text:
Talk:GCSE Science/Newton's second law

The derivation needs work. V1 -V2 is not the acceletartion :-( it's the change in velocity. The acceleration is the rate of change of velocity. i.e.

(V1 -V2)/t

Theresa

Also, there has been no previous definition of force, which makes this supposed proof spurious. At best, it's a poor way of motivating a definition of force, not a derivation. Someone with a GCSE syllabus needs to correct this.

title: Talk:GCSE Science/Newton's second law


text:
Physics Study Guide/Normal force and friction

The Normal Force
Why is it that we stay steady in our chairs when we sit down? According to the first law of motion, if an object is translationally in equilibrium (velocity is constant), the sum of all the forces acting on the object must be equal to zero. For a person sitting on a chair, it can thus be postulated that a normal force is present balancing the gravitational force that pulls the sitting person down. However, it should be noted that only some of the normal force can cancel the other forces to zero like in the case of a sitting person. In Physics, the term normal as a modifier of the force implies that this force is acting perpendicular to the surface at the point of contact of the two objects in question. Imagine a person leaning on a vertical wall. Since the person does not stumble or fall, he/she must be in equilibrium. Thus, the component of his/her weight along the horizontal is balanced or countered (opposite direction) by an equal amount of force -- this force is the normal force on the wall. So, on a slope, the normal force would not point upwards as on a horizontal surface but rather perpendicular to the slope surface.

The normal force can be provided by any one of the four fundamental forces, but is typically provided by electromagnetism since microscopically, it is the repulsion of electrons that enables interaction between surfaces of matter. There is no easy way to calculate the normal force, other than by assuming first that there is a normal force acting on a body in contact with a surface (direction perpendicular to the surface). If the object is not accelerating (for the case of uniform circular motion, the object is accelerating) then somehow, the magnitude of the normal force can be solved. In most cases, the magnitude of the normal force can be solved together with other unknowns in a given problem.

Sometimes, the problem does not warrant the knowledge of the normal force(s). It is in this regard that other formalisms (e.g. Lagrange method of undertermined coefficients) can be used to eventually solve the physical problem.

Friction
When there is relative motion between two surfaces, there is a resistance to the motion. This force is called friction. Friction is the reason why people could not accept Newton's first law of Motion, that an object tends to keep its state of motion. Friction acts opposite to the direction of the original force. The frictional force is equal to the frictional coefficient times the normal force.

Friction is caused due to attractive forces between the molecules near the surfaces of the objects. If two steel plates are made really flat and polished and cleaned and made to touch in a vacuum, it bonds together. It would look as if the steel was just one piece. The bonds are formed as in a normal steel piece. This is called cold welding. And this is the main cause of friction.

The above equation is an empirical one--in general, the frictional coefficient is not constant. However, for a large variety of contact surfaces, there is a well characterized value. This kind of friction is called Coulomb friction. There is a separate coefficient for both static and kinetic friction. This is because once an object is pushed on, it will suddenly jerk once you apply enough force and it begins to move.

Also, the frictional coefficient varies greatly depending on what two substances are in contact, and the temperature and smoothness of the two substances. For example, the frictional coefficients of glass on glass are very high. When you have similar materials, in most cases you don't have Coulomb friction.

For static friction, the force of friction actually increases proportionally to the force applied, keeping the body immobile. Once, however, the force exceeds the maximum frictional force, the body will begin to move. The maximum frictional force is calculated as follows:

The static frictional force is less than or equal to the coefficient of static friction times the normal force. Once the frictional force equals the coefficient of static friction times the normal force, the object will break away and begin to move.

Once it is moving, the frictional force then obeys:

The kinetic frictional force is equal to the coefficient of kinetic friction times the normal force. As stated before, this always opposes the direction of motion.

Variables
Definition of Terms

It's important to note, that in real life we often have to deal with viscose and turbulent friction - they appear when you move the body through the matter.

Viscose friction is proportional to velocity and takes place at approximately low speeds. Turbulent friction is proportional to $$V^2$$ and takes place at higher velocities.

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


text:
User:R3m0t

r3m0t (w:User:R3m0t), 15, mainly editing Lucid dreaming, High school extensions and GCSE Science.

testing my sig: r3m0t (cont) (talk) 04:59, 31 Jan 2004 (UTC)

Latest LD failure (night of 08 April): My school bus was actually falling 5 metres! What do I do? A Reality Check. Does it work? No. What do I do? Another Reality Check. Does it work? Yes! What do I do? Forget in all the excitement of a school bus falling down into a train station.

Wahey!

Memo: Check User:Paddu has finished his/her pear soup.

title: User:R3m0t


text:
High School Mathematics Extensions/Mathematical Programming

Before we begin
 * This chapter will not attempt to teach you how to program rigorously. Therefore a basic working knowledge of the C programming language is highly recommended. It is recommended that you learn as much about the C programming language as possible before learning the materials in this chapter.

 * Please read the first 7 lessons of "C Programming Tutorial" at About.com if you are unfamiliar with programming or the C programming language.

As you gain programming experience you will appreciate the more specific explanations like The C Programming Wikibook.

Introduction to programming
Programming has many uses. Some areas where programming and computer science in general are extremely important include artificial intelligence and statistics. Programming allows you to use computers flexibly and process data very quickly.

When a program is written, it is written into a textual form that a human can understand. However, a computer doesn't directly understand what a human writes. It needs to be transformed into a way that the computer can directly understand.

For example, a computer is like a person who reads and speaks German. You write and speak in English. The letter you write to the computer needs to be translated for the computer to speak. The program responsible for this work is referred to as the compiler.

You need to compile your English-like instructions, so that the computer can understand it. Once a program has been compiled, it is hard to "un-compile" it, or transform it back into English again. A programmer writes the program (to use our analogy, in English), called source code, which is a human-readable definition of the program, and then the compiler translates this into "machine code". We recommend using the widely available gcc compiler.

When we look at mathematical programming here, we will look at how we can write programs that will solve some difficult mathematical problems that would take us normally a lot of time to solve. For example, if we wanted to find an approximation to the root of the polynomial x5+x+1 - this is very difficult for a human to solve. However a computer can do this no sweat -- how?

Programming language basics
We will be using the C programming language throughout the chapter, please learn about the basics of C by reading the first 7 lessons of "C Programming Tutorial" at About.com.

Sample C Program

 * /Sample C Program/

Data Types
Size Constraints: Header files limits.h, and float.h

Computers are machines based on Boolean logic. This means that the computer is based on some method of differentiating a state as true or false, or set and not set. Abstractly we think of computers as using 1's for true or set and 0's for false or not set. We refer to these 1's and 0's as bits. In computers we don't keep track of information as bits. Instead information in a computer is stored in addressable blocks called bytes. A byte is the smallest piece of memory that can be accessed in the computer that is not a bit. When we declare a [scalar] variable in a C program that memory has an address and a length. The address says where the memory starts, and the length states how many bytes are used to express the variable.

The include file  is used to define the size of addressable integer types and the include file  is used to define the size of addressable floating point types. The values in these files are compiler and computer dependent. This means that if you change compilers or compile your program on a different type of computer it may execute differently.

Here are some of the values defined in these two files:

 Exercises

Programming With Integers
Discrete programming deals with integers and how they are manipulated using the computer.

Understanding integer division
In C, the command
 * int number;
 * number = 3 / 2;

will set aside some space in the computer memory, and we can refer to that space by the variable name number. In the computer's mind, number is an integer, nothing else. After
 * number = 3 / 2;

numbers equals 1, not 1.5, this is due to that fact that / when applied to two integers will give only the integer part of the result. For example in C:
 * 5 / 2 equals 2
 * 353 / 3 equals 117
 * -5 / 2 equals -2
 * 353 / -3 equals -117
 * -5 / - 2 equals 2

If the number you are testing is between one and negative one - for example 2 / 5 or -2 / 5 then the result is undefined, although most compilers return 0.

The modular operator, %, returns the remainder resulting from integer division. For example in C:

 * 5 % 2 equals 2
 * 353 % 3 equals 2
 * -5 % 2 equals -2
 * 353 % - 3 equals 2
 * -5 % -2 equals -1

The sign of the result takes on the sign of the dividend as you would expect. For fractions that are between one and negative one the result is the same as the numerator.

 Exercises

Exercise 1

Write down your thoughts on what a program to explore division and modulus should do.
 * What processing has to occur?
 * What types of input do you have?
 * What does your output look like?

The following example will walk you through this exercise.

C Program Example For Exercise 1

 * /Program to Explore Integer Division And Integer Modulus/

Modeling Recursively defined functions
The factorial function n! is recursively defined:
 * 0! = 1
 * n! = n×(n-1)! if n ≥ 1

In C, if fact(n) is the functions as described above we want
 * $$fact(0) = 1;$$
 * $$fact(n) = n * fact(n - 1)$$; if $$n \ge 1$$

we should note that all recursively defined functions have a terminating condition, it is the case where the function can give a direct answer to, e.g. fact(0) = 1.

We can model the factorial functions easily with the following code and then execute it:
 * int fact (int n)
 * if (n == 0)
 * return 1;
 * if (n >= 1)
 * return n * fact(n - 1);
 * }
 * }

The C function above models the factorial function very naturally. To test the results, we can compile the following code:
 * #include  /* Standard Input & Output Header file */
 * int fact (int n)
 * if (n == 0)
 * return 1;
 * if (n >= 1)
 * return n * fact(n - 1);
 * }
 * int main(void)
 * int n = 5;
 * printf("%d", fact(n)); /* printf is defined in stdio.h */
 * }
 * printf("%d", fact(n)); /* printf is defined in stdio.h */
 * }

We can also model the Fibonacci number function. Let fib(n) return the (n + 1)th Fibonacci number, the following should be clear
 * fib(0) should return 1
 * fib(1) should return 1
 * fib(n) should return fib(n - 1) + fib(n - 2); for n ≥ 2

we can model the above using C:
 * int fib (int n)
 * if (n == 0 || n == 1) /* if n = 0 or if n = 1 */
 * return 1;
 * if (n >= 2)
 * return fib(n - 1) + fib(n - 2);
 * }
 * }

Again, you shall see that modeling a recursive function is not hard, as it only involves translating the mathematics in C code.

Modeling non-recursive functions
There are functions that involve only integers, and they can be modelled quite nicely using functions in C. The factorial function
 * f(n) = n! = n(n - 1)(n - 2) ... 3×2×1

can be modeled quite simply using the following code

int n = 10; //get factorial for 10 int f = 1;  //start f at 1 while(n > 0) //keep looping if n bigger then 0 {   f = n * f; //f is now product of f and n    n = n - 1; //n is one less (repeat loop) }

Floating point Programming
Programs can not only be written with integer values, but also with various forms of floating-point values. You should normally use the double keyword to define a floating point number; the reason for this is that in many cases, the intuitive way to write an expression in floating point arithmetic is suboptimal. Floating point arithmetic is non-associative - in base-10, a system that has 2 places of accuracy has (1.0 + 0.02) + 0.04 = 1.0 (rounded down because 1.02 rounds to 1.0, and then 1.04 rounds down to 1.0), but 1.0 + (0.02 + 0.04) = 1.0 + 0.06 = 1.1. There also exists a float type, which uses 4 bytes instead of 8, but you should not use it unless you know what you are doing, since there is only 24 bits of accuracy, or roughly 9 base-10 significant digits.

Beware: If you use 2 integer operands, it still performs integer arithmetic, so this prints 1, not 1.5 as you'd expect:

 * double number;
 * number = 3/2;
 * printf("%f\n",number);

An example of a definition of a floating-point number and then calculating 3/2:

 * double number = 3;
 * number /= 2;
 * printf("%f\n",number);

A caveat: floating point numbers do not perfectly represent all decimal numbers. Obviously, because the memory consumed by a variable is finite, it cannot represent an infinite number, but only an approximation of it. In addition, some numbers cannot be perfectly represented in floating point. In this code:

 * double number;
 * number = 1/10;

the value of number is not 0.1, but actually 0.10000000000000001.

An analogy of this limitation is the value of 1/3. In the decimal system, this value cannot be exactly represented with a finite number of 3s, as in 0.333... Since 2 and 5 are the only prime factors of 10 (the base of the decimal system), only fractions with denominators comprising products of 2s and 5s, such as 5/8 $$\left ( \tfrac{5}{2^3} \right )$$ or 231/250 $$\left ( \tfrac{231}{2\times5^3} \right )$$ (but not 1/3 or 5/14) can be exactly represented by decimal numbers (0.625 and 0.924, respectively).

Computers use base 2 arithmetic, so only fractions with denominators comprising products of 2s (powers of 2), such as 5/8 $$\left ( \tfrac{5}{2^3} \right )$$ or 231/256 $$\left ( \tfrac{231}{2^8} \right )$$ (but not 231/250, 1/3 or 1/10, as above) can be exactly represented by a floating point number.


text:
Physics Study Guide/Fields

=Fields=

A field is one of the more difficult concepts to grasp in physics. A field is an area or region in which an influence or force is effective regardless of the presence or absence of a material medium. Simply put, a field is a collection of vectors often representing the force an object would feel if it were placed at any particular point in space. With gravity, the field is measured in newtons, as it depends solely on the mass of an object, but with electricity, it is measured in newtons per coulomb, as the force on an electrical charge depends on the amount of that charge. Typically these fields are calculated based on canceling out the effect of a body in the point in space that the field is desired. As a result, a field is a vector, and as such, it can (and should) be added when calculating the field created by TWO objects at one point in space.

Fields are typically illustrated through the use of what are called field lines or lines of force. Given a source that exerts a force on points around it, sample lines are drawn representing the direction of the field at points in space around the force-exerting source.

There are three major categories of fields:

 * 1) Uniform fields are fields that have the same value at any point in space.  As a result, the lines of force are parallel.
 * 2) Spherical fields are fields that have an origin at a particular point in space and vary at varying distances from that point.
 * 3) Complex fields are fields that are difficult to work with mathematically (except under simple cases, such as fields created by two point object), but field lines can still typically be drawn.  Dipoles are a specific kind of complex field.

Magnetism also has a field, measured in Tesla, and it also has field lines, but its use is more complicated than simple "force" fields. Secondly, it also only appears in a two-pole form, and as such, is difficult to calculate easily.

The particles that form these magnetic fields and lines of force are called electrons and not magnetons. A magneton is a quantity in magnetism.

title: Physics Study Guide/Fields


text:
User:ゐ~enwikibooks

ウィキペディアの利用者:ゐ

2003年12月6日より活動開始.

「物理学」「生物学」「フランス語」「ラテン語」などの翻訳・作成を予定しています.

title: User:ゐ~enwikibooks


text:
Romanian/Basics

Common phrases

 * Romanian (person): "Român" (Ro-MOOHN), "Româncă" (Ro-MOOHN-kuh)
 * Romanian (language): "Română" (Ro-MOOH-nuh)
 * please: "Vă rog" (Vuh Rohg) (formal)
 * please: "Te rog" (Teh Rohg)   (informal / familiar)
 * sorry: "Îmi pare rău" (Oohm pah-reh ruh)
 * thank you: "Mulțumesc" (Mool-tzu-mesk)
 * you're welcome: Cu plăcere (koo pluh-cheh-reh)
 * yes: "Da" (Da)
 * no: "Nu" (Noo)
 * Excuse me: Scuză-mă (/'sku.zə.mə/) (informal)
 * I don't understand: "Nu înțeleg" (Noo uhnts-eleg)
 * Where's the bathroom?: "Unde este toaleta?" (Oon-deh yes-teh twah-leta)
 * Do you speak English?: "Vorbiți engleza?" (Vor-beets eng-lez-ahh)

Greetings

 * hello: "Salut" (Sa-loot)
 * hello: "Bună" (Boo-Nuh) (informal / familiar)
 * good-bye: "La revedere" (Lah reh-veh-DEH-reh)
 * bye: "Pa" (in informal occasions in Transylvania - "Servus" [sehr-voos])
 * good morning: Bună dimineața (boo-nuh dee-mee-nats-ah)
 * good day: Bună ziua (boo-nuh zee-wah)
 * good evening: Bună seara (boo-nuh seah-rah)
 * good night: Noapte Bună (nwap-teh boo-nuh)
 * How are you?: "Ce faci?" (/↗ t͡ʃe ↘'fat͡ʃʲ/)
 * ok: "bine" (/'bi.ne/)

Simple questions

 * who: Cine (chee-neh)
 * what: Ce (cheh)
 * where: Unde (oon-deh)
 * when: Când (kuhnd)
 * why: De ce (deh cheh)
 * how: Cum (koom)
 * How many or How much: Cât (/kɨt/), Câte (/'kɨ.te/)

title: Romanian/Basics


text:
Romanian/Numbers and colors

Numbers

 * 1 : unu (unu)
 * 2 : doi (doy)
 * 3 : trei (tray)
 * 4 : patru (pah-troo)
 * 5 : cinci (chinch)
 * 6 : șase (sha-seh)
 * 7 : șapte (shap-teh)
 * 8 : opt (opt)
 * 9 : nouă (nou-uh)
 * 10 : zece (zeh-cheh)
 * 11 : unsprezece (oon-spreh-zeh-cheh), unșpe (informal)
 * 12 : doisprezece (doy-spreh-zeh-cheh), doișpe (informal)
 * 13 : treisprezece (tray-spreh-zeh-cheh), treișpe (informal)
 * 14 : paisprezece (pie-spreh-zeh-cheh), paișpe (informal)
 * 15 : cincisprezece (chinch-spreh-zeh-cheh), cinșpe (informal)
 * 16 : șaisprezece (shy-spreh-zeh-cheh), șaișpe (informal)
 * 17 : șaptesprezece (shap-teh-spreh-zeh-cheh), șapteșpe (informal)
 * 18 : optsprezece (opt-spreh-zeh-cheh), optâșpe (informal)
 * 19 : nouăsprezece (nou-uh-spreh-zeh-cheh), nouășpe (informal)
 * 20 : douăzeci (...)
 * 21 : douăzeci și unu (...)
 * 22 : douăzeci și doi (...)
 * 23 : douăzeci și trei (...)
 * 30 : treizeci (...)
 * 40 : patruzeci (...)
 * 50 : cincizeci (...)
 * 60 : șaizeci (...)
 * 70 : șaptezeci (...)
 * 80 : optzeci (...)
 * 90 : nouăzeci (...)
 * 100 : o sută (...)
 * 101 : o sută unu (...)
 * 110 : o sută zece (...)
 * 200 : două sute (...)
 * 300 : trei sute (...)
 * 1,000 : o mie (...)
 * 2,000 : două mii (...)
 * 1,000,000 : un milion (...)
 * number _____ (train, bus, etc.) : numărul _____ (...)
 * half : jumătate (...), jumate (informal)
 * less (than) : mai mic (decât) (...)
 * more (than) : mai mare (decât) (...)

See ../Numbers from 0 to 999/

Colors
NOTE: ah in English represents the sound 'a' as in "far"

 * black : negru (neh-groo)
 * white : alb (ahlb)
 * gray : gri (gree)
 * red : roșu (roh-shoo)
 * blue : albastru (ahl-bahs-troo)
 * yellow : galben (gahl-ben)
 * green : verde (vehr-deh)
 * orange : portocaliu (pohr-toh-ka-leeu), oranj
 * purple : mov (mohv), purpuriu
 * brown : maro (mah-roh)

title: Romanian/Numbers and colors


text:
Romanian/Months and days of the week

Months

 * January : ianuarie (ya-nooh-ah-ree-yeh)
 * February : februarie (fe-bruh-ah-ree-yeh)
 * March : martie (mahr-tee-yeh)
 * April : aprilie (ah-pri-lee-yeh)
 * May : mai (mai)
 * June : iunie (yoo-nee-yeh)
 * July : iulie (yoo-lee-yeh)
 * August : august (ow-goost)
 * September : septembrie (sehp-tehm-bree-yeh)
 * October : octombrie (oc-tohm-bree-yeh)
 * November : noiembrie (no-iehm-bree-yeh)
 * December : decembrie (de-chehm-bree-yeh)

Days

 * today : astăzi (ast-a-z), azi (informal)
 * yesterday : ieri (yehr)
 * tomorrow : mâine (muy-neh)
 * this week : săptămâna asta (sehp-teh-muh-na ahs-tah)
 * last week : săptămâna trecută (sehp-teh-muh-na treh-coo-ta)
 * next week : săptămâna viitoare (sehp-teh-muh-na vee-ee-toah-reh)

Days of the week

 * Sunday : duminică (doo-mee-nee-ka)
 * Monday : luni (loohn)
 * Tuesday : marți (mahrtz)
 * Wednesday : miercuri (myehr-coor)
 * Thursday : joi (joy with the 'j' like the French "Bonjour")
 * Friday : vineri (vee-nehr)
 * Saturday : sâmbătă (suhm-buh-tuh)

Note that as in most languages, but unlike English, days of the week and months are written with lower-case letters except when the word appears at the start of a sentence: luni not Luni.

title: Romanian/Months and days of the week


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User talk:R3m0t

I'm glad you liked HSE so much! I'm trying my best to do a good job, but as so often is the case in my first drafts, glaring spelling and grammar mistakes seem to plague my work. Lucky you pick them out for me.

I do not know enough about Lisp, Prolog, Scheme (never heard of it) or Haskell to decide whether they are suitable for high school students or not. If you think a coherent course can be taught using one of those languages, then please go ahead! Create a new chapte if you have to. But I was thinking of teaching a more conventional language like Pascal. Anyway we are not just writing about programming, but mathematical programming. Haskell seems to be a good choice, but where can I find a good programming IDE/compiler for it? I simply haven't done the research.

And nice to meet you. How do you do?

Zhuo Jia Dai

 * Scheme is a simpler version of Lisp designed for teaching theoretical computer science. It's also used in some professional applications such as GIMP, the image editor. I would highly recommend it as it is fairly similar to Lisp. Note that the version taught at How To Design Programs isn't quite the standard. r3m0t 11:52, 7 Dec 2003 (UTC)

Are you 14 years old ? If so I'm well impressed! Theresa knott 11:35, 14 Jan 2004 (UTC)

OK next question - are you in the uk? Are you doing you GCSE's if so can you help out with the coursework section of GCSE science? (Actually 3 questions I know) Theresa knott 09:40, 16 Jan 2004 (UTC)

Thanks man I am trying not to have a coronary about it ... --Karl Wick - (contr)... Dysprosia 23:19, 30 Jan 2004 (UTC)
 * Thanks :) r3m0t (cont) (talk) 05:00, 31 Jan 2004 (UTC)

Lucid Dreaming deletion
Hi r3m0t, I don't know of a way to delete the version from the history except delete the whole page and the whole history. I think the page should be OK in the old versions list because it is not actively used and people will tend not to refer to it. Thanks for your concern and if there is a way to delete just that version I'd be happy to do it for you. --Karl Wick

image ?
What image are you asking about my friend ? --Karl Wick

PS Thanks for fixing my typos when you see them .. -Karl Wick

nowiki tag
Wow, I never noticed. Thanks for the help though and I will remember that "no wiki" tag that you just taught me. --Karl Wick

auto TOC
The TOC I just made for Cell Biology does not seem to be working on this page .. can you tell what am I doing wrong ? --Karl Wick
 * Umm.. well I got it figured out r3m0t ... I spaced and named the TOC page wrong. Thanks anyway... --Karl Wick

kirbymeister
No, I just didnt wanna keep updating it every LD i get. (btw my cound is 2) KirbyMeister 21:35, 5 May 2004 (UTC)

I got another LD today! Also, I created LD RC Time, please check for errors. KirbyMeister 01:28, 21 Apr 2004 (UTC)

gcse science
Hi thanks for your offer of help. I suppose we could go at movement and change, most of the stuff comes under forces, but I've kinda got bored with physics after doing electricity and was thinking of starting a chemistry module. Would you be willing to help out with that ? What are you studying in chemistry at the moment ? (says Theresa Knott 26 March)

Lucid Dreaming FAQ
On the subect of copyrights, I can only point out the bold text on the bottom of the submit page... And, yes I have read the Lucid Dreaming module, and have been trying unsuccessfully to have an LD for the past couple of weeks. any tips? -- Gabriel Beecham 01:25, 3 Apr 2004 (UTC)

I took out the nowiki tags because they show up when the message is inculded in other pages. look at Los verbos reflexivos (Spanish). I think this is a bug. Perl 13:59, 4 Apr 2004 (UTC)

Hey Remot,
I saw a little mention of the LD wikibook in alt.dreams.lucid. Good job so far! Thanks for the tips, I'll continue writing stuff whenever I feel I have some info to share.

KC 12:43, 5 Apr 2004 (UTC)

Tharenthel
No problem, didn't have much else to do anyway. And good luck on the dream interpretation section, it sounds like a lot of — ummm... fun. :-) —Tharenthel 03:23, 10 Aug 2004 (UTC)

Remot
A long time ago, you asked if I'd like to move the material in the "Science in Simple Steps" development hypertextbook to Wikibooks. Having finally gotten enough time to pay at least a bit of attention to the projec again, I would love to. Especially since I don't know when work and other factors may swallow me whole again, and I'd want the start we've made to go somewhere where others can carry on the good work without having months go by with their messages unanswered. :P  So I would like to ask - beg, even - that you and the others with you help guide me through the process of setting things up and getting them copied over in a way that is consistent with the WikiBooks organization. I would be endlessly grateful for the help. User:April Follies

14??!!
You're 14? And started punishing others? OMG! -- Paddu 22:41, 21 Oct 2004 (UTC)

LoF
Are you doing anything else around the Lord of the Flies page thing?

Oh, there's nothing at all wrong with being 14 :P. I'm 15.... Though I use Wikipedia more. Estel 13:02, 2 Nov 2004 (UTC)

 * Erm, I just about started that page when we stopped studying it in class. However, lucky for you (assuming you want to see the page grow but are too lazy to do it yourself) it's an exam text, not a coursework, so I'll have to revise it. I'll put up notes then. :) Oh yes, and I'm 15 by now. Better change mentions... r3m0t (cont) (talk) 17:34, 2 Nov 2004 (UTC)

Wiki Junior Project
We are currently in the process of deciding what the first topics will be. We have already decided that the first humanities topic will be Countries of the World:South America. We need to decide what our first science topic will be. We already have plenty of pictures available for Big Cats, The Solar System and Human Flight. We're having a little vote to decide which one we should work on first. Please come to Wikijunior project first topics. Cheers! Theresa Knott (Not the skater) 07:45, 6 Nov 2004 (UTC)

 * I put dinosaurs on that page previously. What happened? r3m0t (cont) (talk) 09:15, 6 Nov 2004 (UTC)
 * They were removed as a non starter because of difficulties getting suitable media. The trouble with dinosuars is that it's only possible to easily get photos of bones. Bones are fine, we should certainly have them but we also need good quality artists impressions. It'll be much easier to get artists once the project is up and running. I had to narrow the choice down to a managable list. i opted for the three where we had plenty of photographs already, so that we wouldn't be held back by lack of media. Theresa knott 19:39, 7 Nov 2004 (UTC)

message from User:166.121.36.7 found in wikibook main text
The following message was posted by User:166.121.36.7 in the Lucid Dreaming wikibook. Like an idiot, I replied in the main text too instead of sending you this message. I also replied to the user on the talkpage, but feel free to send him additional comments if you have any.

--> to whoever messaged me... hmm, i have no idea how to message you back, but since you deleted my test, i guess i can talk to you from here. just wondering, isn't the concept of a freely editable "wikibook" something dangerous? if anybody was feeling malicious, he could just go and delete all the contents...

 * (note: I'm not the one who deleted your test) When an edit is made to a page, even an entire content deletion, the version of the page before that is saved so someone can revert (i.e. reverse) it. By clicking on the "history" tab at the top of the page you can view all the edits that have been made to it, latest edit first. To send a message to a user you go to the page "User talk:[username]" (no quotes, replace [username] with the username of the person you want to talk to) and edit that page. For more information see Help:Wikibooks. Exabyte (talk) 01:41, 10 Nov 2004 (UTC)

Exabyte (talk) 04:58, 10 Nov 2004 (UTC)

 * Thanks. :) I'll just tell them about Talk pages. r3m0t (cont) (talk) 12:16, 10 Nov 2004 (UTC)

I'm sorry. It was a mistake. I was going to move my own page.

Book of the month
Hi R3m0t! Congratulations for the book of the month! Thank you for helping me to change all pages connected to "Book of the month". I wanted to do it by myself, but fell asleep (00:00 UTC is 1:00 over here), and didn't get up until now. Anyway, would you help me also in the future to organize things? Your help is highly appreciated, and I think you do things in a very professional way. You also did a good job in talking to people who did not vote correctly. B.t.w. I put in a picture on the main page temporarily, but you can replace it with yours once you are allowed to. Cheers, --Andreas Ipp 08:01, 1 Feb 2005 (UTC)

 * I'm proud. :) I live in London, so I was up till about 00:30 closing the vote and updating pages related to it. Today I was really sleepy at school, but never mind. I'm happy to help with any more administrivia ;) you would like done. Thanks for putting that picture up. No email from the person with the evil license yet. --r3m0t (cont) (talk) 17:31, 1 Feb 2005 (UTC)

Ido Wikibooks
Hi R3m0t, this is mithridates from the Ido wikipedia. I put up a request for an Ido wikibooks a few days ago but here hasn't been any response as of yet and I thought it might be best to ask you directly. Could get this done? Thank you.

Hey people i've been trying my BEST to get a lucid dream,i cant,help me!!!

REMask device
Hi out there, has anyone used an REmask and are they as good , better , worse than the lifetools nova dreamer. Nova dreamers are so dear and hard to get that for $130 i'm considering one. Any views. 45 never been lucid ( that I can recall ), read about it and am dying to experience it too. Any dietery triggers knownabout or worth a go. Tried sleep - wake - sleep with SUGAR but still nothing. Can you try too hard ?

Addition to Lucid Dreaming
I placed my mattress on my bedroom's heating vent (you have no idea how comfortable that can be). Although I've never been lucid in a dream, I have noticed that the dreams themselves are more clear than they used to be (or maybe it improved my recall for some reason, who knows). Maybe you can start a chapter on external stimulation.

(I don't understand this editor too well, I hope this comes out ok)

Translation in french language.
Hello,

I'm very interesting in Lucid dreaming. However, french lucid dreaming community isn't very large.

We need documentation about lucid dreaming but we're not enough in order to redact them.

That's why I ask you if I can translate them into french.

In waiting your approbation, Nicolas.

CC-NC
Why do you upload media under noncommercial license? --Derbeth talk 09:34, 15 September 2006 (UTC)

Appreciation
Hi, I'm reading the Lucid Dreaming book now. It is excellent. Thank you very much.

About Lucid Dreaming
Hello! I've found very useful your book about lucid dreaming... you should go on with this idea! I want to ask you about a problem I have. I find too difficult in concentrating before sleeping, nad most methods needs concentration. Which method requires less concentration? Please, send me an e-mail to gzimize at gmail.com --Gzimize 16:38, 1 December 2006 (UTC)

I'm willing to help
Hey, I'm willing to help you with anything on your Lucid Dreaming book. I know quite a lot myself, and I am willing to do research, etc. I guess just post back on my talk page? -Erik212

Good Resource
Hey I took a look at your lucid dreaming book it's pretty good. I have narcolepsy myself and I have been lucid dreaming almost every night for over 3 years. It's something else I'll tell ya. If you have any questions or anything I'm sure I could offer my insight. I'm somewhat of a self proclaimed sleep expert. You can post on my talk page or email me. -lethargiclown

Fixed copy/paste issue in LD book
^^

title: User talk:R3m0t


text:
User:Brianluft

Brian Luft.

Interested in computer science, specifically C++ and C# programming.

Personal web page: http://www.brianluft.com/

title: User:Brianluft


text:
Calculus/Vectors

Introduction
In most mathematics courses up until this point, we deal with scalars. These are quantities which only need one number to express. For instance, the amount of gasoline used to drive to the grocery store is a scalar quantity because it only needs one number: 2 gallons.

In this unit, we deal with vectors. A vector is a directed line segment -- that is, a line segment that points one direction or the other. As such, it has an initial point and a terminal point. The vector starts at the initial point and ends at the terminal point, and the vector points towards the terminal point. A vector is drawn as a line segment with an arrow at the terminal point:

The same vector can be placed anywhere on the coordinate plane and still be the same vector -- the only two bits of information a vector represents are the magnitude and the direction. The magnitude is simply the length of the vector, and the direction is the angle at which it points. Since neither of these specify a starting or ending location, the same vector can be placed anywhere. To illustrate, all of the line segments below can be defined as the vector with magnitude $$4\sqrt2$$ and angle 45 degrees:

It is customary, however, to place the vector with the initial point at the origin as indicated by the black vector. This is called the standard position.

Component Form
In standard practice, we don't express vectors by listing the length and the direction. We instead use component form, which lists the height (rise) and width (run) of the vectors. It is written as follows:

$$\begin{pmatrix}\text{run}\\\text{rise}\end{pmatrix}$$

Other ways of denoting a vector in component form include:

$$\mathbf{(u_x,u_y)}$$ and $$\mathbf{\left\langle u_x,u_y\right\rangle}$$

From the diagram we can now see the benefits of the standard position: the two numbers for the terminal point's coordinates are the same numbers for the vector's rise and run. Note that we named this vector $$\mathbf{u}$$. Just as you can assign numbers to variables in algebra (usually $$x,y,z$$), you can assign vectors to variables in calculus. The letters $$u,v,w$$ are usually used, and either boldface or an arrow over the letter is used to identify it as a vector.

When expressing a vector in component form, it is no longer obvious what the magnitude and direction are. Therefore, we have to perform some calculations to find the magnitude and direction.

Magnitude
$$|\mathbf{u}|=\sqrt{u_x^2+u_y^2}$$

where $$u_x$$ is the width, or run, of the vector; $$u_y$$ is the height, or rise, of the vector. You should recognize this formula as the Pythagorean theorem. It is -- the magnitude is the distance between the initial point and the terminal point.

The magnitude of a vector can also be called the norm.

Direction
$$\tan(\theta)=\frac{u_y}{u_x}$$

where $$\theta$$ is the counter-clockwise angle made by the vector with the positive $$x$$-axis. This formula is simply the tangent formula for right triangles.

Vector Operations
For these definitions, assume: $$\mathbf{u}=\begin{pmatrix}u_x\\u_y\end{pmatrix},\mathbf{v}=\begin{pmatrix}v_x\\v_y\end{pmatrix}$$

Vector Addition
Vector Addition is often called tip-to-tail addition, because this makes it easier to remember.

The sum of the vectors you are adding is called the resultant vector, and is the vector drawn from the initial point (tip) of the first vector to the terminal point (tail) of the second vector. Although they look like the arrows, the pointy bit is the tail, not the tip. (Imagine you were walking the direction the vector was pointing... you would start at the flat end (tip) and walk toward the pointy end.)

It looks like this:

(Notice, the black lined vector is the sum of the two dotted line vectors!)

Numerically:

$$\binom46+\binom{1}{-3}=\binom53$$

Or more generally: $$\mathbf{u}+\mathbf{v}=\begin{pmatrix}u_x+v_x\\u_y+v_y\end{pmatrix} $$

Scalar Multiplication
Graphically, multiplying a vector by a scalar changes only the magnitude of the vector by that same scalar. That is, multiplying a vector by 2 will "stretch" the vector to twice its original magnitude, keeping the direction the same.

$$2\cdot\binom33=\binom66$$

Numerically, you calculate the resultant vector with this formula:

$$c\mathbf{u}=\begin{pmatrix}cu_x\\cu_y\end{pmatrix}$$, where $$c$$ is a constant scalar.

As previously stated, the magnitude is changed by the same constant:

$$|c\mathbf{u}|=|c||\mathbf{u}|$$

Since multiplying a vector by a constant results in a vector in the same direction, we can reason that two vectors are parallel if one is a constant multiple of the other -- that is, that $$\mathbf{u}||\mathbf{v}$$ if $$\mathbf{u}=c\mathbf{v}$$ for some constant $$c$$.

We can also divide by a non-zero scalar by instead multiplying by the reciprocal, as with dividing regular numbers:

$$\frac{\mathbf{u}}{c}=\frac{1}{c}\mathbf{u},c\ne0$$

Linear Functions

Given a function $$L$$ that accepts a vector as input and returns a vector or scalar as the output, function $$L$$ is considered to be "linear" if the following holds:

 * 1) For any vectors $$\mathbf{u}$$ and $$\mathbf{v}$$, it is the case that $$L(\mathbf{u}+\mathbf{v}) = L(\mathbf{u}) + L(\mathbf{v})$$.
 * 2) For any vector $$\mathbf{u}$$ and scalar $$c$$, it is the case that $$L(c\mathbf{u}) = cL(\mathbf{u})$$.

More generally, when given a function $$M$$ that has multiple vector valued parameters $$\mathbf{u}_1, \mathbf{u}_2, ..., \mathbf{u}_n$$, function $$M$$ is a "multi-linear" function if $$M$$ is linear with respect to each parameter while holding all other parameters constant:

For each $$i = 1, 2, ..., n$$ and vectors $$\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$$:
 * 1) For any vector $$\mathbf{v}$$ from the same vector space as $$\mathbf{u}_i$$, it is the case that $$M(\mathbf{u}_1,...,\mathbf{u}_i+\mathbf{v},...,\mathbf{u}_n) = M(\mathbf{u}_1,...,\mathbf{u}_i,...,\mathbf{u}_n) + M(\mathbf{u}_1,...,\mathbf{v},...,\mathbf{u}_n)$$
 * 2) For any scalar $$c$$, it is the case that $$M(\mathbf{u}_1,...,c\mathbf{u}_i,...,\mathbf{u}_n) = cM(\mathbf{u}_1,...,\mathbf{u}_i,...,\mathbf{u}_n)$$

If $$n = 2$$, then $$M$$ is "bilinear". Bilinear functions include the dot product and the cross product.

Dot Product
The dot product is a way of multiplying two vectors to produce a scalar value. Because it combines the components of two vectors to form a /scalar/, it is sometimes called a scalar product. If you were asked to take the 'dot product of two rectangular vectors' you would do the following: $$\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y$$

It is very important to note that the dot product of two vectors does not result in another vector, it gives you a scalar, just a numerical value.

Another common pitfall may arise if your vectors are not in rectangular ('cartesian') format. Sometimes, vectors are instead expressed in polar coordinates, where the first component is the vector's magnitude (length) and the second is the angle from the $$x$$-axis at which the vector should be oriented. Dot products cannot be performed using the conventional method on these sorts of vectors; vectors in polar format must be converted to their equivalent rectangular form before you can work with them using the formula given above. A common way to convert to rectangular coordinates is to imagine that the vector was projected horizontally and vertically to form a right triangle. You could then use properties of sin and cos to find the length of the two legs the right triangle. The horizontal length would then be the x-component of the rectangular expression of the vector and the vertical length would be the y-component. Remember that if the vector is pointing down or to the left, the corresponding components would have to be negative to indicate that.

With some rearrangement and trigonometric manipulation, we can see that the number that results from the dot product of two vectors is a surprising and useful identity:

$$\mathbf{u}\cdot\mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos(\theta)$$

where $$\theta$$ is the angle between the two vectors. This provides a convenient way of finding the angle between two vectors: $$\cos(\theta)=\frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$$ Notice that the dot product is 'commutative', that is: $$\mathbf{u}\cdot\mathbf{v}=\mathbf{v}\cdot\mathbf{u}$$ Also, the dot product of two vectors will be the length of the vector squared: $$\mathbf{u}\cdot\mathbf{u}=u_xu_x+u_yu_y=(u_x)^2+(u_y)^2$$ and by the Pythagorean theorem, $$(u_x)^2+(u_y)^2=|\mathbf u|^2$$

The dot product can be visualized as the length of a projection of one vector on to the other. In other words, the dot product asks 'how much magnitude of this vector is going in the direction of that vector?'

Deriving the Dot Product
Start with the following definition for the dot product: $$\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos(\theta)$$ where $$\theta$$ is the angle between $$\mathbf{u}$$ and $$\mathbf{v}$$.

The formula $$\mathbf{u}\cdot\mathbf{v} = u_xv_x + u_yv_y$$ can be derived from the above definition through various approaches:

Approach #1

One of the more direct approaches is to use the law of cosines. Create a triangle $$\Delta AOB$$ with vertices $$A(u_x,u_y)$$, $$O(0,0)$$, and $$B(v_x,v_y)$$. The displacement $$\vec{OA} = \mathbf{u}$$, the displacement $$\vec{OB} = \mathbf{v}$$, and the displacement $$\vec{AB} = \mathbf{v}-\mathbf{u}$$. The angle $$\angle AOB = \theta$$.

The lengths of the sides of the triangle are $$|OA| = |\mathbf{u}|$$, and $$|OB| = |\mathbf{v}|$$, and $$|AB| = |\mathbf{v}-\mathbf{u}|$$. Applying the law of cosines gives:

$$|AB|^2 = |OA|^2 + |OB|^2 - 2|OA||OB|\cos(\angle AOB)$$ $$\iff |\mathbf{v}-\mathbf{u}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2|\mathbf{u}||\mathbf{v}|\cos(\theta)$$ $$\iff (v_x-u_x)^2 + (v_y-u_y)^2 = (u_x^2+u_y^2) + (v_x^2+v_y^2) - 2|\mathbf{u}||\mathbf{v}|\cos(\theta)$$ $$\iff - 2u_xv_x - 2u_yv_y = -2|\mathbf{u}||\mathbf{v}|\cos(\theta)$$ $$\iff |\mathbf{u}||\mathbf{v}|\cos(\theta) = u_xv_x + u_yv_y$$

Therefore $$\mathbf{u}\cdot\mathbf{v} = u_xv_x + u_yv_y$$.

Approach #2

A more intuitive derivation of $$\mathbf{u}\cdot\mathbf{v} = u_xv_x + u_yv_y$$ uses the fact that the dot product is a bilinear operator. To establish that the dot product is a bilinear operator, the following must be established:

 * 1) Holding $$\mathbf{u}$$ constant, the dot product $$\mathbf{u} \cdot \mathbf{v}$$ must be linear with respect to $$\mathbf{v}$$.
 * 2) Holding $$\mathbf{v}$$ constant, the dot product $$\mathbf{u} \cdot \mathbf{v}$$ must be linear with respect to $$\mathbf{u}$$.

Since it is readily apparent from the definition $$\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos(\theta)$$ that $$\mathbf{u}\cdot\mathbf{v} = \mathbf{v}\cdot\mathbf{u}$$, linearity with respect to $$\mathbf{v}$$ implies linearity with respect to $$\mathbf{u}$$. It is hence only necessary to establish that the dot product is linear with respect to $$\mathbf{v}$$ to establish bilinearity.

$$\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos(\theta) = |\mathbf{u}|\text{proj}(\mathbf{v}|\mathbf{u})$$ where $$\text{proj}(\mathbf{v}|\mathbf{u}) = |\mathbf{v}|\cos(\theta)$$ is the "orthogonal projection" of vector $$\mathbf{v}$$ onto a line $$L(\mathbf{u})$$ whose direction is the direction of $$\mathbf{u}$$. $$\text{proj}(\mathbf{v}|\mathbf{u})$$ is the component of $$\mathbf{v}$$ that is parallel to $$L(\mathbf{u})$$. By drawing similar triangles, one observes that $$\text{proj}(\mathbf{v}|\mathbf{u})$$ is linear with respect to $$\mathbf{v}$$ while $$\mathbf{u}$$ is held constant. The dot product $$\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}|\text{proj}(\mathbf{v}|\mathbf{u})$$ is linear with respect to $$\mathbf{v}$$, and therefore the dot product is a bilinear operator.

The bilinearity of the dot product now enables the derivation:

$$\mathbf{u}\cdot\mathbf{v} = \begin{pmatrix} u_x \\ u_y \end{pmatrix} \cdot \begin{pmatrix} v_x \\ v_y \end{pmatrix}$$ $$ = \left(u_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + u_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) \cdot \left(v_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + v_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)$$ $$ = \left(\left(u_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + u_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix}\right)v_x + \left(\left(u_x\begin{pmatrix} 1 \\ 0 \end{pmatrix} + u_y\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)v_y$$ $$ = \left(\begin{pmatrix} 1 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix}\right)u_xv_x + \left(\begin{pmatrix} 0 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix}\right)u_yv_x + \left(\begin{pmatrix} 1 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)u_xv_y + \left(\begin{pmatrix} 0 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right)u_yv_y$$ $$ = 1u_xv_x + 0u_yv_x + 0u_xv_y + 1u_yv_y$$ $$ = u_xv_x + u_yv_y$$

Therefore $$\mathbf{u}\cdot\mathbf{v} = u_xv_x + u_yv_y$$.

Unit Vectors
A unit vector is a vector with a magnitude of 1. The unit vector of u is a vector in the same direction as $$\mathbf{u}$$, but with a magnitude of 1:

The process of finding the unit vector of $$\mathbf{u}$$ is called normalization. As mentioned in scalar multiplication, multiplying a vector by constant $$c$$ will result in the magnitude being multiplied by $$c$$. We know how to calculate the magnitude of $$\mathbf{u}$$. We know that dividing a vector by a constant will divide the magnitude by that constant. Therefore, if that constant is the magnitude, dividing the vector by the magnitude will result in a unit vector in the same direction as $$\mathbf{u}$$ :

$$\mathbf{w}=\frac{\mathbf{u}}{|\mathbf{u}|}$$, where $$\mathbf{w}$$ is the unit vector of $$\mathbf{u}$$

Standard Unit Vectors
A special case of Unit Vectors are the Standard Unit Vectors $$\mathbf{i},\mathbf{j}$$ : $$\mathbf{i}$$ points one unit directly right in the $$x$$ direction, and $$\mathbf{j}$$ points one unit directly up in the $$y$$ direction:

$$\mathbf{i}=\binom10$$

$$\mathbf{j}=\binom01$$

Using the scalar multiplication and vector addition rules, we can then express vectors in a different way: $$\binom{x}{y}=x\mathbf{i}+y\mathbf{j}$$

If we work that equation out, it makes sense. Multiplying $$x$$ by $$\mathbf{i}$$ will result in the vector $$\binom{x}{0}$$. Multiplying $$y$$ by $$\mathbf{j}$$ will result in the vector $$\binom0y$$. Adding these two together will give us our original vector, $$\binom{x}{y}$$. Expressing vectors using $$\mathbf{i},\mathbf{j}$$ is called standard form.

Projection and Decomposition of Vectors
Sometimes it is necessary to decompose a vector $$\mathbf{u}$$ into two components: one component parallel to a vector $$\mathbf{v} $$, which we will call $$\mathbf{u}_\parallel$$ ; and one component perpendicular to it, $$\mathbf{u}_\perp$$.

Since the length of $$\mathbf{u}_\parallel$$ is $$|\mathbf{u}|\cdot\cos(\theta)$$, it is straightforward to write down the formulas for $$\mathbf{u}_\perp$$ and $$\mathbf{u}_\parallel$$ :

$$\mathbf{u_\parallel}=|\mathbf{u}|*\frac{(\mathbf{u}\cdot\mathbf{v})}{(|\mathbf{u}||\mathbf{v}|)}*\frac{\mathbf{v}}{|\mathbf{v}|}=\frac{ \mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}$$ and $$\mathbf{u}_\perp=\mathbf{u}-\mathbf{u}_\parallel$$

Length of a vector
The length of a vector is given by the dot product of a vector with itself, and $$\theta=0\ rad$$:

$$\mathbf{u}\cdot\mathbf{u}=|\mathbf{u}||\mathbf{u}|\cos(\theta)=|\mathbf{u}|^2$$

Perpendicular vectors
If the angle $$ \theta $$ between two vectors is 90 degrees or $$\tfrac{\pi}{2}$$ (if the two vectors are orthogonal to each other), that is the vectors are perpendicular, then the dot product is 0. This provides us with an easy way to find a perpendicular vector: if you have a vector $$ \mathbf{u}=\begin{pmatrix}u_x\\u_y\end{pmatrix}$$, a perpendicular vector can easily be found by either $$\mathbf{v}=\begin{pmatrix}-u_y\\u_x\end{pmatrix}=-\begin{pmatrix}u_y\\-u_x\end{pmatrix}$$

Polar coordinates
Polar coordinates are an alternative two-dimensional coordinate system, which is often useful when rotations are important. Instead of specifying the position along the $$x$$ and $$y$$ axes, we specify the distance from the origin, $$r$$, and the direction, an angle $$\theta$$.

Looking at this diagram, we can see that the values of $$x,y$$ are related to those of $$r$$ and $$\theta$$ by the equations $$\begin{matrix}x=r\cos(\theta)&r=\sqrt{x^2+y^2}\\y=r\sin(\theta)&\theta=\arctan\left(\frac{y}{x}\right)\end{matrix}$$

Because tan-1 is multivalued, care must be taken to select the right value.

Just as for Cartesian coordinates the unit vectors that point in the $$x$$ and $$y$$ directions are special, so in polar coordinates the unit vectors that point in the $$r$$ and $$\theta$$ directions are also special.

We will call these vectors $$\hat{\mathbf{r}}$$ and $$\hat{\boldsymbol{\theta}}$$, pronounced r-hat and theta-hat. Putting a circumflex over a vector this way is often used to mean the unit vector in that direction.

Again, on looking at the diagram we see,

$$\begin{matrix} \mathbf{i}=\hat{\mathbf{r}}\cos(\theta)-\hat{\boldsymbol{\theta}}\sin(\theta)&\hat{\mathbf{r}}=\frac{x}{r}\mathbf{i}+\frac{y}{r}\mathbf{j}\\ \mathbf{j}=\hat{\mathbf{r}}\sin(\theta)+\hat{\boldsymbol{\theta}}\cos(\theta)&\hat{\boldsymbol{\theta}}=-\frac{y}{r}\mathbf{i}+\frac{x}{r}\mathbf{j} \end{matrix}$$

Basic definition
Two-dimensional Cartesian coordinates as we've discussed so far can be easily extended to three-dimensions by adding one more value: $$z$$. If the standard $$(x,y)$$ coordinate axes are drawn on a sheet of paper, the $$z$$-axis would extend upwards off of the paper.

Similar to the two coordinate axes in two-dimensional coordinates, there are three coordinate planes in space. These are the $$xy$$-plane, the $$yz$$-plane, and the $$xz$$-plane. Each plane is the "sheet of paper" that contains both axes the name mentions. For instance, the $$yz$$-plane contains both the $$y$$ and $$z$$ axes and is perpendicular to the $$x$$-axis.

Therefore, vectors can be extended to three dimensions by simply adding the $$z$$ value.

$$\mathbf{u}=\begin{pmatrix}x\\y\\z\end{pmatrix}$$

To facilitate standard form notation, we add another standard unit vector:

$$\mathbf{k}=\begin{pmatrix}0\\0\\1\end{pmatrix}$$

Again, both forms (component and standard) are equivalent.

$$\begin{pmatrix}1\\2\\3\end{pmatrix}=1\mathbf{i}+2\mathbf{j}+3\mathbf{k}$$

Magnitude: Magnitude in three dimensions is the same as in two dimensions, with the addition of a $$z$$ term in the radicand.

$$|\mathbf{u}|=\sqrt{u_x^2+u_y^2+u_z^2}$$

Three dimensions
The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate systems, both of which include two-dimensional or planar polar coordinates as a subset. In essence, the cylindrical coordinate system extends polar coordinates by adding an additional distance coordinate, while the spherical system instead adds an additional angular coordinate.

Cylindrical coordinates
The cylindrical coordinate system is a coordinate system that essentially extends the two-dimensional polar coordinate system by adding a third coordinate measuring the height of a point above the plane, similar to the way in which the Cartesian coordinate system is extended into three dimensions. The third coordinate is usually denoted $$h$$ (or simply $$z$$, i.e., same notation as Cartesian coordinate system), making the three cylindrical coordinates $$(r,\theta,h)$$ (or $$(r,\theta,z)$$).

The three cylindrical coordinates can be converted to Cartesian coordinates by
 * $$\begin{align}x&=r\cos(\theta)\\y&=r\sin(\theta)\\z&=h\end{align}$$

Spherical coordinates

Polar coordinates can also be extended into three dimensions using the coordinates $$(\rho,\phi,\theta)$$ (or $$(\rho,\varphi,\theta)$$), where $$\rho$$ is the distance from the origin, $$\phi$$ (or $$\varphi$$) is the angle from the $$z$$-axis (called the colatitude or zenith and measured from 0 to 180°) and $$\theta$$ is the angle from the $$x$$-axis (as in the polar coordinates). This coordinate system, called the spherical coordinate system, is similar to the latitude and longitude system used for Earth, with the origin in the centre of Earth, the latitude $$\delta$$ being the complement of $$\phi$$, determined by $$\delta=90^\circ-\phi$$ , and the longitude $$l$$ being measured by $$l=\theta-180^\circ$$.

The three spherical coordinates are converted to Cartesian coordinates by
 * $$\begin{align}x&=\rho\sin(\phi)\cos(\theta)\\y&=\rho\sin(\phi)\sin(\theta)\\z&=\rho\cos(\phi)\end{align} $$

 * $$\rho=\sqrt{x^2+y^2+z^2}$$

 * $$\theta=\arctan\left(\frac{y}{x}\right)$$

 * $$\phi=\arccos\left(\frac{z}{\rho}\right)$$

Cross Product
The cross product of two vectors is a determinant:

 * $$\mathbf{u}\times\mathbf{v}=

\begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ u_x&u_y&u_z\\ v_x&v_y&v_z \end{vmatrix}$$

and is also a pseudovector.

The cross product of two vectors is orthogonal to both vectors. The magnitude of the cross product is the product of the magnitude of the vectors and the sin of the angle between them.
 * $$|\mathbf{u}\times\mathbf{v}|=|\mathbf{u}||\mathbf{v}|\sin(\theta)$$

This magnitude is the area of the parallelogram defined by the two vectors.

The cross product is linear and anticommutative. For any numbers $$a,b$$ ,
 * $$\mathbf{u}\times\left(a\mathbf{v}+b\mathbf{w}\right)=a\mathbf{u}\times\mathbf{v}+b\mathbf{u}\times\mathbf{w}\qquad\mathbf{u}\times\mathbf{v}=-\mathbf{v}\times\mathbf{u}$$

If both vectors point in the same direction, their cross product is 0.

Deriving the Cross Product

Start with the following definition for the cross product: $$\mathbf{u}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}}$$. Vector $$\hat{\mathbf{n}}$$ is a unit length vector that is perpendicular to both $$\mathbf{u}$$ and $$\mathbf{v}$$ and oriented according to the "right hand rule". Angle $$\theta$$ is the counterclockwise angle from $$\mathbf{u}$$ to $$\mathbf{v}$$ within the plane that is orthogonal to $$\hat{\mathbf{n}}$$.

The formula $$\mathbf{u}\times\mathbf{v} = (u_yv_z-u_zv_y)\mathbf{i} + (u_zv_x-u_xv_z)\mathbf{j} + (u_xv_y-u_yv_x)\mathbf{k}$$ can be derived by exploiting the bilinearity of the cross product. To establish the cross product as a bilinear operator, the following must be established:

 * 1) Holding $$\mathbf{u}$$ constant, the cross product $$\mathbf{u}\times\mathbf{v}$$ must be linear with respect to $$\mathbf{v}$$.
 * 2) Holding $$\mathbf{v}$$ constant, the cross product $$\mathbf{u}\times\mathbf{v}$$ must be linear with respect to $$\mathbf{u}$$.

From the definition $$\mathbf{u}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}}$$, the anticommutative property of the cross product can be inferred: $$\mathbf{u}\times\mathbf{v} = -\mathbf{v}\times\mathbf{u}$$ (vector $$\hat{\mathbf{n}}$$ reverses direction when $$\mathbf{u}$$ and $$\mathbf{v}$$ are swapped). This means that linearity with respect to $$\mathbf{v}$$ implies linearity with respect to $$\mathbf{u}$$. It is hence only necessary to establish that the cross product is linear with respect to $$\mathbf{v}$$ to establish bilinearity.

$$\mathbf{u}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}} = |\mathbf{u}|\textbf{rotate}(\textbf{perp}(\mathbf{v}|\mathbf{u})|\mathbf{u})$$ where:
 * $$\textbf{perp}(\mathbf{v}|\mathbf{u})$$ is the component of $$\mathbf{v}$$ that is perpendicular to a line $$L(\mathbf{u})$$ whose direction is the direction of $$\mathbf{u}$$. $$\textbf{perp}(\mathbf{v}|\mathbf{u}) = (|\mathbf{v}|\sin\theta)\hat{\mathbf{m}}$$ where vector $$\hat{\mathbf{m}}$$ is a unit length vector that is parallel to $$\textbf{perp}(\mathbf{v}|\mathbf{u})$$.
 * $$\textbf{rotate}(\mathbf{w}|\mathbf{u})$$ is a 90 degree counterclockwise rotation of $$\mathbf{w}$$ around $$L(\mathbf{u})$$. Note that $$\textbf{rotate}(\hat{\mathbf{m}}|\mathbf{u}) = \hat{\mathbf{n}}$$.

It can easily be observed that $$\textbf{perp}(\mathbf{v}|\mathbf{u})$$ is linear with respect to $$\mathbf{v}$$ with $$\mathbf{u}$$ held constant, and that $$\textbf{rotate}(\mathbf{w}|\mathbf{u})$$ is linear with respect to $$\mathbf{w}$$ with $$\mathbf{u}$$ held constant. The cross product $$\mathbf{u}\times\mathbf{v} = (|\mathbf{u}||\mathbf{v}|\sin\theta)\hat{\mathbf{n}} = |\mathbf{u}|\textbf{rotate}(\textbf{perp}(\mathbf{v}|\mathbf{u})|\mathbf{u})$$ is linear with respect to $$\mathbf{v}$$, and therefore the cross product is a bilinear operator.

The bilinearity of the cross product now enables the derivation:

$$\mathbf{u}\times\mathbf{v} = (u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times(v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k})$$ $$ = v_x((u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times\mathbf{i}) + v_y((u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times\mathbf{j}) + v_z((u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k})\times\mathbf{k})$$ $$ = u_xv_x(\mathbf{i}\times\mathbf{i}) + u_yv_x(\mathbf{j}\times\mathbf{i}) + u_zv_x(\mathbf{k}\times\mathbf{i}) + u_xv_y(\mathbf{i}\times\mathbf{j}) + u_yv_y(\mathbf{j}\times\mathbf{j}) + u_zv_y(\mathbf{k}\times\mathbf{j}) + u_xv_z(\mathbf{i}\times\mathbf{k}) + u_yv_z(\mathbf{j}\times\mathbf{k}) + u_zv_z(\mathbf{k}\times\mathbf{k})$$ $$ = u_xv_y\mathbf{0} + u_yv_x(-\mathbf{k}) + u_zv_x\mathbf{j} + u_xv_y\mathbf{k} + u_yv_y\mathbf{0} + u_zv_y(-\mathbf{i}) + u_xv_z(-\mathbf{j}) + u_yv_z\mathbf{i} + u_zv_z\mathbf{0}$$ $$ = (u_yv_z-u_zv_y)\mathbf{i} + (u_zv_x-u_xv_z)\mathbf{j} + (u_xv_y-u_yv_x)\mathbf{k}$$

Therefore $$\mathbf{u}\times\mathbf{v} = (u_yv_z-u_zv_y)\mathbf{i} + (u_zv_x-u_xv_z)\mathbf{j} + (u_xv_y-u_yv_x)\mathbf{k}$$.

Triple Products
If we have three vectors we can combine them in two ways, a triple scalar product, $$\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})$$ and a triple vector product $$\mathbf{u}\times(\mathbf{v}\times\mathbf{w})$$

The triple scalar product is a determinant
 * $$\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})=\begin{vmatrix}u_x&u_y&u_z\\v_x&v_y&v_z\\w_x&w_y&w_z\end{vmatrix}$$

If the three vectors are listed clockwise, looking from the origin, the sign of this product is positive. If they are listed anticlockwise the sign is negative.

The order of the cross and dot products doesn't matter.
 * $$\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})=(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}$$

Either way, the absolute value of this product is the volume of the parallelepiped defined by the three vectors $$\mathbf{u},\mathbf{v},\mathbf{w}$$

The triple vector product can be simplified $$\mathbf{u}\times(\mathbf{v}\times\mathbf{w})=(\mathbf{u}\cdot\mathbf{w})\mathbf{v}-(\mathbf{u}\cdot\mathbf{v})\mathbf{w}$$ This form is easier to do calculations with.

The triple vector product is not associative. $$\mathbf{u}\times(\mathbf{v}\times\mathbf{w})\ne(\mathbf{u}\times\mathbf{v})\times\mathbf{w}$$

There are special cases where the two sides are equal, but in general the brackets matter. They must not be omitted.

Three-Dimensional Lines and Planes
We will use $$\mathbf r$$ to denote the position of a point.

The multiples of a vector, $$\mathbf a$$ all lie on a line through the origin. Adding a constant vector $$\mathbf b$$ will shift the line, but leave it straight, so the equation of a line is,
 * $$\mathbf{r}=s\mathbf{a}+\mathbf{b}$$

This is a parametric equation. The position is specified in terms of the parameter $$s$$.

Any linear combination of two vectors, $$\mathbf{a},\mathbf{b}$$ lies on a single plane through the origin, provided the two vectors are not colinear. We can shift this plane by a constant vector again and write
 * $$\mathbf{r}=s\mathbf{a}+t\mathbf{b}+\mathbf{c}$$

If we choose $$\mathbf{a},\mathbf{b}$$ to be orthonormal vectors in the plane (i.e. unit vectors at right angles) then $$s,t$$ are Cartesian coordinates for points in the plane.

These parametric equations can be extended to higher dimensions.

Instead of giving parametric equations for the line and plane, we could use constraints. E.g., for any point in the $$xy$$ plane $$z=0$$

For a plane through the origin, the single vector normal to the plane, $$\mathbf n$$, is at right angle with every vector in the plane, by definition, so
 * $$\mathbf{r}\cdot\mathbf{n}=0$$

is a plane through the origin, normal to $$\mathbf n$$.

For planes not through the origin we get
 * $$(\mathbf{r}-\mathbf{a})\cdot\mathbf{n}=0\qquad\mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n}$$

A line lies on the intersection of two planes, so it must obey the constraint for both planes, i.e.
 * $$\mathbf{r}\cdot\mathbf{n}=a\qquad\mathbf{r}\cdot\mathbf{m}=b$$

These constraint equations con also be extended to higher dimensions.

Introduction to Vector-Valued Functions
Vector-Valued Functions are functions that instead of giving a resultant scalar value, give a resultant vector value. These aid in the creation of direction and vector fields, and are therefore used in physics to aid with visualizations of electric, magnetic, and many other types of fields. They are of the following form:

$$\mathbf{F(t)}=\begin{pmatrix}\mathbf{a_1(t)}\\\vdots\\\mathbf{a_n(t)}\end{pmatrix}$$

Limits, Derivatives, and Integrals
Put simply, the limit of a vector-valued function is the limit of its parts.

Suppose $$\lim_{t\to c}\mathbf{F}(t)=\mathbf{L}=\begin{pmatrix}\mathbf{a_1}\\\vdots\\\mathbf{a_n}\end{pmatrix}$$
 * Proof

Therefore for any $$\varepsilon>0$$ there is a $$\phi>0$$ such that

$$0<|t-c|<\phi\implies|\mathbf{F}(t)-\mathbf{L}|<\varepsilon$$

But by the triangle inequality $$|a_1|\le|\mathbf{F}|\le|a_1|+\cdots+|a_n|$$

$$|a_1(t)-a_1|\le|\mathbf{F}(t)-\mathbf{L}|$$

So

$$0<|t-c|<\phi\implies|a_1(t)-a_1|<\varepsilon$$

Therefore $$\lim_{t\to c}a_1(t)=a_1$$

A similar argument can be used through parts $$a_n(t)$$.

Now let $$\lim_{t\to c}\mathbf{F}(t)=\mathbf{L}=\begin{pmatrix}\mathbf{a_1}\\\vdots\\\mathbf{a_n}\end{pmatrix}$$ again, and that for any $$\varepsilon>0$$ there is a corresponding $$\phi>0$$ such $$0<|t-c|<\phi$$ implies

$$|a_n(t)-a_n|<\frac{\varepsilon}{n}$$

Then

$$0<|t-c|<\phi\implies|\mathbf{F}(t)-\mathbf{L}|\le\frac{\varepsilon_1}{n}+\cdots+\frac{\varepsilon_n}{n}=\varepsilon$$

therefore:

$$\lim_{t\to c}\mathbf{F}(t)=\mathbf{L}=\begin{pmatrix}\mathbf{a_1}\\\vdots\\\mathbf{a_n}\end{pmatrix}=\begin{pmatrix}\lim\limits_{t\to c}\mathbf{a_1(t)}\\\vdots\\\lim\limits_{t\to c}\mathbf{a_n(t)}\end{pmatrix}$$

From this we can then create an accurate definition of a derivative of a vector-valued function:

$$\begin{align} \mathbf{F}'(t)&=\lim_{h\to0}\frac{\mathbf{F}(t+h)-\mathbf{F}(t)}{h}=\begin{pmatrix}\mathbf{a_1(t)}\\\vdots\\\mathbf{a_n(t)}\end{pmatrix}\\ &=\lim_{h\to0}\frac{\begin{pmatrix}\mathbf{a_1(t+h)}\\\vdots\\\mathbf{a_n(t+h)}\end{pmatrix}-\begin{pmatrix}\mathbf{a_1(t)}\\\vdots\\\mathbf{a_n(t)}\end{pmatrix}}{h}\\ &=\begin{pmatrix}\lim\limits_{h\to0}\dfrac{a_1(t+h)-a_1(t)}{h}\\\vdots\\\lim\limits_{h\to0}\dfrac{a_n(t+h)-a_n(t)}{h}\end{pmatrix} \end{align}$$

The final step was accomplished by taking what we just did with limits.

By the Fundamental Theorem of Calculus integrals can be applied to the vector's components.

In other words: the limit of a vector function is the limit of its parts, the derivative of a vector function is the derivative of its parts, and the integration of a vector function is the integration of it parts.

Velocity, Acceleration, Curvature, and a brief mention of the Binormal
Assume we have a vector-valued function which starts at the origin and as its independent variables changes the points that the vectors point at trace a path.

We will call this vector $$\mathbf{r}(t)$$, which is commonly known as the position vector.

If $$\mathbf{r}$$ then represents a position and t represents time, then in model with Physics we know the following:

 * $$\mathbf{r}(t+h)-\mathbf{r}(t)$$ is displacement.
 * $$\mathbf{r}'(t)=\mathbf{v}(t)$$ where $$\mathbf{v}(t)$$ is the velocity vector.
 * $$|\mathbf{v}(t)|$$ is the speed.
 * $$\mathbf{r}''(t)=\mathbf{v}'(t)=\mathbf{a}(t)$$ where $$\mathbf{a}(t)$$ is the acceleration vector.

The only other vector that comes in use at times is known as the curvature vector.

The vector $$\mathbf{T}(t)$$ used to find it is known as the unit tangent vector, which is defined as $$\frac{\mathbf{v}(t)}{|\mathbf{v}(t)|}$$ or shorthand $$\mathbf{\hat v}$$.

The vector normal $$\mathbf{N}$$ to this then is $$\frac{\mathbf{T}'(t)}{|\mathbf{v}(t)|}$$.

We can verify this by taking the dot product

$$\mathbf{T}\cdot\mathbf{N}=0$$

Also note that $$|\mathbf{v}(t)|=\frac{ds}{dt}$$

and

$$\mathbf{T}(t)=\frac{v}{|v|}=\frac{\frac{dr}{dt}}{\frac{ds}{dt}}=\frac{dr}{ds}$$

and

$$\mathbf{N}=\frac{\mathbf{T}'(t)}{|\mathbf{v}(t)|}=\frac{\frac{dT}{dt}}{\frac{ds}{dt}}=\frac{dT}{ds}$$

Then we can actually verify:

$$\frac{d}{ds}(\mathbf{T}\cdot\mathbf{T})=\frac{d}{ds}(1)$$

$$\frac{dT}{ds}\cdot\mathbf{T}+\mathbf{T}\cdot\frac{dT}{ds}=0$$

$$2\mathbf{T}\cdot\frac{dT}{ds}=0$$

$$\mathbf{T}\cdot\frac{dT}{ds}=0$$

$$\mathbf{T}\cdot\mathbf{N}=0$$

Therefore $$\mathbf{N}$$ is perpendicular to $$\mathbf{T}$$

What this gives rise to is the Unit Normal Vector $$\frac{\frac{dT}{ds}}{\left|\frac{dT}{ds}\right|}$$ of which the top-most vector is the Normal vector, but the bottom half $$(|\frac{dT}{ds}|)^{-1}$$ is known as the curvature. Since the Normal vector points toward the inside of a curve, the sharper a turn, the Normal vector has a large magnitude, therefore the curvature has a small value, and is used as an index in civil engineering to reflect the sharpness of a curve (clover-leaf highways, for instance).

The only other thing not mentioned is the Binormal that occurs in 3-d curves $$\mathbf{T}\times\mathbf{N}=\mathbf{B}$$, which is useful in creating planes parallel to the curve.

title: Calculus/Vectors


text:
Talk:Calculus/Vectors

Yo, I added in Green's Theorem and order of integration into Multivariable Calculus, and Projection/derivatives/limits/integral/vector-valued functions/position, velocity, acceleration, tangent, normal, binormal, vectors/curvature into Vector Calculus. However, I'm not sure how much of this scope from Vector Calculus spills into this Multivariable Calculus, what else should be added (as I don't really want to edit for grammatical errors and ordering nicety, just adding articles, since I suck at ordering and English), and where I can put independence of path or if you have that in here, at one point it looks like you do, but to be honest I can't decipher what you're trying to teach; but then again I just randomly put stuff in here also. Anyways, what's going on/where can I read about, thanks, feedback? Fephisto 05:47, 14 January 2006 (UTC)

Just letting you know that there's already stuff/going to be stuff on Linear Algebra/Vectors for the vector stuff. If you wish to do them again here, that's ok. In my opinion, stuff on divergence and curl and all that funky stuff, Green's theorem et al, and vector-valued functions and others should be put here. Dysprosia 05:25, 6 Dec 2003 (UTC)

I was thinking I'd quickly cover the things we need for the following sections so that this "book" would be sorta self-contained. I'm new to wikibooks though, so perhaps that's not the norm? Brianluft 12:23, 6 Dec 2003 (CST)

 * There aren't any concrete policies about redundant information yet. For now, do what you feel is best :) Could you please fix up the header problem too? Or let me know when you're done and I'll do it. It looks nicer if we start at == and work down.  Dysprosia 06:26, 6 Dec 2003 (UTC)

 * Done. :-) Brianluft 12:30, 6 Dec 2003 (CST)

 * By the way, should we use the same bracket notation across articles? We need to maintain consistency across mathematics pages. Dysprosia 08:06, 6 Dec 2003 (UTC)

 * Yeah, I think so. I must admit I've never seen the notation in Linear Algebra/Vectors, but I'll use that instead of my angled brackets notation for consistency.  Brianluft 12:34, 6 Dec 2003 (CST)
 * Thanks :) For a good reason why, see Linear transformations Dysprosia 22:50, 6 Dec 2003 (UTC)
 * I disagree, that notation (the large vertical parantheses) is commonly used in combinatorics, and angled brackets are much more widely used. Also, does the Wiki system have support for overbars/underbars, because that notation is more commonly used than bold also. Hao 11:45 13 July 2005 (JST)

How far is the scope of this page going to cover? There aren't any sections on curl or divergence or what-have-you. I'd be glad to fill them in, but I'm not sure exactly how far it should be taken. It seems to me that there should really be a section on scalar fields before we hop into vector fields, though. --Eienmaru 00:39, 15 May 2005 (UTC)

deleted image for polar coordinates
I deleted an unlicensed image you were using in section "Polar Coordinates" but I replaced it with a similar one from Commons; you'll have to change the text in that section to match the image (ie. it doesn't use x and y as variables etc.)  – Mike.lifeguard  | talk 02:23, 1 September 2007 (UTC)

What about torsion?
Maybe the importance of the binormal should be mentioned in respect to measuring the torsion of a curve?

Is a conversational tone OK?
I find many parts of this wikibook are kind of hard to follow, solely because of their lack of descriptive prose. Is it alright to write in a conversational tone, as an instructor would speak to students? (using 'you', for example) or is this too inconsistent and hard to reproduce for the wiki format? Enthdegree (discuss • contribs) 05:26, 3 March 2012 (UTC)

title: Talk:Calculus/Vectors


text:
User talk:Brianluft

Hi Brian :) Welcome to Wikibooks.

We generally start headers with == and work down from there. The = header is for the title of the page, up the top. And there's a question/message for you waiting at Talk:Calculus:Multivariable calculus.

Thanks, and keep contributing :) Dysprosia 05:41, 6 Dec 2003 (UTC)

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

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

Please signify the copyright information on all images you have uploaded or will upload. Remember that images without this important information or giving false information can be deleted by an administrator. If you have any questions, ask them at the Staff lounge. Thank you.--Az1568(Talk) 05:01, 20 January 2007 (UTC)

title: User talk:Brianluft


text:
Lisp Programming

Lisp is a programming language. It is named after the collapsed phrase List Processing.

If you have programmed before and would like to see a little bit of how Lisp works and is different from other programming languages, you can get an overview.

Dialects
Because Lisp itself is, technically, just seven operators, to become a useful language, much more needs to be implemented atop it. Common Lisp and Scheme are two such designs to create a useful programming language.

Common Lisp is an ANSI standard, and features an extensive array of library functions. It is the more widely used of the two.

Scheme is designed in a minimalistic fashion, with a very small amount of built in functions. This is probably true, but Scheme lacks many of the time-saving built-in functions of Common Lisp.

Emacs Lisp is an implementation of Lisp in Emacs.

newLISP is a Lisp-like, general-purpose scripting language

Contents

 * /Introduction/
 * /History/
 * /Overview/
 * /Beginning Lisp/
 * /Syntax/
 * /Features/
 * /Lisp and data/
 * /Functions and interfaces/
 * /Advanced Lisp/
 * /External Links/

title: Lisp Programming


text: Finnish/coverpage

title: Finnish/coverpage


text:
User:Tarquin~enwikibooks

Tarquin from Wikipedia.

Oh no. Another wikimedia project is sucking me in. HELP!!!! HELP!!!!!

title: User:Tarquin~enwikibooks


text:
Finnish/Aakkoset

^^Contents^^ | <>

In short, the basic alphabet of the Finnish language:

Finnish is a very phonetic language, as every pronunciation has its own letter. That is to say that things are "pronounced exactly as they are written" so SAMPA and IPA notations of Finnish words are almost identical to the written language. However, do not take this too literally; there are certainly many details in speech that cannot be easily expressed in written language, and Finnish is no exception.

The glyphs 'ä' and 'ö' have been borrowed from Swedish. They are independent letters and phonemes (sounds), not modified nor accented letters. Changing 'Ä' into 'A' or 'Ö' into 'O' is akin to changing 'Q' into 'O'.

 * Ä is similar to the following bolded sounds in English: "A fat pancake man sat on a cat."
 * Ö is similar to the following bolded sounds in British English: "Girls all over the world tried to figure out what all the words mean until a consensus was reached that it cannot be worked out."

Go through each of the letters of the alphabet and practice saying them:

title: Finnish/Aakkoset


text:
General Genetics

Contents

 * 1) /Introduction/
 * 2) /Basics of Heredity/
 * 3) /Genetic Principles/
 * 4) /Dominant and Recessive Genes/
 * 5) /Mendelian Inheritance/
 * 6) /Non-Mendelian Genetics/
 * 7) /Molecular Genetics/
 * 8) /Genomes/
 * 9) /Chromosomes/
 * 10) /The DNA Molecule/
 * 11) /The Discovery of the DNA Molecule/
 * 12) /Structure of the DNA Molecule/
 * 13) /DNA Replication/
 * 14) /DNA Replication in Prokaryotes: E. coli/
 * 15) /Gene Expression/
 * 16) /Gene Expression in Eukaryotes/
 * 17) /Gene Expression in Prokaryotes/
 * 18) /Transcription/
 * 19) /RNA processing/
 * 20) /Capping/
 * 21) /Splicing/
 * 22) /Polyadenylation/
 * 23) /Editing/
 * 24) /Nuclear Export/
 * 25) /RNA surveillance//
 * 26) /Translation/
 * 27) /Regulation of Gene Expression/
 * 28) /Gene Control of Proteins/
 * 29) /Recombinant DNA Cloning Technology/
 * 30) /Molecular Evolution/
 * 31) /Transposition/
 * 32) /Developmental Genetics/
 * 33) /Population Genetics/
 * 34) Hardy-Weinberg Equilibrium
 * 35) Fisher-Wright Model
 * 36) Neutral Theory of Molecular Evolution
 * 37) /Genetics, Ecology, and Modern Synthesis Theory/
 * 38) /Behavioral Genetics/
 * 39) /Quantitative Genetics/
 * 40) /Mutations and Evolution/
 * 41) /Diseases and Genes/
 * 42) /Linking Diseases with Associated Genes/
 * 43) /Genetic Predispositions for Diseases/
 * 44) Case studies:
 * 45) /Cystic Fibrosis/
 * 46) /Obesity and the Leptin Protein/
 * 47) /Cancer/
 * 48) /Introduction to the Genetics of Viruses/
 * 49) /Pharmacogenetics/
 * 50) /Ethical Issues/
 * 51) /Reproductive Cloning/
 * 52) /Cloning for Research/

Related books

 * General Biology/Genetics
 * Animal Behavior/Behavioral Genetics
 * Animal Behavior/Population Genetics
 * Handbook of Genetic Counseling
 * Genes, Technology and Policy
 * Genetic Information

title: General Genetics


text:
Polish/Plural

= Plural = < Order of words in a sentence < ^ Polish ^ > Noun cases >

Like most other Indo-European languages, and unlike Chinese or Japanese, in Polish, the grammatical number of nouns, pronouns, adjectives, and verbs can be either singular or plural (pojedyncza and mnoga).

Let's first examine how verbs change in the singular and plural forms.

Now, let's look at how some nouns change in the singular and plural forms:

As you can see, with the exception of person-masculine, the nominative and accusative are the same in the plural (in the person masculine accusatives are the same as genitives).

Like in other Indo-European languages, the verb agrees with the subject.
 * Basia ma chomiki. - Basia has (owns) hamsters.
 * Dziewczyny mają komputer. - The girls have a computer.
 * Koty piją mleko. - The cats are drinking milk.

< Order of words in a sentence < ^ Polish ^ > Noun cases >

title: Polish/Plural


text:
Polish/Noun cases

= Noun cases =

There are seven cases of noun and adjective declension in Polish. So far we have only introduced the nominative (mianownik) and the accusative (biernik), in Polish:Basic sentences 1. The whole list, in traditional order, is:
 * Nominative (mianownik)
 * Genitive (dopełniacz)
 * Dative (celownik)
 * Accusative (biernik)
 * Instrumental (narzędnik)
 * Locative (miejscownik)
 * Vocative (wołacz)

Nominative
No need to memorize endings as nouns and modifiers are in this case. Use the basic, unmarked, dictionary forms.

 * Indicates the subject of a sentence.
 * Gosia pisze książkę. - Gosia is writing a book. (Gosia in nominative because she is the subject of the sentence)

 * Used for most lone adjectives and sentences of the type "To jest X".
 * Piotr jest przystojny. - Piotr is handsome. (przystojny is in the nominative)
 * To jest moja żona. - This is my wife. (moja żona is in the nominative)

Genitive

 * Used to indicate possession. Often corresponds to English 's or of:
 * Adam widzi kota Agnieszki - Adam sees Agnieszka's cat (Agnieszka in genitive)
 * Kot Piotra pije mleko - Piotr's cat drinks milk (Piotr in genitive)

 * Used with units or amounts:
 * puszka piwa - a can of beer (beer in genitive)

 * Used after certain prepositions such as dla, do, od.

 * Used for the direct object in a negative sentence (Only if the positive sentence would use the accusative!):
 * Kot pije mleko - The cat drinks milk. (Milk in the accusative)
 * Kot nie pije mleka - The cat doesn't drink milk. (Milk in genitive)
 * Agnieszka jest dziewczyną - Agnieszka is a girl. (Girl in instrumental)
 * Adam nie jest dziewczyną - Adam is not a girl. (Girl remains in the instrumental)
 * Agnieszka daje kotu mleko - Agnieszka gives milk to the cat (Milk in the accusative, cat in the dative)
 * Agnieszka nie daje kotu mleka - Agnieszka doesn't give milk to the cat (Milk is now in the genitive, cat still in the dative).

Dative

 * Indicates the indirect object. To whom something is given, or action for somebody:
 * Basia daje kotu mleko - Basia gives milk to the cat. (Cat is in the dative.)
 * Znajdę Ci dom. - I'll find you a house (for you).

 * After certain verbs, like dziękować komuś (to thank someone), pomóc komuś (to help someone), wierzyć komuś (to believe someone)

 * expressions of the type:
 * Jest mi zimno - I'm cold
 * Jest nam smutno - We're feeling sad
 * Będzie wam trudniej... - It will be more difficult for you guys...
 * Śniło jej się, że... - She dreamt that...

 * when something is taken away or something befalls someone
 * Zdechł im pies - Their dog died
 * Wzięli mi komputer - They took away my computer
 * Zgasła mojemu sąsiadowi nadzieja - My neighbor's hopes were extinguished.

Accusative

 * Indicates the direct object of the sentence
 * Piotr kocha Kasię. - Piotr loves Kasia.
 * Bożena śpiewa piosenkę. - Bożena is singing a song.

 * Used with some prepositions to indicate a motion/change (Note that the same preposition + locative or, more rarely the instrumental, indicates a state/condition)
 * "na stole" - on the table (a state/condition, so you need to use the locative)
 * "na stół" - onto the table (motion, so accusative)
 * "na poczcie" - at the post office (state/condition so locative)
 * "na pocztę" - to the post office (motion so accusative)
 * "pod stołem" - under the table (a state/condition, pod is one of those prepostions which take the instrumental, not the locative)
 * "pod stół" - towards and under the table (motion so accusative)

Exception: The preposition "w" (in) takes the locative when indicating a state/condition, but to imply change you must use the preposition "do" + genitive.
 * "Jestem w sklepie" (I am in the shop), but "Idę do sklepu" (I am going to the shop) instead of "Idę w sklep".

Instrumental

 * Indicates the tool or means
 * Basia jedzie samochodem - Basia is driving a car. (The car is the tool or means so it needs to be in the instrumental)

 * A few prepositions, like "z" ("with") or "za":
 * Adam jest z Basią - Adam is with Basia (May either mean "in this moment", or more permanently, for example as boy- and girlfriend.)

 * With certain verbs "być" (to be), "zostać" (to become), "interesować się czymś/kimś" (to be interested in something):
 * Basia jest dziewczyną - Basia is a girl.
 * Adam jest chłopakiem - Adam is a boy.
 * Marzena interesuje się modą - Marzena is interested in fashion.
 * Piotr jest przystojnym chłopakiem. - Piotr is a handsome guy.

Exceptions: Most lone adjectives and "To jest X".
 * Piotr jest przystojny. - Piotr is handsome. (przystojny is in the nominative)
 * To jest moja żona. - This is my wife. (moja żona is in the nominative)

Locative

 * Never used alone; only in connection with certain prepositions, like "w" (in) and "o" (about).
 * Chłopak myśli o dziewczynie - The boy is thinking about the girl.
 * Adam jest w banku. - Adam is in the bank.

Vocative

 * Used to address someone directly

 * Often replaced by the nominative in colloquial language, however the vocative remains common in the following situations:
 * To address someone in a rude or offensive manner (A phrase like ty kretynie! ("you cretin!") or ty idioto ("you idiot!") can only be expressed in the vocative, never in the nominative).
 * In formal situations, Panie Kazimierzu, niech pan przyniesie te klucze.
 * In affectionate contexts, such as Kocham Cię Krzysiu! ("I love you, Chris!")

Only masculine and feminine singulars have a separate form.

Before declension tables are provided, let's take a look at adjectives:

< Polish pronunciation < ^ Polish ^

title: Polish/Noun cases


text:
Polish/Adjectives

= Adjectives =

The adjective (przymiotnik) is a significant part of speech in Polish. It declines very regularly depending on case, number and gender. It can also be used as a noun. For example Polish adjectives "bogaty" (rich) and "czarny" (black) are often used as "nouns" (taking forms "bogata" and "czarna" in feminine gender).

While there is only one pattern, the final consonant group takes different endings.

The vocative always has the same form as the nominative.

Fields, by colour:
 * White - basic form (feminine singular nominative)
 * Green – second form (with "i") in the case of adjectives ending in "ka", "ga" or "ia" in the base form, first form (without "i") otherwise
 * Purple – second form (with "i") in the case of adjectives ending in "ia" in the basic form, first form (without "i") otherwise
 * Red – final consonant group must be palatized (or "softened"), and "i" or "y" must be added, depending on result

Some softening rules are (compared to the singular masculine nominative):

See polski for sample "-ki" declension.

Adjective forms can also get comparative and superlative forms.

title: Polish/Adjectives


text:
Polish/Feminine noun declension

< Hard and soft consonants < ^ Polish ^ > Masculine noun declension >

= Feminine noun declension =

The declension of nouns in Polish is less regular than of adjectives, but follows a pattern that is in many ways similar to adjective declension.

Let's take a look at a few typical feminine declension nouns in singular and are equal to nominative in plural): As you can see it's quite regular. Possible changes are:

 * softening of final consonant group in the singular dative and locative (which have the same form)
 * change of "o" to "ó" (pronounced /u/) in the plural genitive
 * insertion of "e" between the two final consonants in the plural genitive. You may think of this "e" as a kind of aid to pronunciation. "mrówk" would be pretty hard to say without this "e". The Polish language allows quite complicated consonant groups at the beginning of a syllable, but it tries to avoid complex syllable endings.

These changes aren't specific to the feminine noun declension - they happen throughout the Polish language, so you'd better get used to them.

A bit less typical are feminine nouns that end in "-ia":

Notice that in the genitive and the accusative the pronunciation is the same in the singular and plural. This is not usual in Polish, and may cause some problems if number is not obvious from context. One solution is to overemphasise difference between "e" and "ę" in speech (which are usually pronounced the same at word endings). A better solution is to use some adjective or pronoun, for example:
 * "tej komedii" (singular genitive, of this comedy)
 * "tych komedii" (plural genitive, of these comedies)
 * "tę komedię" (singular accusative, this comedy)
 * "te komedie" (plural accusative, these comedies)

And abstract feminine nouns ending in "-ść" (note vocative forms): There is some magic here with "ść" changing to "śc" but it's only spelling. You never write the softened version of a consonant before a vowel - you change it to the "normal" version and add "i" to mark it as "soft". You don't have to add "i" if it's already there.

Without this magic endings would look like:

Let's try to use that knowledge in practice.
 * Agnieszka myśli o miłości - Agnieszka thinks about love (love in locative)
 * Agnieszka nie myśli o miłości - Agnieszka doesn't think about love (as above)
 * Dziewczyny lubią komedie - The girls like comedies (comedies in accusative)
 * Kobieta pije kawę - The woman is drinking coffee. (coffee in accusative)
 * Dziewczyna nie pije kawy - The girl doesn't drink coffee (coffee in genitive)
 * Basia nie widzi krów - Basia doesn't see cows (cows in genitive)
 * Marta nie lubi mrówek - Marta doesn't like ants (ants in genitive)

 * myśli - third person singular form of "to think"
 * lubi - third person singular form of "to like"
 * lubią - third person plural form of "to like"

< Hard and soft consonants < ^ Polish ^ > Masculine noun declension >

title: Polish/Feminine noun declension


text:
Polish/Masculine noun declension

< Feminine noun declension < ^ Polish ^ > Neuter noun declension >

= Masculine noun declension =

There are three masculine genders in Polish – masculine personal, masculine animate and masculine inanimate. The differences between them are regular:

 * In the case of masculine personal, the accusative has the same form as the genitive
 * In the case of masculine animate, the accusative has the same form as the genitive in the singular, but as the nominative in the plural
 * In the case of masculine inanimate, the accusative has the same form as the nominative
 * Vocative singular is always the same as locative singular and vocative plural is the same as nominative plural.

But there are also some other differences in declension. Before they get explained, let's take a look at some examples.

Phonetic changes and irregularities are:

 * All "e" (or "ie") and "o"/"ó" changes in the singular nominative, and also the accusative if it has the same form as the nominative.
 * The plural nominative isn't very regular. In the case of some nouns softening of the final consonant may be required. Some nouns even have two correct plural nominative forms – one with "owie" and another in a soft "e".
 * Some nouns have a singular locative in "u", other have in "e", and in the case of the latter the final consonant group must be softened.
 * Very few masculine nouns have a singular dative in "u" instead in "owi". However, they are really common words - Pan (literally lord, nobleman, master - a very important word because it is used exactly as the English pronoun "You" to express politeness e.g "Podać Panu kawę? Could you take some coffee, sir?) brat (brother), kot (cat), pies (dog) and Bóg (God). As you see, apart from unusual dative ending in all of them declension is connected with unpleasant stem alternations, resulting from various historical processes e.g. consonant cluster softening, compensatory lengthening, development of Slavonic semi-vowels (jery)
 * Some inanimate-masculine nouns have a singular genitive in "u", most others in "a".

Now some sentences:

 * Adam jest rolnikiem - Adam is a farmer. (farmer in instrumental)
 * Facet daje kotu mleko - The guy gives milk to the cat. (cat in dative)
 * Chłopak nie ma psa - The boy doesn't have a dog. (dog in genitive)
 * Pies jest pod stołem - The dog is under the table. (table in instrumental)

< Feminine noun declension < ^ Polish ^ > Neuter noun declension >

title: Polish/Masculine noun declension


text:
Polish/Neuter noun declension

= Neuter noun declension =

The neuter gender in Polish is least common. Many neuter words end in "-o" and their declension is somewhat similar to inanimate-masculine.

The archaic instrumental oczyma doesn't have much to do with the neuter gender - it was the Indoeuropean dual number (for a pair of things), which disappeared in almost all modern Indo-European languages. You can still find it in some expressions.

In the singular locative "krześle" you can see how softening works on a group of consonants – all consonants are softened, in that case "s" changes to "ś" and "ł" to "l".

As with other declensions, "e" may appear between two final consonants if there is no vowel after them ("null" ending).

The plural instrumental is "-ami"/"-iami" in almost all nouns, and "-imi"/"-ymi" in almost all adjectives. Here you can see one of the exceptions. For some historical reasons "a" disappeared and we have "-ćmi" instead of the expected "-ciami".

In the case of "dziecko", "oko" and "ucho" you can see that a slightly different form is used as the base for the singular and plural forms. This is also quite archaic.

If you find some neuter noun, try to decline it rather like "piwo", "krzesło" and "ciastko" rather than like "dziecko", "oko" or "ucho".

Examples:
 * Dziecko je ciastka - child eats cakes (cake in accusative)
 * Dzieci nie piją piwa - children don't drink beer (beer in genitive)
 * Oczy Piotra są zielone - Piotr's eyes are green (eyes in nominative, Piotr in genitive, green in nominative)
 * Kot jest na krześle - cat is on chair (chair in locative)

 * zielone - plural neuter form of adjective "green"

Neuter nouns ending in "-e" or "-ę"
The most common template is: Other templates are:

title: Polish/Neuter noun declension


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Polish/More on adjectives

= More on Adjectives =

Adjectives in Polish can be used in various ways, including:
 * In constructs "X is Y"
 * As nouns
 * As descriptors for nouns, usually placed before nouns

The English construct "X is Y" has multiple meanings. If Y is a noun, it typically implies that X is a member of a certain class, for example, "Kasia jest dziewczyną" (Kasia is a girl). In this case, Y should be used in the instrumental case. However, if Y is an adjective, it implies that Y has a certain feature. For instance, "Kot jest czarny" (the cat is black) doesn't mean that the cat is a member of the class of black things. To distinguish this situation, the nominative is used for Y. In the case of adjectives that can be used as nouns, both cases may be used, but the nominative is much more common.

Here is the conjugation of the verb "to be":

Examples:
 * Agnieszka jest ładna - Agnieszka is pretty
 * Kot Basi jest czarny - Basia's cat is black
 * Krowy rolnika są białe - The farmer's cows are white
 * Jestem wysoki - I'm tall
 * Jesteś gruby - You're fat (singular you)
 * Jesteśmy młodzi - We're young
 * Jesteście bogaci - You're rich (plural you)

Here are some adjectives and their antonyms, in the feminine singular nominative:
 * ładna - pretty
 * brzydka - ugly
 * czarna - black
 * biała - white
 * wysoka - tall/high
 * niska - short (height)
 * gruba - thick/fat
 * chuda - thin
 * młoda - young
 * stara - old
 * bogata - rich
 * biedna - poor

Here are some examples with adjectives used as nouns:
 * Czarni lubią muzykę - Black people like music (the fact that it's black people discussed must be found out from context)
 * Basia lubi czarną - Basia likes black coffee (of course it could have also been in the context of "pornography")
 * Bogaci lubią golf - Rich people like golf

If an adjective describes a noun, it has to be used in the same case, number, and gender as the noun. Here are some examples:
 * Biały kot pije mleko - A white cat drinks milk (white in nominative)
 * Ładna dziewczyna ma białego kota - The pretty girl has a white cat (pretty in nominative, white in accusative)
 * Gruby chłopak nie ma czarnego kota - The fat boy doesn't have a black cat (fat in nominative, black in genitive)
 * Czarne oczy Marty są ładne - Marta's black eyes are pretty (black and pretty in nominative)
 * Dorota jest stara - Dorota is old (old in nominative)
 * Dorota jest starą kobietą - Dorota is an old woman (old in instrumental)

You can also stack adjectives:
 * Basia ma grubego czarnego kota - Basia has a fat black cat (fat and black in accusative)

title: Polish/More on adjectives


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Polish/Grouping expressions

We often want to group nouns or sentences, to say "X and Y" or "X or Y".

In Polish the commonly used word for "and" is "i":
 * Marta i Basia mają koty - Marta and Basia have cats
 * Kobieta je ciastko i pije kawę - The woman is eating a cookie and drinking coffee
 * Kot jest gruby i czarny - The cat is fat and black
 * Adam lubi Martę i Basię - Adam likes Marta and Basia

The word "oraz" may also be used to express English "and": It sounds a little more sophisticated. However, you shouldn't use oraz inbetween adjectives - "Kot jest gruby oraz czarny" doesn't sound very natural. When in doubt, say "i".
 * Basia ma kota oraz psa - Basia has a cat and a dog.

You can also use "z" + instrumental (with) and "bez" (without) + genitive
 * Magda z Tomkiem są małżeństwem - Magda and Tomek are married (Tomek in instrumental)
 * Magda i Tomek są małżeństwem - Same.
 * Dziecko lubi kanapki z serem - The child likes sandwiches with cheese (cheese in instrumental) - meaning "The child likes cheese sandwiches"
 * Dziecko lubi kanapki i ser - The child likes sandwiches and cheese - this one meaning "The child likes both sandwiches and cheese"
 * Kobieta lubi kawę bez cukru - The woman likes her coffee without sugar (sugar in genitive) - we don't say "her coffee" in Polish, just "coffee"

 * małżeństwo - married couple
 * kanapka - a sandwich
 * ser - cheese
 * cukier - sugar
 * Tomek - popular male name, diminuitive of Tomasz - the English equivalent is Thomas

When you group a few nouns, the result will always be plural, and if any of them was person-masculine, so will the whole expression be.

There are many words for "or" - we mostly use "albo" or "lub" (the same meaning), but "czy" (which usually has a different meaning) may be used for this purpose too, especially in questions.

 * Masz kota albo psa? - "Do you have a cat or dog ?" Proper answers are "tak" ("yes", if you have either or even both) or "nie" ("no", if you have neither)
 * Masz kota czy psa? - "Do you have a cat, or a dog?", meaning "Which do you have, cat or dog?". Proper answers are "kota" ("a cat", if you have a cat but not a dog), or "psa" ("a dog", if you have a dog but not a cat). If you have both or neither some more complex answer will be needed. But "tak"/"nie" ("yes"/"no") will leave the other party rather confused.
 * Marta albo Basia ma kota - Marta or Basia has a cat
 * Adam ma kota albo psa - Adam has a cat or a dog
 * Dziecko je ciasto albo pije mleko - The child is eating cake or drinking milk.

^ Polish ^ > Polish pronunciation >

title: Polish/Grouping expressions