you did not make any unconstructive edits. Az1568(Talk) 21:34, 10 January 2007 (UTC)

title: User talk:75.68.100.2


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User talk:Dwallen

Leave a new message

Global Text
Though I wish I could say "No Problem!", I'm afraid I can't say that. The incompatibility of the two licenses might be fixed someday, but for now they're incompatible (as far as I know... I'm not a lawyer). I would be interested to see what they had to say... please email it to me.

You also might want to bring this up on textbook-l, our mailing list. -- SB_Johnny | talk 21:48, 10 January 2007 (UTC)

New student-written text about educational psychology?
Dear Dwight,

I teach educational psychology at the University of Manitoba, and I am thinking about creating a student-written textbook about educational psychology. It would be reminiscent of your Wikibook about the social and cultural foundations of education.

I've read most of the behind-the-scenes advice that I could find about doing this (including your course syllabus for ECI 301, and the discussion pages for your Wikibook). My overall impression is that I should be able to do something along these lines, but it will have to be more modest, at least at the beginning. The reasons are that I have only about 70-75 students per year, only a handful of graduate students, and no grant to support this project at the moment. Nonetheless I would like to see what can be done.

I believe that you may have published an article or two about your experiences and/or assessment of the course, but I haven't been able to locate them. Can you point me in the right direction, and/or send me appropriate reprints (electronically is fine)? Any advice, ideas, or cautions would also be most welcome.

Thanks, Kelvin Seifert Faculty of Education University of Manitoba Winnipeg, Manitoba, Canada email: seifert@cc.umanitoba.ca

Klseifert 21:38, 19 July 2007 (UTC)

title: User talk:Dwallen


text:
User talk:87.254.78.203

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

On of the Wikibooks users have nominated the Manx Youth Games book for deletion. Please be aware that Wikibooks is for the creation of books and that articles of an encyclopedic nature should be created on Wikipedia which is more suitable. I'm from the Isle of Man also so would be delighted to help in any way over on Wikipedia.

Xania talk 22:26, 10 January 2007 (UTC)

title: User talk:87.254.78.203


text:
Animal Behavior/Contributors

Contributors
While Wikibooks offers somewhat clearer opportunities for "authorship" than Wikipedia, there remains the fact that anything put here is really a contribution. Content is thus developed by contributors rather than authors. If you are interested in developing a particular subject as part of this textbook, please add your material to an existing chapter or start a new page and fit it into the book's structure. The chapter arrangement can be fine-tuned at a later time.

 * Robert Huber is a biologist at Bowling Green State University. This book started as an online, html-based, modular textbook in 1998. This material was transferred into the Wikibook to start things off. Since then content continues to develop for use in a college course in Animal Behavior.
 * C. Jackson. Contributed to the Nature vs. Nurture section.
 * C. Nordal. Contributed to the Sensory Memory section.
 * G. Grecco. Contributed to the Curare section.
 * S. Laing. Contributed to the Epigenetics section.
 * R. Goldsmith. Contributed to Pituitary Gland, Oxytocin sections.
 * ... (add your name and contribution)
 * Case Studies were contributed by ... (add your name if you have contributed a case study, research summary, or example)
 * Wikibook contributors (see individual page histories)
 * Wikipedia contributors

<<Return to Contents Page

title: Animal Behavior/Contributors


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User talk:140.163.0.5

Radiation Oncology wikibook
Thank you for helping out with the Radiation Oncology wikibook! Brim 23:35, 10 January 2007 (UTC)

title: User talk:140.163.0.5


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User talk:Alfie the Goom

You have been temporarily blocked from editing for vandalism of Wikibooks. If you wish to make useful contributions, you are welcome to come back after the block expires.

Xania talk 00:45, 11 January 2007 (UTC)

title: User talk:Alfie the Goom


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Template:PWP

/Transcluded Modules

title: Template:PWP


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Adventist Youth Honors Answer Book/Nature/Poisonous plants/Amanitas

|


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User talk:202.69.57.8

Regarding edits made during January 10 2007
Please do not add inappropriate external links to Wikibooks. Wikibooks is intended to be read without the need for external reference and requires content rather than external links particularly those which are for advertising or promotion. Inappropriate links may include (but are not limited to) links to personal web sites, links to web sites with which you are affiliated, and links that exist to attract visitors to a web site or promote a product. If you feel the link should be added to the article, then please discuss it on the article's talk page rather than re-adding it. See the welcome page to learn more about Wikibooks. Thank you. Az1568(Talk) 01:34, 11 January 2007 (UTC)

title: User talk:202.69.57.8


text:
User:Kesh

=Welcome!=

My user page will be a bit sparse at first. I'm familiar with Wikipedia, but I'm fairly new to WikiBooks. Time to learn some new templates and formatting!

In the meantime, I'm using this page as a repository for links I find useful

 * Wikibooks for Wikipedians
 * What is Wikibooks
 * Manual of Style
 * Policies and guidelines
 * Naming policy
 * Study help desk
 * Staff lounge
 * Help:Template
 * Wiki Science

title: User:Kesh


text:
User talk:Mleex037

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
 * You should sign your name on Talk and vote pages using four tildes, like this: ~~~~.
 * Remember to conduct any editing experiments in the sandbox.
 * If you're a Wikipedian, see Wikibooks for Wikipedians for a primer on how things work here (it's a little different).
 * If you want to base your work here on materials from Wikipedia, please use WB:RFI to bring the material over in compliance with the GFDL.
 * If you're an instructor and plan on using wikibooks for a class project, see Guidelines for class projects
 * Please say hello at the Staff Lounge with any questions or ideas.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Please take a look at Naming policy before starting a new book.
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck!. Az1568(Talk) 01:39, 11 January 2007 (UTC)

title: User talk:Mleex037


text:
User talk:Kesh

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
 * You should sign your name on Talk and vote pages using four tildes, like this: ~~~~.
 * Remember to conduct any editing experiments in the sandbox.
 * If you're a Wikipedian, see Wikibooks for Wikipedians for a primer on how things work here (it's a little different).
 * If you want to base your work here on materials from Wikipedia, please use WB:RFI to bring the material over in compliance with the GFDL.
 * If you're an instructor and plan on using wikibooks for a class project, see Guidelines for class projects
 * Please say hello at the Staff Lounge with any questions or ideas.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Please take a look at Naming policy before starting a new book.
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck!. Az1568(Talk) 01:39, 11 January 2007 (UTC)

title: User talk:Kesh


text:
Talk:Operating System Design/Case Studies/NTFS

This guy doesn't know what he's talking about. "In his opinion NTFS is faster...", IN HIS OPINION??? FAT32 is faster, FACT. NTFS is a much more robust and advanced filesystem, but it's definately not faster than the much simpler FAT32. —The preceding unsigned comment was added by 76.5.191.2 (talk • contribs) 06:45, 11 Jan 2007 (UTC)

Agreed. The author of this article has no idea what he's talking about, or even what he should be talking about. He is describing NTFS from a user's standpoint. A windows user's standpoint never the less. This article should be talking about the actual filesystem, not how a specific operating system's installer uses it.


text:
Cookbook:Maida

| Flour

Maida, also maida flour, maida mavu, is super-refined wheat flour used in Indian cuisine. It is mainly used to make pastries and other bakery items like breads and biscuits. Some maida may have tapioca starch added.

title: Cookbook:Maida


text:
General Mechanics/Mach's Principle

Mach's principle is the name given by Einstein to a vague hypothesis first supported by the physicist and philosopher Ernst Mach.

title: General Mechanics/Mach's Principle


text:
Non-Programmer's Tutorial for Python 2.6/Print version

= Authors = = Front matter =

= Intro =

= Hello, World =

= Who Goes There? =

= Count to 10 =

= Decisions =

= Debugging =

= Defining Functions =

= Advanced Functions Example =

= Lists =

= For Loops =

= Boolean Expressions =

= Dictionaries =

= Using Modules =

= More on Lists =

= Revenge of the Strings =

= File IO =

= Dealing with the imperfect =

= The End =

= FAQ =

= License =

The Non-Programmer's Tutorial for Python is licensed under the GNU Free Documentation License. All programming examples in the text are granted to the public domain.


text:
Modern Physics/Brans-Dicke theory

In theoretical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan-Brans-Dicke theory) is a minority viewpoint in physics. The Brans-Dicke theory is a well-known competitor of Einstein's more popular theory of general relativity. The theory was developed by Robert H. Dicke and Carl H. Brans building upon, among others, the earlier work of Pascual Jordan. It is an example of a scalar-tensor theory, a gravitational theory in which the gravitational interaction is mediated by a scalar field as well as the tensor field of general relativity.

At present, both Brans-Dicke theory and general relativity are generally held to be in agreement with observation, although the experiments of the golden age of general relativity have considerably constrained the allowed parameters of Brans-Dicke theory.

title: Modern Physics/Brans-Dicke theory


text:
Non-Programmer's Tutorial for Python 2.6/Authors

 * Josh Cogliati
 * User:Jrincayc, jjcogliati-jan2007 AT yahoo.com

Contributors

 * Kiah Morante (User:greenmanwitch)
 * Elizabeth Cogliati
 * James A. Brown
 * Joe Oppegaard
 * Benjamin Hell (User:Siebengang)


text:
Cookbook:Tapioca

Tapioca can mean several different things in cuisine:

 * Tapioca Starch — a refined flour derived from the cassava root, sometimes called tapioca flour
 * Tapioca Pearls — small spheres of tapioca starch, often used in puddings such as tapioca pudding, and as a substitute for sago
 * Tapioca Pudding — a thick, sweet pudding made from tapioca pearls (and sometimes, cooked rice)

title: Cookbook:Tapioca


text:
Wikibooks:WikiProject Volunteer editors

Volunteer Editors is a project to compile a list of wikibookians by ability. Wikibookians on this list could be asked to help proofread related books, especially in conjunction with Wikiproject editorial board. List your names below along with your competencies in various subjects. This is optional.

List of volunteer editors
Please enter your name and qualifications or userid and contributions. Indicate the specialist areas where you feel qualified to act as a member of an editorial board. Make a separate entry in each section where you feel you can contribute.

Natural Sciences
RobinH 11:56, 11 January 2007 (UTC) Special relativity, Consciousness studies. Undergraduate level: Physiology, biophysics, physics, neurophysiology, philosophy - metaphysics. High school level: science, biology, zoology, botany, chemistry, maths, physics.

Whiteknight. BS in Electrical Engineering, Graduate-level work in the same field. Background in chemistry, physics.

Mike.lifeguard: Psychology, neuroscience (esp. cognition and language) and the intersecting related fields; biology (esp. evolutionary, cell and some biochemistry).

Social Sciences
RobinH 12:02, 11 January 2007 (UTC) Consciousness studies. High school level: psychology

Kijog 18:55, 27 January 2007 (UTC) Languages and Language teaching, especially Romance. Professional Journalist.

Computing
RobinH 12:02, 11 January 2007 (UTC) High school level: computing, C, C++, fortran

Whiteknight. Knowledge of C, C++, x86 ASM, Java, Perl, others. Windows, Linux. Computer hardware design.

xixtas. 10+ years of professional experience as a networking and security consultant.

within focus. Extensive C#, PHP, Ruby/Rails, some C/C++, proficient with Windows and Linux, various computer engineering including VLSI, network design, and security.

Humanities and arts
RobinH 12:02, 11 January 2007 (UTC) Consciousness studies Undergraduate level: philosophy of mind. High school level: History of Art

xixtas Erstwhile professional musician, some background in musical theory.

within focus. Annotated texts and literature study guides.

Kijog 18:55, 27 January 2007 (UTC) Languages and Language teaching, especially Romance. Professional Journalist.

Mike.lifeguard Foreign policy, geopolitics and its history (esp. as relating to the US and terrorism); philosophy (esp. morality, theory of mind, religion). 2021 march (fmt) stert to make human to how live wite there gose and his retavie and other to unederstind each other how use money it bigging to exampnd for one place to anther how to live in they community and understand they  community live

Miscellaneous
Mike.lifeguard Teaching and learning theories (esp. concerning the underlying psychology & esp. practical learning); lifeguarding, water safety, swimming & other water sports, first aid (and related "medical" stuff like resuscitation and physiology).

title: Wikibooks:WikiProject Volunteer editors


text:
Animal Behavior/Cover

Animal Behavior

A guide to the hows and whys of animals as they interact with each other and with the world around them.

This textbook focuses on the mechanisms and evolution of animal behavior, including neural, hormonal, and genetic substrates of behavior; foraging; anti-predator defenses; mating systems and sexual selection; social behavior; communication; parental care; kin selection and recognition; and territoriality. Associated laboratory exercises provide hands-on experience for many of these concepts.

Table of contents

Comportament Animal

title: Animal Behavior/Cover


text:
Template:Haskell/General Practices

Debugging Testing Packaging your software (Cabal) Using the Foreign Function Interface (FFI) Generic programming: Scrap your boilerplate

title: Template:Haskell/General Practices


text:
Template:Haskell/Specialised Tasks

Graphical user interfaces (GUI) Databases Working with XML Parsing mathematical expressions Writing a basic type checker

title: Template:Haskell/Specialised Tasks


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Template:Haskell/Libraries Reference

Introduction MaybeIORandom Data structures primer ArraysMaps

title: Template:Haskell/Libraries Reference


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Talk:Non-Programmer's Tutorial for Python 2.6

Interactive code examples
Even better than syntax highlighting would be actual in-browser interactive examples like those that can be seen at trinket.io. I saw trinkets in an edX course and they seem really useful, especially for beginners. This would require wikibooks to allow their embed codes but it looks like they are adding more subjects (HTML, music theory, python console) so it could be a cool partnership.

Not every example could be converted but it would be a huge win for those that could!

Passing variables to os.system
This book should cover the os.system call and state how to pass a varible to this call for example I have >>> os.system ("ls -l ,dir ") and that does not work 70.70.255.225 (talk) 00:00, 5 December 2010 (UTC)

Syntax highlighting
I think the tutorial could get even more attractive with some syntax highlighting of the code. Wikibooks supports the mw:Extension:SyntaxHighlight GeSHi extension, which actually looks really nice. In order to use it, code examples have to be put into   tags, which could actually replace the   ...   tags used almost everywhere in the book so far. This works only for code blocks and not for in-line code, but I think that's a minor problem. Any objections? Siebengang 12:58, 27 September 2007 (UTC)

 * I could implement this change quickly using my bot, if nobody disagrees with the change. I think it would be a big improvement over the existing code formatting. --Whiteknight (Page) (Talk) 14:17, 27 September 2007 (UTC)

 * Jrincayc 02:41, 28 September 2007 (UTC)

 * Automating dirty work is great:
 * Go for it! I can have a look at the result and fine-tune things if necessary. Siebengang 08:29, 28 September 2007 (UTC)
 * Go for it! I can have a look at the result and fine-tune things if necessary. Siebengang 08:29, 28 September 2007 (UTC)

 * ✅. I didnt realize there were so many special cases, so my bot ended up making a big mess. I've fixed it all as best I could, and i think it all looks pretty good. Let me know if we need anything else. --Whiteknight (Page) (Talk) 17:15, 28 September 2007 (UTC)
 * Thank you! I went through the book and fixed some more small things. Should be more or less bug free now. --Siebengang 10:15, 1 October 2007 (UTC)

Refurbishing Code examples according to Python Style Guide (PEP 8)
During the next weeks, I am going to check the formatting of the code examples so that they get compliant with the Python Style Guide (PEP 8, http://www.python.org/dev/peps/pep-0008/). Apart from that, there are some LaTeX style references in the text (\ref{...something...}) and some other minor bugs which I'll fix, too. --Siebengang 12:33, 11 January 2007 (UTC)

 * Done up to Debugging --Siebengang 15:37, 11 January 2007 (UTC)
 * Done up to Lists --Siebengang 15:27, 15 January 2007 (UTC)
 * Done up to Boolean Expressions --Siebengang 10:45, 17 January 2007 (UTC)
 * Done until Using Modules --Siebengang 11:49, 22 January 2007 (UTC)
 * Did More on Lists, Revenge of the Strings, The End and FAQ --Siebengang 11:38, 23 January 2007 (UTC)
 * Finally done: Dealing with the imperfect and File IO --Siebengang 12:08, 23 January 2007 (UTC)

Before You Begin Your Quest chapter
I merged the information from Before You Begin Your Quest with Intro and took away the link from the table of contents, to get rid of the redundency. --Siebengang (talk) 11:21, 20 February 2008 (UTC)

Redundant solution set info?
Is there a particular reason why the same text from module exercise descriptions is included in the solution set as well? It seems redundant to me with the current (description of exercise) (solution bar) (description of exercise) setup the text has now. --Tehdrago (talk) 20:20, 15 November 2008 (UTC)

break and continue
It's probably worth introducing the 'break' and 'continue' statements here. I don't think they are described anywhere at all in the book at the moment Dooglus 19:37, 1 May 2006 (UTC) dooglus

None
Might want to have something on assigning a value of "None" to variables in the Count to 10 module or in the previous lesson, Who Goes There?". The Exercise section in Count to 10 uses it without a prior introduction, I was a bit confused when trying to figure out the solution.

Maybe something like this?

I'm not sure about the exacts details here as I'm learning python myself, maybe someone more experienced could clean this up and add it in? --Tehdrago (talk) 10:18, 15 November 2008 (UTC)

Merge talk pages?
I added the "Count to 10" talk page contents here to the main page. I think it might be easier if all the talk pages for the wikibook were in the same location. As there only seems to be a small number of contributors currently working on the wiki, I think it would make things easier to have all the suggestions in the same place. Maybe create a "master" talk page that had the contents of all the individual talk similar to the "printable version" style page? It might get messy if all the module talk pages were redirected here. --Tehdrago (talk) 09:06, 16 November 2008 (UTC)

Archiving
Would anyone mind if I moved the older, implemented suggestion discussions here to an "archive" page, so only the not discussed / unresolved issues are shown? --Tehdrago (talk) 09:06, 16 November 2008 (UTC)
 * Hmm, I found a to-do list in the module "The End", I'll use that for keeping track of suggested changes for now. --Tehdrago (talk) 18:18, 16 November 2008 (UTC)

Python 3.0
I am going to start creating Non-Programmer's Tutorial for Python 3.0 soon. Since most of the examples in the existing tutorial will be invalid in python 3.0, and python 2.6 will be here for a while, I figure we can have two tutorials for awhile. Jrincayc (talk) 04:49, 30 December 2008 (UTC)
 * I started it. Jrincayc (talk) 15:29, 31 December 2008 (UTC)

 * Should we move this tutorial for Python 2.6 to "Non-Programmer's Tutorial for Python 2.6"? --DavidCary (talk) 00:11, 5 January 2009 (UTC)

 * Well, I think any python 2.2 to 2.6 will work with this version. Almost all the examples will work in python 1.6 - 2.6.  That said I have no objection to moving it to something like "Non-Programmer's Tutorial for Python 2".  Jrincayc (talk) 04:20, 5 January 2009 (UTC)

I moved it...something is wrong...its all red links? I even moved the subpages. --33rogers (talk) 11:36, 20 June 2009 (UTC)
 * nevermind I just purged the wiki cache and its proper now :) --33rogers (talk) 11:40, 20 June 2009 (UTC)

Exercises
I noticed that many solutions to excersises were much messier than they needed to be. Should they at least note that their are other solutions?
 * They can note that there are other solutions. Some of the exercises already have multiple solutions.  They also can be rewritten, but be careful to only use features of python that have already been discussed (For example don't use boolean logic until after it has been discussed in the tutorial.) Jrincayc (talk) 12:17, 9 June 2009 (UTC)

I just love WIKI
I just Love WIKI .. GREAT WORK. ...Thanks to all of You ..

username/password exercise
The first solution given, just keeps on asking you your name and doesn't go to the end of the program (for me at least). The second solution works fine.

How to contribute to future versions?
I would like to help make this document available for up-to-date versions of Python (2.7 and 3.x). However, it looks like this project is generated automatically out of a LaTeX source, meaning it would not be helpful to edit it here. Is that correct? How should others contribute to newer versions?

Python 2 - Outdated
While this is still a good tutorial, Python 2 has reached it's end-of-life as of 2020. I would recommend removing this tutorial from the 'Featured', and adding a disclaimer mentioning that Python 2 is outdated.


text:
Talk:When It Hits the Fan/Rebuilding

This wikibook is kind of a how-to for restoring civilization from scratch.

You might read Lord of the Flies by William Golding, Tunnel in the Sky by Robert A. Heinlein and possibly Lucifer's Hammer by Larry Niven and Jerry Pournelle for ideas and/or inspiration.

Basicly, I created this book to flesh out a thought experiment of mine (what happens if you drop a plane full of people on an island with no chance of return), and decided to open it up to the public in a general sense ((re)building civilization in any remote location). -DavesTA 12:56, 1 March 2007 (UTC)

Really rebuilding civilization
I believe the object of the document should be to list all steps necessary to actually rebuild civilization from scratch. All technologies necessary and the foundation technologies necessary for rebuilding civilization to the current level need to be included or referenced.

When the document is completed, a group of people should be able to be dropped into the desert or a habitable wilderness and be able to create a functioning high-tech civilization using materials at hand.

The founding technologies of survival (how to start a fire, how to kill animals for food, how to identify edible plants, how to build a shelter, how to purify water) should form the first sections. Next should come the sections describing the next level of technology and how to create the tools to build it, from iron all the way to microchips. The sum total of all human knowledge needs to be collected into one place which is what Wikipedia is all about. There should of course be a discussion of the various forms of human government and which are more effective during each technological phase, and the process by which the next phase can be implemented.

The text should ultimately include video files demonstrating how to perform various useful acts.

The most important aspect would be the "how to safeguard the information in this document" along with various methods. Perhaps a priesthood (like the "technomages" in Babylon 5) should be created to protect and copy the document, and ensure that it survives any nuclear or other catastrophe that befalls the human race.

This is a topic I have thought about for years. How could civilization quickly recover from any massive catastrophe in a short period of time without being reset back to the stone age? Think of this article as a disaster recovery document.

Methods of preserving the document should also be considered: hard copy, CD/DVD, a video player embedded in a stone monolith ala 2001 (my favorite ;-) etc.

-jjb

Golf Course?
Wouldn't that imply: "Oh, we're stranded and we're playing golf, then we've 'planned' to be stranded for an extended amount of time?" That might raise hopelessness for some people, projects that should be done while stranded should be projects that is useful to the survival of the group (or at least something that looks useful).

Week long trips should be avoided to avoid exhaustion and wounds when medical treatments is a scarce resource, unless there is something that needs immediate investigation or there is a hope that something that is helpful could be found on the exploration. In short, blind exploration is a no-no, the exploration should be done with a clear goal (like exploration to find the other side of the island, which might be inhabited by tribes, after seeing some evidence that there might be other human civilization nearby other than the group).

 * Well, technically, this is about rebuilding civilization. I'm not nit-picking, because I don't care all that much, but a golf course doesn't seem all that productive either. As for long trips, I can see your point. Would short, day-long trips be acceptable, for the purpose of exploration or finding new sources of food/water/supplies? -DavesTA 14:14, 6 April 2007 (UTC)

Additions
Wow, more text here than in the book... I took the liberty of adding some stuff I know from other survival works such as the SAS survival manual concerning water and food and some stuff I know about government because I am a freaking genius.

I agree with jjb although it should suffice with the all important technologies such as science, fire, basic tools, weapons, powder, metalurgi, paper, printing press, electricity, steam engines, windmills, ships, sails/wings, farming, animal/plant breeding, a simple and very basic computer if it gets really advanced maybe. Just enough to kick things off, though just instilling the scientific principle would go a long way by itself though.

-Realpra

Err details anybody?
Jeez, did anyone think to oh i dont know... put details on how to do stuff... i mean you say "light a fire" and how does one do that... what you have put so far (24/6/10) is great but your talking way below the level to skip on survival details (by this i mean, your talking about starting a fire and securing water/food supplies... both valid points, but if you have to tell the reader that thats important then odds are they wont know how to start a fire... civilization starts at the most basic with governance (not needed at survival stage) farming (not needed at survival stage) defence (possibly needed at survival stage, but instinct will provide the level of this for survival)) you havent got a secure food or water supply, but lets get the internet going? no... your book reads like a survival guide... civilization should start way beyond that...

so, basically, im gonna re-write the whole book, and when someone else comes along, they are more than welcome to restore the original if they think its better The seraphim (talk) 17:46, 24 June 2010 (UTC)

ok i just became too busy to do this now, so i reverted the changes i made, someone else do it... lol im not ur mom The seraphim (talk) 18:05, 24 June 2010 (UTC)

title: Talk:When It Hits the Fan/Rebuilding


text:
User talk:Traci506

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title: User talk:Traci506


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User talk:Camacho4940~enwikibooks

Hello and welcome to Wikibooks!

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You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck!--Sundance Raphael 13:43, 11 January 2007 (UTC)

Your account will be renamed
Hello,

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

Unfortunately, your account clashes with another account also called Camacho4940. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Camacho4940~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:01, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:45, 19 April 2015 (UTC)

title: User talk:Camacho4940~enwikibooks


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User talk:69.7.4.10

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Herby talk thyme 13:47, 11 January 2007 (UTC)

title: User talk:69.7.4.10


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User talk:136.182.158.129

Please do not add inappropriate external links to Wikibooks. Wikibooks is intended to be read without the need for external reference and requires content rather than external links particularly those which are for advertising or promotion. Inappropriate links may include (but are not limited to) links to personal web sites, links to web sites with which you are affiliated, and links that exist to attract visitors to a web site or promote a product. If you feel the link should be added to the article, then please discuss it on the article's talk page rather than re-adding it. See the welcome page to learn more about Wikibooks. Thank you. -- Herby talk thyme 14:00, 11 January 2007 (UTC)

title: User talk:136.182.158.129


text:
User talk:62.171.194.41

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 14:37, 11 January 2007 (UTC)

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Herby talk thyme 14:03, 24 May 2007 (UTC)

title: User talk:62.171.194.41


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User talk:71.52.42.67

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 14:41, 11 January 2007 (UTC)

title: User talk:71.52.42.67


text:
User talk:Ctisd001

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
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You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be an idea if you add this page to your "watchlist" so that you can see when any new information is posted there.

Good luck!--Sundance Raphael 15:16, 11 January 2007 (UTC)

title: User talk:Ctisd001


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User talk:38.116.192.91

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 16:42, 11 January 2007 (UTC)

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User talk:172.197.103.139

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 16:44, 11 January 2007 (UTC)

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User talk:209.80.150.218

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 18:59, 11 January 2007 (UTC)

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text:
Taxation in the United Kingdom/Legislation/Section 35 of the Income and Corporation Taxes Act 1988

Subsection (1) says that this section applies to any lease of duration not exceeding 50 years where the grantor of a lease could have, based on arm's length rates at the time the lease was granted, required the tenant to pay additional amounts by way of premium. Such amounts are referred to as the amount foregone.

Subsection (2) provides that where this section applies to a lease, and the lease is later assigned for a consideration greater than any premiums for the grant of the lease (if it has not previously been assigned), or for a consideration greater than any premium for the last assignment (otherwise), the lower of the amount of the excess and the amount foregone less excesses on previous assignments, is treated as additional rent by the assignor.

Subsection (2A) says that such amounts are deemed to be received when the consideration for the assignment becomes payable and is taken into account when determining rental income in the accounting period in which it is received.

Subsection (3) allows the grantor, assignor or assignee of a lease to make a statement to the Inspector of Taxes of what corporation tax, if any, is due under this section. If the Inspector is satisfied that the statement is accurate, he shall certify it as accurate.


text:
User talk:72.95.166.2

Xania talk 21:29, 11 January 2007 (UTC)

title: User talk:72.95.166.2


text:
Super NES Programming/Pointers

Required Tools

 * The WLA-65816 Macro Assembler
 * A SNES Emulator
 * An initialization routine and a header.

Overview
Pointers are an important part of a game or ROM. In the Super Nintendo, these pointers are usually made up of various addresses organized in a specific order. Some examples of pointer tables may include, for example, the list of locations in a ROM of a game's levels, or pointers to where certain sections of code are. A pointer could be related to a file name; C:\WINDOWS\explorer.exe, for example, points to the file explorer.exe in the folder WINDOWS.

Example
Say you have a game with three levels. You've inserted them into your ROM, but you need your game to be able to read the levels. This could be solved by using pointers to the three levels in a table, somewhat like this:

This table would now contain the addresses in the ROM of level 1, level 2, and level 3. The game could then read from this table and know where to look for those level files.

db/dw/dl
.dl, .dw, and .db are three assembler directives for WLA-65816. This means that they tell the assembler to do something, in this case they tell it to create a byte, two bytes, or three bytes that together make up an address. .db creates a one-byte number, .dw a two-byte number, and .dl a three-byte number. Note that although they are used for pointers in this tutorial, these three commands can also be used to hold absolute values.

Accessing the table
In 65816 ASM, tables are relatively easy to access. If you are already familiar with A and X in the 65c816, then this next section should not be a problem for you.

This code will take X, and then read from (Table + X) to get a value, and then store that to the Accumulator (A). Note that X would need to be doubled in order to read the correct entry. Having X be one would read from the second byte of the first entry and the first byte of the second entry, which is not the intended operation of the code.

title: Super NES Programming/Pointers


text:
Horticulture/Mealybugs

Mealybug is the common name of insects in Pseudococcidae, a family of unarmored ../Scale Insects/ found in moist, warm climates. They feed on plant juices of greenhouse plants, house plants and some outdoor plants.

Description
Mealybugs are sexually dimorphic, meaning that the sexes have distinct morphological differences. Females are nymphal, exhibit reduced morphology, and are wingless. The females do not change completely and are likely to be neotonic (exhibiting nymphal characteristics). Males are winged and do change completely during their lives. Since mealybugs (as well as all other Hemiptera) are hemimetabolous insects, they do not undergo complete metamorphosis in the true sense of the word. I.e. there are no clear larval, pupal and adult stages, and the wings do not develop internally. However, male mealybugs do exhibit a radical change during their life cycle, changing from wingless, ovoid nymphs to "wasp-like" flying adults.

Symptoms and Signs
Large infestations may induce leaf drop.

Ecology
Mealybug females feed on plant sap, normally in roots or other crevices. They attach themselves to the plant and secrete a powdery wax layer (therefore the name mealybug) used for protection while they suck the plant juices. The males on the other hand, are short-lived as they do not feed at all as adults and only live to fertilize the females. Male citrus mealy bugs fly to the females and resemble fluffy gnats.

Some species of mealybug lay their eggs in the same waxy layer used for protection in quantities of 50–100; other species are born directly from the female.

The most serious pests are mealybugs that feed on citrus; other species damage sugarcane, grapes, pineapple (Jahn et al. 2003), coffee trees, cassava, ferns, and orchids. Mealybugs only tend to be serious pests in the presences of ants because the ants protect them from predators and parasites. Mealybugs also infest some species of carnivorous plant such as ../Sarracenia/ (pitcher plants), in such cases it is difficult to eradicate them without repeated applications of insecticide such as ../Diazinon/. .

title: Horticulture/Mealybugs


text:
Horticulture/Hemiptera

Hemiptera is a large, cosmopolitan order of insects, comprising some 67,500 known species in three suborders. Traditionally these taxa were treated as two separate orders, Homoptera (Auchenorrhyncha and Sternorrhyncha) and Heteroptera; the former name is now obsolete (the group was paraphyletic), and the latter name is falling into disuse, often replaced by Prosorrhyncha. Referring latest investigations the name Auchenorrhyncha should also be replaced by Archeorrhyncha and Clyeporrhyncha. Members of the "Heteroptera" are typically called true bugs, and this name sometimes is used for members of "Hemiptera" as a whole. The name heteroptera comes from their forewings having both membranous and hard portions. It is also essentially this same feature which gives the order its name, hemiptera, coming from the Greek for half-wing. Members of the Hemiptera are distinguished from all other insects by both adults and nymphs having a proboscis that includes a salivary channel as well as a food channel. The proboscis is usually specialized to suck the juices from various parts of plants, including seeds, although some species are predatory (on arthropods and sometimes small animals), and a few are adapted to suck blood from mammals. In addition, the space between the overlapping wings in the members of this order forms a triangular shape near the head which is commonly used to identify a true bug.

Suborder Prosorrhyncha includes 25,000 known species in over 60 families.

Suborder Archaeorrhyncha:Some authors prefer this name to refer to the Fulgoromorpha, or Planthoppers.

Suborder [Clypeorrhyncha is sometimes used to refer to the Cicadomorpha.

Suborder Sternorrhyncha contains 12,500 species.

title: Horticulture/Hemiptera


text:
User talk:The enemies of god

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

Xania talk 22:06, 11 January 2007 (UTC)

title: User talk:The enemies of god


text:
User talk:67.86.7.255

Welcome!
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title: User talk:67.86.7.255


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User talk:76.19.244.225

Welcome!
Welcome to Wikibooks, a collection of open-content textbooks! You don't have to log in to read or edit articles on Wikibooks, but creating an account is quick, free and non-intrusive, requires no personal information, and gives you many benefits, including:
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title: User talk:76.19.244.225


text:
Advanced Mathematics for Engineers and Scientists/Parallel Plate Flow: Realistic IC

The Steady State
The initial velocity profile chosen in the last problem agreed with intuition but honestly came out of thin air. A more realistic development follows.

The problem stated that (to come up with an IC) the fluid was under a pressure difference for some time, so that the flow became steady aka flowing steadily. "Steady" is another way of saying "not changing with time", and "not changing with time" is another way of saying that:

\frac{\partial u}{\partial t} = 0\, $$

Putting this into the PDE from the previous section:

\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial y^2} - \frac{P_x}{\rho}\, $$

 * $$\Big\Downarrow$$

0 = \nu \frac{\partial^2 u}{\partial y^2} - \frac{P_x}{\rho} \quad \Rightarrow \quad \frac{P_x}{\nu \rho} = \frac{d^2 u}{d y^2}\, $$

Independent of $$t$$, the PDE became an ODE with variables separated and thus we can integrate.

\frac{d^2 u}{d y^2} = \frac{P_x}{\nu \rho} \quad \Rightarrow \quad u = \frac{P_x}{2 \nu \rho}y^2 + C_1 y + C_0\, $$

The no slip condition results in the following BCs: $$u = 0$$ at $$y = 0$$ and $$y = 1$$. We can plug the BC values into the integrated ODE and resolve the Cs.

0 = \frac{P_x}{2 \nu \rho} \cdot 0^2 + C_1 \cdot 0 + C_0 \quad \Rightarrow \quad C_0 = 0 \qquad \mbox{(Bottom plate BC: u = 0, y = 0)}\, $$

0 = \frac{P_x}{2 \nu \rho} \cdot 1^2 + C_1 \cdot 1 \quad \Rightarrow \quad C_1 = -\frac{P_x}{2 \nu \rho} \qquad \mbox{(Top plate BC: u = 0, y = 1)}\, $$

Inserting the Cs and and simplifying yields:

u = \frac{P_x}{2 \nu \rho}(y^2 - y)\, $$

For the sake of example, take $$P_x/(2 \nu \rho) = -4$$ (recall that a negative pressure gradient causes left to right flow). Also note that this is a constant gradient or slope. This gives a parabola which starts at $$0$$, increases to a maximum of $$1$$ at $$y = 1/2$$, and returns to $$0$$ at $$y = 1$$.

This parabola looks pretty much identical to the sinusoid previously used (you must zoom in to see a difference). However, even more so on the narrow domain of interest, the two are very different functions (look at their taylor expansions, for example). Using the parabola instead of the sine function results in a much more involved solution.

So this derives the steady state flow, which we will use as an improved, realistic IC. Recall that the problem is about a fluid that's initially in motion that is coming to a stop due to the absence of a driving force. The IBVP (Initial Boundary Value Problem) is now subtly different:

\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial y^2} \qquad \mbox{(PDE)} \, $$

u(y, 0) = 4 y - 4 y^2 \qquad \mbox{(IC)} \, $$

u(0, t) = 0\, $$

u(1, t) = 0 \qquad \mbox{(BCs)} \, $$

Separation
Since the only difference from the problem in the last section is the IC, the variables may be separated and the BCs applied with no difference, giving:

u(y, t) = e^{-(n \pi)^2 \nu t} B \sin(n \pi y)\, $$

But now we're stuck (after applying the BCs)! Applying the IC makes the $$e^{-(n \pi)^2 \nu t}$$ term go away as t = 0, which is the IC. However, then the IC function can't be made to match:

u(y, 0) = 4 y - 4 y^2\, $$

B \sin(n \pi y) = 4 y - 4 y^2\, $$

What went wrong? It was the assumption that $$u(y, t) = Y(y)T(t)$$. The fact that the IC couldn't be fulfilled means that the assumption was wrong. It should be apparent now why the IC was chosen to be $$\sin(\pi y)$$ in the previous section.

We can proceed however, thanks to the linearity of the problem. Another detour is necessary, it gets long.

Linearity (the superposition principle specifically) says that if $$u_1$$ is a solution to the BVP (not the whole IBVP, only the BVP, Boundary Value Problem, the BCs applied) and so is another $$u_2$$, then a linear combination, $$C_1 u_1 + C_2 u_2$$, is also a solution.

Let's take a step back and suppose that the IC was

u(y, 0) = \sin(\pi y) + 1/5 \sin(3 \pi y). $$

This is no longer a realistic flow problem but it contains the first two terms of what is called a Fourier sine expansion, see these examples of Fourier sine expansions. We are going to generalize this below. Let's now use this expression and equate it to the half way solution (BCs applied) with $$e^{-(n \pi)^2 \nu t}$$ being eliminated as t = 0:

B \sin(n \pi y) = \sin(\pi y) + \frac{1}{5} \sin(3 \pi y)\, $$

And it still can't match. However, observe that the individual terms in the IC can. We simply set the constants to values making both sides match:

B_1 \sin(n_1 \pi y) = \sin(\pi y) \Rightarrow n_1 = 1 \ \mbox{and} \ B_1 = 1\, $$

B_3 \sin(n_3 \pi y) = \frac{1}{5} \sin(3 \pi y) \Rightarrow n_3 = 3 \ \mbox{and} \ B_3 = \frac{1}{5}\, $$

Note the subscripts are used to identify each term: they reflect the integer $$n$$ from the separation constant. Solutions may be obtained for each individual term of the IC, identified with $$n$$:

u_1(y, t) = e^{-(1 \pi)^2 \nu t} 1 \sin(1 \pi y) = e^{-\pi^2 \nu t} \sin(\pi y) $$       $$n = 1 \ \mbox{and} \ B = 1\, $$

u_3(y, t) = e^{-(3 \pi)^2 \nu t} \frac{1}{5} \sin(3 \pi y) = e^{-9 \pi^2 \nu t} \frac{1}{5} \sin(3 \pi y)\,$$       $$n = 3 \ \mbox{and} \ B = \frac{1}{5}\, $$

Linearity states that the sum of these two solutions is also a solution to the BVP (no need for new constants):

u_{1 + 3}(y, t) = e^{-\pi^2 \nu t} \sin(\pi y) + \frac{1}{5} e^{-9 \pi^2 \nu t} \sin(3 \pi y)\, $$

So we added the solutions and got a new solution... what is this good for? Try setting $$t = 0$$:

u_{1 + 3}(y, 0) = e^{-\pi^2 \nu \cdot 0} \sin(\pi y) + \frac{1}{5} e^{-9 \pi^2 \nu \cdot 0} \sin(3 \pi y) = \sin(\pi y) + \frac{1}{5} \sin(3 \pi y)\, $$

Each component solution satisfies the BVP, and the sum of these just happened to satisfy our surrogate IC. The IBVP with IC $$u(y, 0) = \sin(\pi y) + 1/5 \sin(3 \pi y)$$ is now solved. It would work the same way for any linear combination of sine functions whose half frequencies are $$n \pi$$. "Linear combination" means a sum of terms, each multiplied by a constant. The sum is assumed to converge and be term by term differentiable.

Let's do what we just did in a more generalized fashion. First, we make our IC a linear combination of sines (with $$e^{-(n \pi)^2 \nu t}$$ eliminated as t = 0), in fact, infinitely many of them. But each successive term has to 'converge', it can't stray wildly all over the place.

u(y, 0) = \sum_{n=1}^{\infty} B_n \sin(n \pi y) \qquad \mbox{(IC: an arbitrary linear combination of sines)} \, $$

Second, find the n and B for each term assuming t = 0 (the IC), then plug them back into each term making no assumptions about t, leaving t as is.

u_n(y, t) = e^{-(n \pi)^2 \nu t} B_n \sin(n \pi y) \qquad \mbox{(solution of} \ n^{th} \ \mbox{term)} \, $$

Third, sum up all the terms with their individual n and Bs.

u(y, t) = \sum_{n=1}^{\infty} u_n(y, t) = \sum_{n=1}^{\infty} e^{-(n \pi)^2 \nu t} B_n \sin(n \pi y) \qquad \mbox{(sum of solutions)} \, $$

Fourth, plug t = 0 into the sum of terms and recover the IC from the first step.

u(y, 0) = \sum_{n=1}^{\infty} e^{-(n \pi)^2 \nu \cdot 0} B_n \sin(n \pi y) = \sum_{n=1}^{\infty} B_n \sin(n \pi y) \qquad \mbox{(IC recovered)} \, $$

So we went full circle on this example but found the n and Bs because we were able to equate/satisfy each term with the IC. Now we can solve the problem if the IC is a linear combination of sine functions. But the IC for this problem isn't such a sum, it's just a stupid parabola. Or is it?

Series Construction
In the 19th century, a man named Joseph Fourier took a break from helping Napoleon take over the world to ask an important question while studying this same BVP (concerning heat flow): can a function be expressed as a sum of sinusoids, similar to a taylor series? The short answer is yes, if a few reasonable conditions apply as we have already indicated. The long answer follows, and this section is a longer answer.

A function meeting certain criteria may indeed be expanded into a sum of sines, cosines, or both. In our case, all that is needed to accomplish this expansion is to find the coefficients $$B_n$$. A little trick involving an integral makes this possible.

The sine function has a very important property called orthogonality. There are many flavors of this, which will be served in the next chapter. Relevant to this problem is the following:

\int_{0}^{1} 2 \sin(m \pi y) \sin(n \pi y)\, dy = \begin{cases} 1, & m = n \\ 0, & m \ne n \end{cases} $$

A quick hint may help. Orthogonality literally means two lines at a right angle to each other. These lines could be vectors, each with its own tuple of coordinates. If those two vectors are at a right angle to each other, multiplying and summing their coordinate tuples always yields zero (in Euclidean space). The method of multiplying and summing is also used to determine whether two functions are orthogonal. Using this definition, our multiplied and integrated functions above are orthogonal most of the time, but not always.

Let's call the IC $$\phi(y)$$ to generalize it. We equate the IC with its expansion, meaning the linear combination of sines, and then apply some craftiness. And remember that our goal is to reproduce a parabolic function from linearly combined sines:

\sum_{n=1}^{\infty} B_n \sin(n \pi y) = \phi(y)\, $$

2 \sin(m \pi y) \cdot \sum_{n=1}^{\infty} B_n \sin(n \pi y) = 2 \sin(m \pi y) \cdot \phi(y)\, $$

\sum_{n=1}^{\infty} B_n \cdot 2 \sin(m \pi y) \sin(n \pi y) = 2 \sin(m \pi y) \phi(y)\, $$

\int_{0}^{1} \sum_{n=1}^{\infty} B_n \cdot 2 \sin(m \pi y) \sin(n \pi y) dy = \int_{0}^{1} 2 \sin(m \pi y) \phi(y)\ dy\, $$

\sum_{n=1}^{\infty} B_n \int_{0}^{1} 2 \sin(m \pi y) \sin(n \pi y) dy = \int_{0}^{1} 2 \sin(m \pi y) \phi(y)\ dy\, $$

B_m = \int_{0}^{1} 2 \sin(m \pi y) \phi(y)\ dy\, $$

In the last step, all of the terms in the sum became $$0$$ except for the $$m^{th}$$ term where $$m = n$$, the only case where we get $$1$$ for the otherwise orthogonal sine functions. This isolates and explicitly defines $$B_m$$ which is the same as $$B_n$$ as m = n. The expansion for $$\phi(y)$$ is then:

\phi(y) = \sum_{m=1}^{\infty} ( \int_{0}^{1} 2 \sin(m \pi y) \phi(y)\ dy ) \cdot \sin(m \pi y)\, $$

Or equivalently:

\phi(y) = \sum_{m=1}^{\infty} B_m \sin(m \pi y) \qquad ; \ B_m = \int_{0}^{1} 2 \sin(m \pi y) \phi(y)\ dy\, $$

Many important details have been left out for later in a devoted chapter; one noteworthy detail is that this expansion is only approximating the parabola (very superficially) on the interval $$0 \leq y \leq 1$$, not say from $$-\infty$$ to $$+\infty$$.

This expansion may finally be combined with the sum of sines solution to the BVP developed previously. Note that the last equation looks very similar to $$u(y, 0)$$. Following from this:

u(y, 0) = \sum_{n=1}^{\infty} B_n \sin(n \pi y)\, $$

u(y, 0) = \sum_{n=1}^{\infty} \int_{0}^{1} 2 \sin(n \pi y) \phi(y)\ dy \cdot \sin(n \pi y) \qquad \, $$

So the expansion will satisfy the IC given as $$\phi(y)$$ (surprised?). The full solution for the problem with arbitrary IC is then:

u(y, t) = \sum_{n=1}^{\infty} e^{-(n \pi)^2 \nu t} B_n \sin(n \pi y)\, $$

u(y, t) = \sum_{n=1}^{\infty} e^{-(n \pi)^2 \nu t} \int_{0}^{1} 2 \sin(n \pi y) \phi(y)\ dy \cdot \sin(n \pi y)\, $$

In this problem specifically, the IC is $$\phi(y) = 4 y - 4 y^2$$, so:

B_n = \int_{0}^{1} 2 \sin(n \pi y) (4 y - 4 y^2) dy = \frac{8}{n^3 \pi^3} (2 - 2 \cos(n \pi) - n \pi \sin(n \pi))\, $$

Sines and cosines appear from the integration dependent only on $$n \pi$$. Since $$n$$ is an integer, these can be made more aesthetic.

\sin(n \pi) = 0 \qquad ; \ n \mbox{ is an integer.}\, $$

\cos(n \pi) = (-1)^n \qquad ; \ n \mbox{ is an integer.}\, $$

 * $$\qquad \Big\Downarrow$$

B_n = \frac{16 - 16 (-1)^n}{n^3 \pi^3}\, $$

Note that for even $$n$$, $$B_n = 0$$. Putting everything together finally completes the solution to the IBVP:

u(y, t) = \sum_{n=1}^{\infty} e^{-(n \pi)^2 \nu t} \frac{16 - 16 (-1)^n}{n^3 \pi^3} \sin(n \pi y)\, $$

There are many interesting things to observe. To begin with, $$u(y, t)$$ is not a product of a function of $$y$$ and a function of $$t$$. Such a solution was assumed in the beginning, proved to be wrong, but eventually happened to yield a solution anyway thanks to linearity and what is called a Fourier sine expansion.

A careful look at the procedure reveals something that may be disturbing: this lengthy solution is strictly valid for the given BCs. Thanks to the definition of $$\phi(y)$$, the solution is generic as far as the IC is concerned (the IC doesn't even need to match the BCs), however a slight change in either BC would mandate starting over almost from the beginning.

The parabolic IC, which looks very similar to the sine function used in the previous section, is wholly to blame (or thank once you understand the beauty of a Fourier series!) for the infinite sum. It is interesting to approximate the first several numeric values of the sequence $$B_n$$:

\begin{align} B_1 &\approx 1.03205\\ B_3 &\approx 0.03822\\ B_5 &\approx 0.00826\\ B_7 &\approx 0.00301\, \end{align} $$

Recall that the even terms are all $$0$$. The first term by far dominates, this makes sense since the first term already looks very, very similar to the parabola. Recall that $$n^2$$ appears in an exponential, making the higher terms even smaller for time not too close to $$0$$.


text:
Advanced Mathematics for Engineers and Scientists/Details and Applications of Fourier Series

Details and Applications of Fourier Series
In the study of PDEs (and much elsewhere), a Fourier series (or more generally, trigonometric expansion) often needs to be constructed.

Preliminaries
Suppose that a function f(x) may be expressed in the following way:

 * $$f(x) = \frac{A_0}{2} + \sum_{n = 1}^{\infty}\left(A_n \cos\left(\frac{n \pi}{L} x\right) + B_n \sin\left(\frac{n \pi}{L} x\right)\right)$$

It can be shown (not too difficult, but beyond this text) that the above expansion will converge to f(x), except at discontinuities, if the following conditions hold:

 * f(x) = f(x + 2L), i.e.  f(x) has period 2L .
 * f(x), f ' (x), and f  (x) are piecewise continuous on the interval -L ≤ x ≤ L''.
 * The pieces that make up f(x), f ' (x), and f  (x'') are continuous over closed subintervals.

The first requirement is most significant; the last two requirements can, to an extent, be partly eased off in most cases without any trouble. An interesting thing happens at discontinuities. Suppose that f(x) is discontinuous at x = a; the expansion will converge to the following value:

 * $$\frac{1}{2}\left(\lim_{x \rightarrow a^-} f(x) + \lim_{x \rightarrow a^+} f(x)\right) \,$$

So the expansion converges to the average of the values to the left and the right of the discontinuity. This, and the fact that it converges in the first place, is very convenient. The Fourier series looks unfriendly but it's honestly working for you.

The information needed to express f(x) as a Fourier series are the sequences An and Bn. This is done using orthogonality, which for the sinusoids may be derived easily using a few identities. The following are some useful orthogonality relations, with m and n restricted to integers:

 * $$\int_{0}^{L} \frac{2}{L} \sin\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right) dx = \delta_{m,n}$$

 * $$\int_{0}^{L} \frac{2}{L} \cos\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) dx = \delta_{m,n}$$

 * $$\int_{-L}^{L} \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right) dx = \delta_{m,n}$$

 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) dx = \delta_{m,n}$$

 * $$\int_{-L}^{L} \sin\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) dx = 0$$

δm,n is called the Kronecker delta, defined by:

\delta_{m,n} = \begin{cases} 1, & m = n \\ 0, & m \ne n \end{cases}$$

The Kronecker delta may be thought of as a discrete version of the Dirac delta "function". Relevant to this topic is its sifting property:

 * $$\sum_{n = -\infty}^{\infty} A_n \delta_{m,n} = A_m\,$$

Derivation of the Fourier Series
We're now ready to find An and Bn.

 * $$f(x) = \frac{A_0}{2} + \sum_{n = 1}^{\infty}\left(A_n \cos\left(\frac{n \pi}{L} x\right) + B_n \sin\left(\frac{n \pi}{L} x\right)\right)$$

 * $$\frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \cdot f(x) =

\frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \cdot \left(\frac{A_0}{2} + \sum_{n = 1}^{\infty}\left(A_n \cos\left(\frac{n \pi}{L} x\right) B_n \sin\left(\frac{n \pi}{L} x\right)\right)\right)\,$$

\sum_{n = 1}^{\infty}\left( A_n \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) + B_n \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right) \right) $$
 * $$\frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) f(x) =$$
 * $$\frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \frac{A_0}{2} +$$
 * $$\frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \frac{A_0}{2} +$$
 * }

\sum_{n = 1}^{\infty}\left(A_n \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) + \frac{B_n}{L} \cos\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right)\right) dx $$
 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) f(x) dx$$
 * $$= \int_{-L}^{L} \frac{A_0}{2 L} \cos\left(\frac{m \pi}{L} x\right) +$$
 * $$= \int_{-L}^{L} \frac{A_0}{2 L} \cos\left(\frac{m \pi}{L} x\right) +$$
 * }

\sum_{n = 1}^{\infty}\left(A_n \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) dx + \frac{B_n}{L} \int_{-L}^{L} \cos\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right) dx \right) $$
 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) f(x) dx$$
 * $$= \int_{-L}^{L} \frac{A_0}{2 L} \cos\left(\frac{m \pi}{L} x\right) dx +$$
 * $$= \int_{-L}^{L} \frac{A_0}{2 L} \cos\left(\frac{m \pi}{L} x\right) dx +$$
 * }

 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) f(x) dx =

\int_{-L}^{L} \frac{A_0}{2 L} \cos\left(\frac{m \pi}{L} x\right) dx + \sum_{n = 1}^{\infty}\left(A_n \delta_{m,n} + \frac{B_n}{L} \cdot 0 \right)$$

This is supposed to hold for an arbitrary integer m. If m = 0, note that the sum doesn't allow n = 0 and so the sum would be zero since in no case does m = n. This leads to:

 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{0 \cdot \pi}{L} x\right) f(x) dx =

\int_{-L}^{L} \frac{A_0}{2 L} \cos\left(\frac{0 \cdot \pi}{L} x\right) dx$$

 * $$\int_{-L}^{L} \frac{1}{L} f(x) dx = A_0 \int_{-L}^{L} \frac{dx}{2 L}$$

 * $$\int_{-L}^{L} \frac{1}{L} f(x) dx = A_0$$

This secures A0. Now suppose that m > 0. Since m and n are now in the same domain, the Kronecker delta will do its sifting:

 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) f(x) dx =

\int_{-L}^{L} \frac{A_0}{2 L} \cos\left(\frac{m \pi}{L} x\right) dx + \sum_{n = 1}^{\infty}\left(A_n \delta_{m,n}\right)$$

 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) f(x) dx =

\frac{A_0 \sin(m \pi)}{m \pi} + A_m$$

 * $$\int_{-L}^{L} \frac{1}{L} \cos\left(\frac{m \pi}{L} x\right) f(x) dx = 0 + A_m$$

 * $$A_n = \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{n \pi}{L} x\right) f(x) dx$$

In the second to the last step, sin(mπ) = 0 for integer m. In the last step, m was replaced with n. This defines An for n > 0. For the case n = 0,

 * $$A_0 = \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{0 \cdot \pi}{L} x\right) f(x) dx$$

 * $$A_0 = \int_{-L}^{L} \frac{1}{L} f(x) dx$$

Which happens to match the previous development (now you know why it's A0/2 and not just A0). So the sequence An is now completely defined for any value of n of interest:

 * $$A_n = \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{n \pi}{L} x\right) f(x) dx$$

To get Bn, nearly the same routine is used.

 * $$f(x) = \frac{A_0}{2} + \sum_{n = 1}^{\infty}\left(A_n \cos\left(\frac{n \pi}{L} x\right) + B_n \sin\left(\frac{n \pi}{L} x\right)\right)$$

 * $$\frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \cdot f(x) =

\frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \cdot \left(\frac{A_0}{2} + \sum_{n = 1}^{\infty}\left(A_n \cos\left(\frac{n \pi}{L} x\right) B_n \sin\left(\frac{n \pi}{L} x\right)\right)\right)\,$$

\sum_{n = 1}^{\infty}\left( A_n \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) + B_n \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right) \right) $$
 * $$\frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) f(x) =$$
 * $$\frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \frac{A_0}{2} +$$
 * $$\frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \frac{A_0}{2} +$$
 * }

\sum_{n = 1}^{\infty}\left(\frac{A_n}{L} \sin\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) + B_n \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right)\right) dx $$
 * $$\int_{-L}^{L} \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) f(x) dx$$
 * $$= \int_{-L}^{L} \frac{A_0}{2 L} \sin\left(\frac{m \pi}{L} x\right) +$$
 * $$= \int_{-L}^{L} \frac{A_0}{2 L} \sin\left(\frac{m \pi}{L} x\right) +$$
 * }

\sum_{n = 1}^{\infty}\left(\frac{A_n}{L} \int_{-L}^{L} \sin\left(\frac{m \pi}{L} x\right) \cos\left(\frac{n \pi}{L} x\right) dx + \frac{B_n}{L} \int_{-L}^{L} \sin\left(\frac{m \pi}{L} x\right) \sin\left(\frac{n \pi}{L} x\right) dx \right) $$
 * $$\int_{-L}^{L} \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) f(x) dx$$
 * $$= \frac{A_0}{2 L} \int_{-L}^{L} \sin\left(\frac{m \pi}{L} x\right) dx +$$
 * $$= \frac{A_0}{2 L} \int_{-L}^{L} \sin\left(\frac{m \pi}{L} x\right) dx +$$
 * }

 * $$\int_{-L}^{L} \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) f(x) dx = \frac{A_0}{2 L} \cdot 0 +

\sum_{n = 1}^{\infty}\left(\frac{A_n}{L} \cdot 0 + B_n \delta_{m,n} \right)$$

 * $$\int_{-L}^{L} \frac{1}{L} \sin\left(\frac{m \pi}{L} x\right) f(x) dx = B_m$$

 * $$B_n = \int_{-L}^{L} \frac{1}{L} \sin\left(\frac{n \pi}{L} x\right) f(x) dx$$

The Fourier series expansion of f(x) is now complete. To have it all in one place:

 * $$f(x) = \frac{A_0}{2} + \sum_{n = 1}^{\infty}\left(A_n \cos\left(\frac{n \pi}{L} x\right) + B_n \sin\left(\frac{n \pi}{L} x\right)\right)$$

 * $$A_n = \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{n \pi}{L} x\right) f(x) dx$$

 * $$B_n = \int_{-L}^{L} \frac{1}{L} \sin\left(\frac{n \pi}{L} x\right) f(x) dx$$

It's finally time for an example. Let's derive the Fourier series representation of a square wave, pictured at the right:

This "wave" may be quantified as f(x):

f(x) = \begin{cases} 1, & -1 < x < 0 \\ -1, & 0 < x < 1 \\ f(x + 2) \end{cases}$$

f(x) has period 2. Since 2L = P, L = 1. Now, finding the Fourier coefficients:

 * $$A_n\,$$
 * $$= \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{n \pi}{L} x\right) f(x) dx$$
 * $$= \int_{-1}^{1} \cos(n \pi x) f(x) dx$$
 * $$= \int_{-1}^{0} \cos(n \pi x) (-1) dx + \int_{0}^{1} \cos(n \pi x) (1) dx$$
 * $$= -\frac{\sin(n \pi)}{n \pi} + \frac{\sin(n \pi)}{n \pi}$$
 * $$= 0\,$$
 * }
 * $$= \int_{-1}^{0} \cos(n \pi x) (-1) dx + \int_{0}^{1} \cos(n \pi x) (1) dx$$
 * $$= -\frac{\sin(n \pi)}{n \pi} + \frac{\sin(n \pi)}{n \pi}$$
 * $$= 0\,$$
 * }
 * $$= -\frac{\sin(n \pi)}{n \pi} + \frac{\sin(n \pi)}{n \pi}$$
 * $$= 0\,$$
 * }
 * $$= 0\,$$
 * }

 * $$B_n\,$$
 * $$= \int_{-L}^{L} \frac{1}{L} \sin\left(\frac{n \pi}{L} x\right) f(x) dx$$
 * $$= \int_{-1}^{1} \sin(n \pi x) f(x) dx$$
 * $$= \int_{-1}^{0} \sin(n \pi x) (-1) dx + \int_{0}^{1} \sin(n \pi x) (1) dx$$
 * $$= \left(\frac{1}{n \pi} - \frac{\cos(n \pi)}{n \pi}\right) + \left(\frac{1}{n \pi} - \frac{\cos(n \pi)}{n \pi}\right)$$
 * $$= \frac{2}{n \pi}\left(1 - \cos(n \pi)\right)\,$$
 * $$= \frac{2}{n \pi}\left(1 - (-1)^n\right)\,$$
 * }
 * $$= \int_{-1}^{0} \sin(n \pi x) (-1) dx + \int_{0}^{1} \sin(n \pi x) (1) dx$$
 * $$= \left(\frac{1}{n \pi} - \frac{\cos(n \pi)}{n \pi}\right) + \left(\frac{1}{n \pi} - \frac{\cos(n \pi)}{n \pi}\right)$$
 * $$= \frac{2}{n \pi}\left(1 - \cos(n \pi)\right)\,$$
 * $$= \frac{2}{n \pi}\left(1 - (-1)^n\right)\,$$
 * }
 * $$= \frac{2}{n \pi}\left(1 - \cos(n \pi)\right)\,$$
 * $$= \frac{2}{n \pi}\left(1 - (-1)^n\right)\,$$
 * }
 * $$= \frac{2}{n \pi}\left(1 - (-1)^n\right)\,$$
 * }
 * }

 * $$f(x) = \sum_{n = 1}^{\infty} B_n \sin(n \pi x)$$

 * $$f(x) = \sum_{n = 1}^{\infty} \frac{2}{n \pi}\left(1 - (-1)^n\right) \sin(n \pi x)$$

 * $$f(x) = \sum_{n = 1}^{\infty} \frac{4 \sin((2n - 1) \pi x)}{(2n - 1) \pi}$$

In the last bit we used the fact that all of the even terms happened to be absent, and the odd numbers are given by 2n - 1 for integer n. The sum will indeed converge to the square wave, except at the discontinuities where it'll converge to zero (the average of 1 and -1).

Graphs of partial sums are shown at right. Note that this particular expansion doesn't converge too quickly, and that as an approximation of the square wave it's poorest near the discontinuities.

There's another interesting thing to note: all of the cosine terms are absent. It's no coincidence, and this may be a good time to introduce the Fourier sine and cosine expansions for, respectively, odd and even functions.

Periodic Extension and Expansions for Even and Odd functions
Two important expansions may be derived from the Fourier expansion: the Fourier sine series and the Fourier cosine series, the first one was used in the previous section. Before diving in, we must talk about even and odd functions.

Suppose that feven(x) is an even function and fodd(x) is an odd function. That is:

 * $$f_\text{even}(-x) = f_\text{even}(x)\,$$

 * $$f_\text{odd}(-x) = -f_\text{odd}(x)\,$$

Some interesting identities hold for such functions. Relevant ones include:

 * $$f_\text{even}(x) \cdot f_\text{even}(x) = \mbox{an even function}\,$$

 * $$f_\text{odd}(x) \cdot f_\text{odd}(x) = \mbox{an even function}\,$$

 * $$f_\text{even}(x) \cdot f_\text{odd}(x) = \mbox{an odd function}\,$$

 * $$\int_{-a}^{a} f_\text{even}(x) dx = 2 \int_{0}^{a} f_\text{even}(x) dx$$

 * $$\int_{-a}^{a} f_\text{odd}(x) dx = 0$$

This is all very relevant to Fourier series. Suppose that an even function is expanded. Recall that sine is odd and cosine is even. Then:

 * $$A_n\,$$
 * $$= \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{n \pi}{L} x\right) f_\text{even}(x) dx$$ (whole integrand is even)
 * $$= \int_{0}^{L} \frac{2}{L} \cos\left(\frac{n \pi}{L} x\right) f_\text{even}(x) dx$$
 * }
 * $$= \int_{0}^{L} \frac{2}{L} \cos\left(\frac{n \pi}{L} x\right) f_\text{even}(x) dx$$
 * }
 * }

 * $$B_n\,$$
 * $$= \int_{-L}^{L} \frac{1}{L} \sin\left(\frac{n \pi}{L} x\right) f_\text{even}(x) dx$$ (whole integrand is odd)
 * $$= 0\,$$
 * }
 * $$= 0\,$$
 * }
 * }

So the Fourier cosine series (note that all sine terms disappear) is just the Fourier series for an even function, given as:

 * $$f_\text{even}(x) = \frac{A_0}{2} + \sum_{n = 1}^{\infty}\left(A_n \cos\left(\frac{n \pi}{L} x\right)\right)$$

 * $$A_n = \int_{0}^{L} \frac{2}{L} \cos\left(\frac{n \pi}{L} x\right) f_{even}(x) dx$$

A Fourier expansion may be similarly built for an odd function:

 * $$A_n\,$$
 * $$= \int_{-L}^{L} \frac{1}{L} \cos\left(\frac{n \pi}{L} x\right) f_\text{odd}(x) dx$$ (whole integrand is odd)
 * $$= 0\,$$
 * }
 * $$= 0\,$$
 * }
 * }

 * $$B_n\,$$
 * $$= \int_{-L}^{L} \frac{1}{L} \sin\left(\frac{n \pi}{L} x\right) f_\text{odd}(x) dx$$ (whole integrand is even)
 * $$= \int_{0}^{L} \frac{2}{L} \sin\left(\frac{n \pi}{L} x\right) f_\text{odd}(x) dx$$
 * }
 * $$= \int_{0}^{L} \frac{2}{L} \sin\left(\frac{n \pi}{L} x\right) f_\text{odd}(x) dx$$
 * }
 * }

And the Fourier sine series is:

 * $$f_\text{odd}(x) = \sum_{n = 1}^{\infty}\left(B_n \sin\left(\frac{n \pi}{L} x\right)\right)$$

 * $$B_n = \int_{0}^{L} \frac{2}{L} \sin\left(\frac{n \pi}{L} x\right) f_\text{odd}(x) dx$$

At this point, the periodic extension may be considered. In the previous chapter, the problem mandated a sine expansion of a parabola. A parabola is by no means a periodic function, and yet a Fourier sine expansion was done on it. What actually happened was that the function was expanded as expected within its domain of interest: the interval 0 ≤ x ≤ 1. Inside this interval, the expansion truly is a parabola. Outside this interval, the expansion is periodic, and as a whole is odd (just like the sine functions it's built on).

The parabola could've been expanded just as well using cosines (resulting in an even expansion) or a full Fourier expansion on, say, -1 ≤ x ≤ 1.

Note that we weren't able to pick which expansion to use, however. While the parabola could be expanded any way we want on any interval we want, only the sine expansion on 0 ≤ x ≤ 1 would solve the problem. The ODE and BCs together picked the expansion and the interval. In fact, before the expansion was even constructed we had:

 * $$u(y, t) = \sum_{n=1}^{\infty} u_n(y, t) = \sum_{n=1}^{\infty} e^{-(n \pi)^2 \nu t} A_n \sin(n \pi y)\,$$

Which is a Fourier sine series only at t = 0. That the IC was defined at t = 0 allowed the expansion. For t > 0, the solution has nothing in common with a Fourier series.

What's trying to be emphasized is flexibility. Knowledge of Fourier series makes it much easier to solve problems. In the parallel plate problem, knowing what a Fourier sine series is motivates the construction of the sum of un. In the end it's the problem that dictates what needs to be done. For the separable IBVPs, expansions will be a recurring nightmare theme and it is most important to be familiar and comfortable with orthogonality and its application to making sense out of infinite sums. Many functions have orthogonality properties, including Bessel functions, Legendre polynomials, and others.

The keyword is orthogonality. If an orthogonality relation exists for a given situation, then a series solution is easily possible. As an example, the diffusion equation used in the previous chapter can, with sufficiently ugly BCs, require a trigonometric series solution that is not a Fourier series (non-integer, not evenly spaced frequencies of the sinusoids). Sturm-Liouville theory rescues us in such cases, providing the right orthogonality relation.