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Discrete Mathematics/Functions and relations

Introduction
This article examines the concepts of a function and a relation.

A relation is any association or link between elements of one set, called the domain or (less formally) the set of inputs, and another set, called the range or set of outputs. Some people mistakenly refer to the range as the codomain(range), but as we will see, that really means the set of all possible outputs—even values that the relation does not actually use. (Beware: some authors do not use the term codomain(range), and use the term range instead for this purpose. Those authors use the term image for what we are calling range.  So while it is a mistake to refer to the range or image as the codomain(range), it is not necessarily a mistake to refer to codomain as range.)

For example, if the domain is a set Fruits = {apples, oranges, bananas} and the codomain(range) is a set Flavors = {sweetness, tartness, bitterness}, the flavors of these fruits form a relation: we might say that apples are related to (or associated with) both sweetness and tartness, while oranges are related to tartness only and bananas to sweetness only. (We might disagree somewhat, but that is irrelevant to the topic of this book.) Notice that "bitterness", although it is one of the possible Flavors (codomain)(range), is not really used for any of these relationships; so it is not part of the range (or image) {sweetness, tartness}.

Another way of looking at this is to say that a relation is a subset of ordered pairs drawn from the set of all possible ordered pairs (of elements of two other sets, which we normally refer to as the Cartesian product of those sets). Formally, R is a relation if

 * $$R \subseteq X \times Y = \{(x, y) | x \in X, y \in Y\}$$

for the domain X and codomain(range) Y. The inverse relation of R, which is written as R-1, is what we get when we interchange the X and Y values:

 * $$R^{-1} = \{(y, x) | (x, y) \in R\}$$

Using the example above, we can write the relation in set notation: {(apples, sweetness), (apples, tartness), (oranges, tartness), (bananas, sweetness)}. The inverse relation, which we could describe as "fruits of a given flavor", is {(sweetness, apples), (sweetness, bananas), (tartness, apples), (tartness, oranges)}. (Here, as elsewhere, the order of elements in a set has no significance.)

One important kind of relation is the function. A function is a relation that has exactly one output for every possible input in the domain. (The domain does not necessarily have to include all possible objects of a given type. In fact, we sometimes intentionally use a restricted domain in order to satisfy some desirable property.)  The relations discussed above (flavors of fruits and fruits of a given flavor) are not functions: the first has two possible outputs for the input "apples" (sweetness and tartness); and the second has two outputs for both "sweetness" (apples and bananas) and "tartness" (apples and oranges).

The main reason for not allowing multiple outputs with the same input is that it lets us apply the same function to different forms of the same thing without changing their equivalence. That is, if f is a function with a (or b) in its domain, then a = b implies that f(a) = f(b). For example, z - 3 = 5 implies that z = 8 because f(x) = x + 3 is a function unambiguously defined for all numbers x.

The converse, that f(a) = f(b) implies a = b, is not always true. When it is, there is never more than one input x for a certain output y = f(x). This is the same as the definition of function, but with the roles of X and Y interchanged; so it means the inverse relation f-1 must also be a function. In general—regardless of whether or not the original relation was a function—the inverse relation will sometimes be a function, and sometimes not.

When f and f-1 are both functions, they are called one-to-one, injective, or invertible functions. This is one of two very important properties a function f might (or might not) have; the other property is called onto or surjective, which means, for any y ∈ Y (in the codomain), there is some x ∈ X (in the domain) such that f(x) = y. In other words, a surjective function f maps onto every possible output at least once.

A function can be neither one-to-one nor onto, both one-to-one and onto (in which case it is also called bijective or a one-to-one correspondence), or just one and not the other. (As an example which is neither, consider f = {(0,2), (1,2)}. It is a function, since there is only one y value for each x value; but there is more than one input x for the output y = 2; and it clearly does not "map onto" all integers.)

Relations
In the above section dealing with functions and their properties, we noted the important property that all functions must have, namely that if a function does map a value from its domain to its co-domain, it must map this value to only one value in the co-domain.

Writing in set notation, if a is some fixed value:
 * |{f(x)|x=a}| ∈ {0, 1}

The literal reading of this statement is: the cardinality (number of elements) of the set of all values f(x), such that x=a for some fixed value a, is an element of the set {0, 1}. In other words, the number of outputs that a function f may have at any fixed input a is either zero (in which case it is undefined at that input) or one (in which case the output is unique).

However, when we consider the relation, we relax this constriction, and so a relation may map one value to more than one other value. In general, a relation is any subset of the Cartesian product of its domain and co-domain.

All functions, then, can be considered as relations also.

Notations
When we have the property that one value is related to another, we call this relation a binary relation and we write it as
 * x R y

where R is the relation.

For arrow diagrams and set notations, remember for relations we do not have the restriction that functions do and we can draw an arrow to represent the mappings, and for a set diagram, we need only write all the ordered pairs that the relation does take: again, by example
 * f = {(0,0),(1,1),(-1,1),(2,2),(-2,2)}

is a relation and not a function, since both 1 and 2 are mapped to two values, (1 and -1, and 2 and -2 respectively) example let A=2,3,5;B=4,6,9 then A*B=(2,4),(2,6),(2,9),(3,4),(3,6),(3,9),(5,4),(5,6),(5,9) Define a relation R=(2,4),(2,6),(3,6),(3,9) add functions and problems to one another.

Some simple examples
Let us examine some simple relations.

Say f is defined by
 * {(0,0),(1,1),(2,2),(3,3),(1,2),(2,3),(3,1),(2,1),(3,2),(1,3)}

This is a relation (not a function) since we can observe that 1 maps to 2 and 3, for instance.

Less-than, "<", is a relation also. Many numbers can be less than some other fixed number, so it cannot be a function.

Properties
When we are looking at relations, we can observe some special properties different relations can have.

Reflexive
A relation is reflexive if, we observe that for all values a:
 * a R a

In other words, all values are related to themselves.

The relation of equality, "=" is reflexive. Observe that for, say, all numbers a (the domain is R):
 * a = a

so "=" is reflexive.

In a reflexive relation, we have arrows for all values in the domain pointing back to themselves:
 * [[Image:Arrow diagram reflexive.png]]

Note that ≤ is also reflexive (a ≤ a for any a in R). On the other hand, the relation < is not (a < a is false for any a in R).

Symmetric
A relation is symmetric if, we observe that for all values of a and b:
 * a R b implies b R a

The relation of equality again is symmetric. If x=y, we can also write that y=x also.

In a symmetric relation, for each arrow we have also an opposite arrow, i.e. there is either no arrow between x and y, or an arrow points from x to y and an arrow back from y to x:
 * [[Image:Arrow diagram symmetric.png]]

Neither ≤ nor < is symmetric (2 ≤ 3 and 2 < 3 but neither 3 ≤ 2 nor 3 < 2 is true).

Transitive
A relation is transitive if for all values a, b, c:
 * a R b and b R c implies a R c

The relation greater-than ">" is transitive. If x > y, and y > z, then it is true that x > z. This becomes clearer when we write down what is happening into words. x is greater than y and y is greater than z. So x is greater than both y and z.

The relation is-not-equal "≠" is not transitive. If x ≠ y and y ≠ z then we might have x = z or x ≠ z (for example 1 ≠ 2 and 2 ≠ 3 and 1 ≠ 3 but 0 ≠ 1 and 1 ≠ 0 and 0 = 0).

In the arrow diagram, every arrow between two values a and b, and b and c, has an arrow going straight from a to c.
 * [[Image:Arrow diagram transitive.png]]

Antisymmetric
A relation is antisymmetric if we observe that for all values a and b:
 * a R b and b R a implies that a=b

Notice that antisymmetric is not the same as "not symmetric."

Take the relation greater than or equal to, "≥" If x ≥ y, and y ≥ x, then y must be equal to x. a relation is anti-symmetric if and only if a∈A, (a,a)∈R

Trichotomy
A relation satisfies trichotomy if we observe that for all values a and b it holds true that: aRb or bRa

The relation is-greater-or-equal satisfies since, given 2 real numbers a and b, it is true that whether a ≥ b or b ≥ a (both if a = b).

Problem set
Given the above information, determine which relations are reflexive, transitive, symmetric, or antisymmetric on the following - there may be more than one characteristic. (Answers follow.) x R y if
 * 1) x = y
 * 2) x < y
 * 3) x2 = y2
 * 4) x ≤ y

Answers

 * 1) Symmetric, Reflexive, and transitive
 * 2) Transitive, Trichotomy
 * 3) Symmetric, Reflexive, and transitive (x2 = y2 is just a special case of equality, so all properties that apply to x = y also apply to this case)
 * 4) Reflexive, Transitive and Antisymmetric (and satisfying Trichotomy)

Equivalence relations
We have seen that certain common relations such as "=", and congruence (which we will deal with in the next section) obey some of these rules above. The relations we will deal with are very important in discrete mathematics, and are known as equivalence relations. They essentially assert some kind of equality notion, or equivalence, hence the name.

Characteristics of equivalence relations
For a relation R to be an equivalence relation, it must have the following properties, viz. R must be: (A helpful mnemonic, S-T-R)
 * symmetric
 * transitive
 * reflexive

In the previous problem set you have shown equality, "=", to be reflexive, symmetric, and transitive. So "=" is an equivalence relation.

We denote an equivalence relation, in general, by $$x \sim y$$.

Example proof
Say we are asked to prove that "=" is an equivalence relation. We then proceed to prove each property above in turn (Often, the proof of transitivity is the hardest).

 * Reflexive: Clearly, it is true that a = a for all values a. Therefore, = is reflexive.
 * Symmetric: If a = b, it is also true that b = a. Therefore, = is symmetric
 * Transitive: If a = b and b = c, this says that a is the same as b which in turn is the same as c. So a is then the same as c, so a  = c, and thus = is transitive.

Thus = is an equivalence relation.

Partitions and equivalence classes
It is true that when we are dealing with relations, we may find that many values are related to one fixed value.

For example, when we look at the quality of congruence, which is that given some number a, a number congruent to a is one that has the same remainder or modulus when divided by some number n, as a, which we write
 * a ≡ b (mod n)

and is the same as writing
 * b = a+kn for some integer k.

(We will look into congruences in further detail later, but a simple examination or understanding of this idea will be interesting in its application to equivalence relations)

For example, 2 ≡ 0 (mod 2), since the remainder on dividing 2 by 2 is in fact 0, as is the remainder on dividing 0 by 2.

We can show that congruence is an equivalence relation (This is left as an exercise, below Hint use the equivalent form of congruence as described above).

However, what is more interesting is that we can group all numbers that are equivalent to each other.

With the relation congruence modulo 2 (which is using n=2, as above), or more formally:
 * x ~ y if and only if x ≡ y (mod 2)

we can group all numbers that are equivalent to each other. Observe:
 * $$0 \equiv 2 \equiv 4 \equiv \ldots \pmod{2}$$
 * $$1 \equiv 3 \equiv 5 \equiv \ldots \pmod{2}$$

This first equation above tells us all the even numbers are equivalent to each other under ~, and all the odd numbers under ~.

We can write this in set notation. However, we have a special notation. We write:
 * [0]={0,2,4,...}
 * [1]={1,3,5,...}

and we call these two sets equivalence classes.

All elements in an equivalence class by definition are equivalent to each other, and thus note that we do not need to include [2], since 2 ~ 0.

We call the act of doing this 'grouping' with respect to some equivalence relation partitioning (or further and explicitly partitioning a set S into equivalence classes under a relation ~). Above, we have partitioned Z into equivalence classes [0] and [1], under the relation of congruence modulo 2.

Problem set
Given the above, answer the following questions on equivalence relations (Answers follow to even numbered questions) $$ x \sim y\ \mbox{iff}\ x \equiv y \pmod{6}$$
 * 1) Prove that congruence is an equivalence relation as before (See hint above).
 * 2) Partition {x | 1 ≤ x ≤ 9} into equivalence classes under the equivalence relation

Answers
2. [0]={6}, [1]={1,7}, [2]={2,8}, [3]={3,9}, [4]={4}, [5]={5}

Partial orders
We also see that "≥" and "≤" obey some of the rules above. Are these special kinds of relations too, like equivalence relations? Yes, in fact, these relations are specific examples of another special kind of relation which we will describe in this section: the partial order.

As the name suggests, this relation gives some kind of ordering to numbers.

Characteristics of partial orders
For a relation R to be a partial order, it must have the following three properties, viz R must be: (A helpful mnemonic, R-A-T)
 * reflexive
 * antisymmetric
 * transitive

We denote a partial order, in general, by $$x \preceq y$$.

Question:
 * 1) Suppose R is a relation on a set of integers Z then prove that R is a partial order relation on Z iff a=b raise to power r.

Example proof
Say we are asked to prove that "≤" is a partial order. We then proceed to prove each property above in turn (Often, the proof of transitivity is the hardest).

Reflexive
Clearly, it is true that a ≤ a for all values a. So ≤ is reflexive.

Antisymmetric
If a ≤ b, and b ≤ a, then a must be equal to b. So ≤ is antisymmetric

Transitive
If a ≤ b and b ≤ c, this says that a is less than b and c. So a is less than c, so a ≤ c, and thus ≤ is transitive.

Thus ≤ is a partial order.

Problem set
Given the above on partial orders, answer the following questions
 * 1) Prove that divisibility, |, is a partial order (a | b means that a is a factor of b, i.e., on dividing b by a, no remainder results).
 * 2)  Prove the following set is a partial order: (a, b) $$\preceq$$ (c, d) implies ab≤cd for a,b,c,d integers ranging from 0 to 5.

Answers
2. Simple proof; Formalization of the proof is an optional exercise.
 * Reflexivity: (a, b) $$\preceq$$ (a, b) since ab=ab.
 * Antisymmetric: (a, b) $$\preceq$$ (c, d) and (c, d) $$\preceq$$ (a, b) since ab≤cd and cd≤ab imply ab=cd.
 * Transitive: (a, b) $$\preceq$$ (c, d) and (c, d) $$\preceq$$ (e, f) implies (a, b) $$\preceq$$ (e, f) since ab≤cd≤ef and thus ab≤ef

Posets
A partial order imparts some kind of "ordering" amongst elements of a set. For example, we only know that 2 ≥ 1 because of the partial ordering ≥.

We call a set A, ordered under a general partial ordering $$\preceq$$, a partially ordered set, or simply just poset, and write it (A, $$\preceq$$).

Terminology
There is some specific terminology that will help us understand and visualize the partial orders.

When we have a partial order $$\preceq$$, such that a $$\preceq$$ b, we write $$\prec$$ to say that a $$\preceq$$ but a ≠ b. We say in this instance that a precedes b, or a is a predecessor of b.

If (A, $$\preceq$$) is a poset, we say that a is an immediate predecessor of b (or a immediately precedes b) if there is no x in A such that a $$\prec$$ x $$\prec$$ b.

If we have the same poset, and we also have a and b in A, then we say a and b are comparable if a $$\preceq$$ b or b $$\preceq$$ a. Otherwise they are incomparable.

Hasse diagrams
Hasse diagrams are special diagrams that enable us to visualize the structure of a partial ordering. They use some of the concepts in the previous section to draw the diagram.

A Hasse diagram of the poset (A, $$\preceq$$) is constructed by
 * placing elements of A as points
 * if a and b ∈ A, and a is an immediate predecessor of b, we draw a line from a to b
 * if a $$\prec$$ b, put the point for a lower than the point for b
 * not drawing loops from a to a (this is assumed in a partial order because of reflexivity)

Operations on Relations
There are some useful operations one can perform on relations, which allow to express some of the above mentioned properties more briefly.

Inversion
Let R be a relation, then its inversion, R-1 is defined by

R-1 := {(a,b) | (b,a) in R}.

Concatenation
Let R be a relation between the sets A and B, S be a relation between B and C. We can concatenate these relations by defining

R • S := {(a,c) | (a,b) in R and (b,c) in S for some b out of B}

Diagonal of a Set
Let A be a set, then we define the diagonal (D) of A by

D(A) := {(a,a) | a in A}

Shorter Notations
Using above definitions, one can say (lets assume R is a relation between A and B):

R is transitive if and only if R • R is a subset of R.

R is reflexive if and only if D(A) is a subset of R.

R is symmetric if R-1 is a subset of R.

R is antisymmetric if and only if the intersection of R and R-1 is D(A).

R is asymmetric if and only if the intersection of D(A) and R is empty.

R is a function if and only if R-1 • R is a subset of D(B).

In this case it is a function A → B. Let's assume R meets the condition of being a function, then

R is injective if R • R-1 is a subset of D(A).

R is surjective if {b | (a,b) in R} = B.

Functions
A function is a relationship between two sets of numbers. We may think of this as a mapping; a function maps a number in one set to a number in another set. Notice that a function maps values to one and only one value. Two values in one set could map to one value, but one value must never map to two values: that would be a relation, not a function.

For example, if we write (define) a function as:
 * $$f(x)=x^2$$

then we say:
 * 'f of x equals x squared'

and we have
 * $$f(-1)=1$$
 * $$f(1)=1$$
 * $$f(7)=49$$
 * $$f(1/2)=1/4$$
 * $$f(4)=16$$

and so on.

This function f maps numbers to their squares.

Range and codomain
If D is a set, we can say
 * $$f(D) = \{f(x) |\ x \in D\}$$

which forms a née of f is usually a subset of a larger set. This set is known as the codomain of a function. For example, with the function f(x)=cos x, the range of f is [-1,1], but the codomain is the set of real numbers.

Notations
When we have a function f, with domain D and range R, we write:
 * $$f : D \longrightarrow R$$

If we say that, for instance, x is mapped to x2, we also can add
 * $$f : D \longrightarrow R;\ x \longmapsto x^2$$

Notice that we can have a function that maps a point (x,y) to a real number, or some other function of two variables -- we have a set of ordered pairs as the domain. Recall from set theory that this is defined by the Cartesian product - if we wish to represent a set of all real-valued ordered pairs we can take the Cartesian product of the real numbers with itself to obtain
 * $$\mathbb{R}\times\mathbb{R}=\mathbb{R}^2=\{(x,y)|\ x\ \mbox{and}\ y \in\mathbb{R}\}$$.

When we have a set of n-tuples as part of the domain, we say that the function is n-ary (for numbers n=1,2 we say unary, and binary respectively).

Other function notation
Functions can be written as above, but we can also write them in two other ways. One way is to use an arrow diagram to represent the mappings between each element. We write the elements from the domain on one side, and the elements from the range on the other, and we draw arrows to show that an element from the domain is mapped to the range.

For example, for the function f(x)=x3, the arrow diagram for the domain {1,2,3} would be:

Another way is to use set notation. If f(x)=y, we can write the function in terms of its mappings. This idea is best to show in an example.

Let us take the domain D={1,2,3}, and f(x)=x2. Then, the range of f will be R={f(1),f(2),f(3)}={1,4,9}. Taking the Cartesian product of D and R we obtain F={(1,1),(2,4),(3,9)}.

So using set notation, a function can be expressed as the Cartesian product of its domain and range. f(x) This function is called f, and it takes a variable x. We substitute some value for x to get the second value, which is what the function maps x to.

Types of functions
Functions can either be one to one (injective), onto (surjective), or bijective.  INJECTIVE Functions  are functions in which every element in the domain maps into a unique elements in the codomain.

 SURJECTIVE Functions  are functions in which every element in the codomain is mapped by an element in the domain.

 'BIJECTIVE Functions  are functions that are both injective and surjective.

---onto functions   a function f form A to B is onto ,


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CACS/Glossary/Aldus

Aldus (Company) a former software vendor, noted for the developement of Pagemaker, which was sold to Adobe in 1994.

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

title: CACS/Glossary/Aldus


text:
CACS/Glossary/ANI

Animated Cursor – (software graphics) a feature that modifies the displayed cursor based on the state of the component the cursor is located in or the properties of a control that the cursor points to. State variation usually alters the cursor when a wait state exists. Property variation identifies controls or text that are accessible or available for end-user manipulation.

(Microsoft Windows) The file extension .ANI is used for the graphics and conditions that control the display.

title: CACS/Glossary/ANI


text:
CACS/Glossary/Apache

Apache (Internet Software) A free, open source Web Server HTTP processing software package from the Apache Software Foundation. Apache runs on OS/2, NetWare, Unix, and Windows NT. Version 2.0 dates from May 2002. The software source code may be downloaded from the Apache Website, which also contains Apache Documentation. For Apache related questions or issues, you can search this large unofficial Apache Forum hosted by Nabble which currently archives all the Apache projects' mailing lists for cross search and browsing. You can also post your question to the appropriate sub forum which will then forward your post to the corresponding Apache mailing list.

title: CACS/Glossary/Apache


text:
CACS/Glossary/ASF

Apache Software Foundation (Organization) A collaborative non-profit working group dedicated to the education, development and delivery of open source web software. Its Web address is the Apache Foundation.

title: CACS/Glossary/ASF


text:
CACS/Glossary/Application

An Application is: Typically, the application uses a single set of libraries and is considered a single project.
 * 1) A bundle of computer software sold, installed, or considered as a whole.
 * 2) (PC Programming) A single program or suite of program modules that may be executed from the desktop.
 * 3) (IDE) An integrated and interrelated set of program modules developed as a whole.

title: CACS/Glossary/Application


text:
CACS/Glossary/Assembler

An Assembler is:
 * 1) a software program that creates an executable module from assembly source language source code.
 * 2) a software component that creates a machine level executable program module from an interim work product produced by a higher level compiler.
 * 3) a shorthand reference for Assembly Language Code.

title: CACS/Glossary/Assembler


text:
CACS/Glossary/Assembly

Assembly is the process of creating an executable module from assembly language source code. Sometimes assembly is used as a shorthand reference for Assembly Language Code.

title: CACS/Glossary/Assembly


text:
CACS/Glossary/ASML

Assembly Language is the lowest level of symbolic progarmming language characterized by the use of a single statement for each machine instruction. The language also permits the use of labels for routines, variables, and storage assignments. There is a different language for each computer CPU. Some versions of Assembly Language support the use of Macro Instructions.

(PC Platforms) Assembley Language source code is usually stored with a file extension of .ASM.

title: CACS/Glossary/ASML


text:
Talk:CACS/Glossary

This page may eventually need to be broken up to keep its size manageable. If this occurs, then it should have a set of pages that use the form: CACS:Glossary/A. That page would get all the terms that start with A, and the index to individual terms would be moved. We should be able to do this without moving the individual pages (except maybe those like the C Programming Language).

title: Talk:CACS/Glossary


text:
Esperanto/Contents

The Wikibooks Esperanto Textbook is a collaborative project to create an online open-content textbook for Esperanto, which we hope will become the definitive Esperanto textbook for English speakers.

Contents

 * An introduction to Esperanto
 * The quick-and-dirty guide to Esperanto

Lessons

 * 1) Introducing Yourself - Grammatical Introduction: nouns, adjectives, adverbs, personal and possessive pronouns, the present tense, direct and indirect objects
 * 2) Traveling - Verb conjugation for active verbs and the forms of all six participles
 * 3) Recreational Activities - Correlatives and participle synthesis
 * 4) Family
 * 5) School - More complex verbs formed with participles
 * 6) Word Synthesis
 * 7) Electronic Communication
 * 8) Typesetting Esperanto Texts

Appendices

 * Alphabet and pronunciation
 * Original 1889 English-Esperanto word list
 * English-Esperanto word list
 * Esperanto-English word list
 * Abbreviations
 * Summary of grammar
 * Web resources
 * Table of correlatives
 * Table of word endings
 * Table of affixes
 * False friends
 * Slang
 * Common phrases
 * Starting vocabulary list

About the book

 * Authors

title: Esperanto/Contents


text:
CACS/User Conundrum

Wiki Note

''This note will be removed later. The entire essay may become a secion within an Introduction Chapter. Or, it might remain here and just have an assigned reading on the introduction. Please use the Talk page for this entry for additional discussion. Thanks. LouI''

=The Myth of the User= There are several terms widely used in the computer industry that are not clearly defined. The result is that everyone understands them and uses them, but the use may tend to complicate rather than illuminate a discussion. This outline looks just two of these terms in the context of an application within a fairly large corporation. Then it draws some conclusions about the impact of context on the use of terminology.

Consider the terms User and Application. The User is defined as the person who uses a computer. The Platform is the hardware or software that supports an application or a system. The suggested application is one of monitoring and use of a particular resource pool across an area of the company.

The Application
Project and unit managers share a common resource pool. They budget and track the use of the pool in their area. They use a series of spreadsheets that show their planned vs. actual use of many resources. The data is extracted for their status reports. The data is also uploaded through the company's networks for use in other summaries of the effective use of the resource pool. These support the manager and planning staff that are responsible for the pool.

To make this work, a set of spreadsheet macros are in place to ensure data validity and consistency. These templates are put in place by the corporate IT staff. The end-user needs to record their work results, using the spreadsheets by an arbitrary cutoff time on Friday.

The Software Hierarchy
Now consider the hierarchy of software engineering that puts this application in place. The example could be extended further. The extensions in Step 3 could be produced by an outside Value Added Reseller to produce a resource planning system. Step 1 could look into the earlier history in Hardware for PC Configuration, BIOS, Operating Systems, The C++ compiler, etc. But these 5 layers are quite complex enough to illustrate the terminology problems.
 * 1) The Spreadsheet application has a programming team that work for the software vendor. They assume an Intel Pentium running Windows NT as their platform. They produce a set of software that integrates the work of the Visual Basic for Applications team with the next release of the operating system software and their own code (written in C++) to produce a working application.
 * 2) The vendors packaging team puts this together with the other desktop embedded applications to produce an integrated version or release of the composite operating system.
 * 3) The corporate IT department takes this software, along with the company standard network, Operating System,, and desktop applications as their platform. They add the VBA macros with  upload and reporting utilities that will make the aggregate set of applications work. Then they lock the results to ensure that no bad input comes from the end-users. They deliver their work to the system administrators.
 * 4) The system administrators in each division install the results. Some of these are on department servers, and some on individual desktops. The entire assembled application is their platform, and the desktops of the affected staff the target domain for their work.
 * 5) The individual manager now uses the entire work product as a seamless platform to do their work.

''For those who are have never worked in this situation, this is NOT an uncommon example. Millions of people work in this environment every day at your bank, your university, or the manufacturer of your automobile.''

The Confusion
The above example is fine when it works. But, what about when it fails? As the end-user, I've got a problem in my spreadsheet, and an error message. I find something from the 1st level, the User's Guide, and try to find a solution. But it is a hopeless quest. I can't use most of the options they discuss. I can't even inspect the VBA code that the guide says my error message comes from. The usually helpful System Admin who put this new update on my desk is equally puzzled, and other users haven't reported a problem. So, what went wrong?

Every layer in the hierarchy has their own viewpoint. Everything in the layers above them is viewed as a Platform and constitutes a given. Every participant in the layers below them is the User and is considered with the same monolithic view.

In some cases, the monolithic view is correct, but even then it is usually distorted. When I buy a new PC for home use, I have the seemingly unfiltered copy of the spreadsheet. But even here, the software was preinstalled by the people I bought the PC from. They chose certain options left by the 1st level spreadsheet team to make the installed software work with this particular configuration. Certain other options may be precluded by their choices.

A really good practitioner working in the hierarchy will be partially aware one layer above and below their own level. They can influence the layer above them, through user groups or other methods. They can consider the needs of those one layer below them, and improve their product and documentation to help out, but only as limited by budgets and competitive factors.

Conclusions
If you begin to understand the problem, then Welcome to the world of Computers!. But you can also understand how many niches in this world open opportunities for integrators and vendors. How a Peoplesoft vendor can package and sell an integrated solution for a business function. How a high-powered consulting firm can charge high prices to help corporate management try to cope. Why the industry keeps looking for something like an XML magic wand to make it fit together.

And most of all, how some kind of mental framework is needed before you can see how the pieces fit together. This kind of a framework is called an Architecture, and several of them are needed to see how the pieces fit together.

title: CACS/User Conundrum


text:
Talk:Calculus/Differentiation/Differentiation Defined

2003
Why use square brackets for the differential operator? IMHO, round brackets are just fine... Dysprosia 09:17, 16 Aug 2003 (UTC)

 * My calc book uses square brackets when differentiating an expression, like $${d \over dx} [x^2]$$. Some is style, some is real theory.  Ex.: what is $$\lim_{\Delta x \to 0} {\Delta y \over \Delta x}$$?  Why it's $${dy \over dx}$$ of course.  It would look wierd to have $${\Delta \over \Delta x} y$$; delta what, right?  So I guess that's the reason.  However, that's also why I think that the equation for the derivative should be $$\lim_{\Delta x \to 0} {f(x + \Delta x) - f(x) \over \Delta x}$$ because it correctly shows what dy/dx really means.Boothinator 02:17, 19 Mar 2004 (UTC)

question
How do the last two steps listed for f'(sin(x)) work? Ugen64 01:09, 18 Dec 2003 (UTC)

 * You mean D sin(x)? The second summand should be clear (from sin(x)/x -> 1 as x->0), and the second is a result from that limit, though I should point that out in the article (and leave it to the Trig page ;). Dysprosia 05:20, 18 Dec 2003 (UTC)

 * Is it worth it to prove the product rule, division rule, or chain rule here? Do prove the mean value theorem in this book?

--Cronian 07:43, 26 Mar 2004 (UTC)

 * OK, am I the only one who can't see the apostrophe in f'(x)? If not, should I put a space after the f, or is there a better way to make it more readable? --Xhad

If you itallicize the 'f' and the 'prime' you can read it a bit more clearly: f'(x). The whole bit itallicized: f'(x). Maybe that's better? ChaoticLlama  14:59, 13 April 2007 (UTC)

question
In the differentiation rules (d/dx)c=0 and (d/dx)mx=m, what exactly does "d" stand for? does it stand for change in x and y??? could somebody please explain these rules more clearly? --Elpenmaster
 * I think of the letter d as meaning "a tiny difference". ..... (d/dx)c means a tiny difference of a constant (obviously there is none, or else it would not be a constant!) divided by a tiny difference in x is zero. ..... And (d/dx)mx means (a tiny difference of the product m times x) divided by (a tiny difference in x) is m, where m, by the way, is the trigonometric tangent of the angle of that line (y = mx + any constant) against the horizontal. QUITTNER 142.150.49.171 18:42, 17 Jun 2005 (UTC)

 * First: The power rule is used in an example (about the addition rule) before it is introduced. Would people new to calculus be confused by the sudden transformation and miss the point of the example as a whole?

 * Second: Regarding proofs, in my opinion, lengthy proofs (read, "90% of proofs") are not overly important to the average reader. I believe that important proofs (chain rule, product rule, etc.) could all be placed on a seperate page in order to keep the text from becoming cluttered. If the goal is to reach people who don't already understand calculus, too much notation all at once becomes confusing and distracting.

--User:Noah 16:14, 30 Jul 2004 (UTC)

power rule
I noticed that the power rule section said $$\frac{d}{dx}\left[x^n\right]=nx^{n-1}, n\ne0$$. If you work it out when $$n=0$$, you get $$\frac{d}{dx}\left[x^0\right]=0x^{0-1}=0x^{-1}=0$$. Since $$x^{0}=1$$, you get $$\frac{d}{dx}\left[1\right]=0$$, which is true. Unless I made a mistake, the $$n\ne0$$ is not needed. Should it be taken out of the equation? DanielLC 00:28, 1 August 2005 (UTC)

While often $$x^{0}=1$$, there is some confusion over the special case of x=0 (0^0).

The $$n\ne0$$ keeps us far, far away from that one point of confusion.

--DavidCary 20:13, 31 October 2005 (UTC)

bold new words
Should new words like "derivative" and "differentiable" (Section 2.1) be emphasized (bold/italic) for clarity? --anonymous

I think the first thing that should be included on this page is the definition of the derivative of a function by First Principles, ie. limit of the gradient of secants, and then possibly explain some of the derivatives pulled, apparently, out of the blue, in terms of that.

Second, for those asking what the d in (d/dx)c is all about - d/dx is a "differential operator", which means it's something that acts on a function to differentiate it, in this case with respect to x. It follows some of the rules of normal fractions, so you have (d/dx)(f(x)), which means "the derivative (with respect to x) operator applied to the function f of x", and write it df/dx, which means "the derivative of f with respect to x" (ie. pretty much the same thing). Then if f(x) = c, ie. a constant function, then you can write either (d/dx)c or dc/dx, both of which equal 0. 203.26.177.2 18:52, 26 September 2005 (UTC)

High school
I'm a bored high school student currently in Precalc, trying to teach myself derivatives, so hopefully I can contribute to this project. I had to read the old Sales Example several times to figure out exactly what was going on, so I tried to make it more clear, based on what I gathered from the article, and also tried to make it more text-bookesque. Someone please check the changes in that section, because I'm not entirely sure I'm correct. --SimRPGman 03:44, 27 October 2005 (UTC)

 * Well done for deciding to contribute, it is a great way to learn the subject.  That example looks correct to me.  My only concern is the sentence

"Because the functions are defined for all n >= 0 and are only differentiable in n > 0, the only points that can be minima or maxima are those above."

 * Actually it is defined and differentiable for all n. But we are only concerned with positive n (as it makes no sense to produce -5 items).
 * Also here are some idead which might make the example clearer

 * Instead of 8.1n maybe use (8.1)n.  When I first read it I thought it mean 8.(1n)
 * It is implicit by using the integer n that we are dealing with whole numbers. (otherwise we should use a letter like x,y,z).    Since the max is 3.97.... the seller really has to produce 4 units (as that is the nearest whole number) Juliusross 11:34, 27 October 2005 (UTC)

 * The previously quoted sentence was like that when I had it.
 * The example states that all amounts are in thousands, so 3.797 thousand units would be a valid amount at 3,797 units.
 * --SimRPGman 23:09, 27 October 2005 (UTC)

 * My mistake, you are right. Juliusross 23:52, 27 October 2005 (UTC)

Confused
Hey folks, the section on the Maxima and Minima confused the heck out of me. I tried to re-word the first paragraph, as I thought I grasped the point, but the second paragraph eluded me. If someone with more math knowledge than me (I'm trying to learn this stuff) could take a look I'd appreciate it. It almost seems as though it were written by someone with English as a second language, or just someone who knows their math but perhaps isn't as smooth with English. Not that I'm criticising the effort! Thanks all.--Dave
 * Agreed. These two paragraphs need to be expanded (it might take a few pages in an ordinary textbook).  One way to learn is to write.  Do you want to get involved? Juliusross 12:43, 31 October 2005 (UTC)
 * (This is me, the same as above--I just got an account) Well, that sounds like a challenge...sure, I'm game. Let me give it a shot and then get a critique. Dubitable 13:52, 31 October 2005 (UTC)

Also on the subject of Maxima and Minima, I "stole" some info from Wikipedia and formed it into a introduction to the concept. A picture would really be nice. Should we use Wikipedia's picture? --SimRPGman 05:41, 7 November 2005 (UTC)
 * Sure, take any material from wikipedia if you think it is suitable. You should maybe add a line at the bottom of the page to say you have taken material from wikipedia, and put in a link (although I am not 100% sure what is really necessary) Juliusross 02:18, 8 November 2005 (UTC)

Ok, I took a swing at that last paragraph on Maxima and Minima and actually made two paragraphs out of it. Please review and see if it's any clearer and if I'm right (because I'm not entirely certain I am). --SimRPGman 04:27, 21 November 2005 (UTC)

Whoever is editing this book needs to stop adding "Example" as a category - it appears in the table of contents and looks very cumbersome. If each subcategory extended for more than page - THEN it might be good to add example to the table of contents - but as it stands it just looks horrible. I fixed a couple of them. Fresheneesz
 * I agree that this is not optimal. But  we need a consistent way of presenting examples.  Until per page css is possible this is the best I can come up with.  Have a look at the discussion page on the main calculus page and you will see some other comments about this Juliusross 18:40, 5 November 2005 (UTC)
 * You changed my mind. I will not longer use example as a category. Juliusross 02:18, 8 November 2005 (UTC)

I rewrote the Maxima and Minima section (and renamed it, though I'm not sure that it shouldn't be "Stationary Points"), as it was confusing and incorrect in several places. It is possible to classify a stationary point without doubt. My browser crashed and I forgot to log in so I am 87.194.203.205. Sorry. Inductiveload 20:40, 23 October 2007 (UTC)

Better exercises needed
I've already been through the mill on learning derivatives and can practically do them in my sleep (I'm a AP Calculus C student), but I had to read the exercise at the bottom of the page twice to understand what it's asking. To someone who's just learning differentiation it's a horribly ambiguous question to ask. I've added some simpler straightforward differentiation exercises above the existing one, which is now at the bottom of the list (if someone could clean up my LaTEX equation syntax, I'd appreciate it; I've never used it myself). It's better to go through the process itself a few times from a few different approaches before getting that in-depth into the workings of it--asking a student to use the definition of derivative to prove the theories discussed in the article before having them actually do any differentiation independently of running it through the definition of derivative is a bit of a stretch. I also reworded the exercise that was already there to make it a little more clear in what it's asking. --Jesse Hannah 05:38, 24 February 2007 (UTC)

What is Differentiation?
This section is a little bit in depth for someone who is just being introduced to derivatives. While those of us who have already taken calculus would understand it fine, I am a little worried that those who have not taken calculus will be lost as soon as you start introducing syntax. I think this section should just mention that differentiation is a way of solving the tangent problem, and leave it at that. Maybe mentioning the difference quotient would be good. Tapioca 19:23, 5 March 2007 (UTC)

Overall Structure
I've been reading through the differentiation article several times making edits as I go, however I've noticed there is little flow in this article; Even from sentence to sentence. As I finish finalizing the notes I am creating, I will post the structure and teaching method that I believe to be most effective. I would appreciate any comments in modifying this layout, yet do not reply if you merely wish to comment on this post.

ChaoticLlama 17:51, 20 March 2007 (UTC)

Section Removed
I don't see how this hand-waving argument adds to the article. Any comments otherwise? ChaoticLlama 14:53, 13 April 2007 (UTC)

Moving Sections Around
IMO, this article should be focused on describing the theory of differention, and its techniques. Topics such as Maxima & Minima or the Sales Example are clearly applications of differentiation. I'd move Max & Mins to applications, and put the sales example under Mean Value Theorem. Thoughts? ChaoticLlama  15:09, 13 April 2007 (UTC)

 * Well, I agree sales and extrema are application, but the concept of local extrema, and their relation to differentiation, is very important. So if and when they are moved, they should receive their own clearly defined section, preferably at the very top of the application section. cheers. --Cronholm144 07:56, 18 June 2007 (UTC)

 * I agree with both. The extrema section is (now) well written, but it doesn't flow with the rest of this page. More importantly it relies on Calculus/Higher Order Derivatives, which is a later module. --Mrwojo (talk) 01:16, 29 December 2008 (UTC)

addition and subtraction
In the very first line of the section on addition and subtraction rules, the brackets in the expression in the numerator are imbalanced. There are 2 open brackets and 1 close bracket. I would think we need a close bracket just to the left of the first minus sign. $$ \lim_{\Delta x \to 0}\left[\frac{[f(x+\Delta x) \pm g(x + \Delta x) - [f(x) \pm g(x)]}{\Delta x}\right] $$ Johnor (talk) 02:39, 16 October 2008 (UTC)

 * Fixed. --Mrwojo (talk) 01:04, 29 December 2008 (UTC)

Unusual Terminology
Hi, I find the use of the term "Gradient" a bit unusual. I have not often encountered the phrase, and most calculus books I have encountered stick with the term slope. I am fine with keeping the first sentence of section "The Definition of Slope" but I think after that it would be wise to use the term slope. Does anyone mind if I make this change? Thenub314 (talk) 11:08, 21 January 2009 (UTC)

Proof of the Power Rule
I don't have time to implement this now, but if someone else would like to, it would be nice to have a brief discussion of where the power rule for non-integer exponents comes from. There is a proof given at PlanetMath that may be useful. --Greenbreen (discuss • contribs) 17:48, 11 June 2011 (UTC)


text:
Calculus/Ordinary differential equations

Ordinary differential equations involve equations containing: and their solutions.
 * variables
 * functions
 * their derivatives

In studying integration, you already have considered solutions to very simple differential equations. For example, when you look to solving
 * $$\int f(x) \,dx=g(x)$$

for g(x), you are really solving the differential equation
 * $$ g'(x) = f(x) \,$$

Notations and terminology
The notations we use for solving differential equations will be crucial in the ease of solubility for these equations.

This document will be using three notations primarily:
 * f' to denote the derivative of f
 * D f to denote the derivative of f
 * $${df \over dx}$$ to denote the derivative of f (for separable equations).

Terminology
Consider the differential equation
 * $$3 f^{\prime \prime}(x)+5xf(x)=11$$

Since the equation's highest derivative is 2, we say that the differential equation is of order 2.

Some simple differential equations
A key idea in solving differential equations will be that of integration.

Let us consider the second order differential equation (remember that a function acts on a value).
 * $$f''(x)=2 \,$$

How would we go about solving this? It tells us that on differentiating twice, we obtain the constant 2 so, if we integrate twice, we should obtain our result.

Integrating once first of all:

 * $$\int f''(x) \,dx = \int 2 \,dx$$
 * $$f'(x)=2x+C_1 \,$$

We have transformed the apparently difficult second order differential equation into a rather simpler one, viz.

 * $$f'(x)=2x+C_1 \,$$

This equation tells us that if we differentiate a function once, we get $$2x+C_1$$. If we integrate once more, we should find the solution.

 * $$\int f'(x) \,dx = \int 2x+C_1 \,dx$$
 * $$f(x)=x^2+C_1x+C_2 \,$$

This is the solution to the differential equation. We will get $$f=2 \,$$ for all'' values of $$C_1$$ and $$C_2$$.

The values $$C_1$$ and $$C_2$$ are related to quantities known as initial conditions.

Why are initial conditions useful? ODEs (ordinary differential equations) are useful in modeling physical conditions. We may wish to model a certain physical system which is initially at rest (so one initial condition may be zero), or wound up to some point (so an initial condition may be nonzero, say 5 for instance) and we may wish to see how the system reacts under such an initial condition.

When we solve a system with given initial conditions, we substitute them after our process of integration.

Example
When we solved $$f''(x)=2 \,$$ say we had the initial conditions $$f'(0)=3 \,$$ and $$f(0)=2 \,$$. (Note, initial conditions need not occur at f(0)).

After we integrate we make substitutions:
 * $$f'(0)=2(0)+C_1 \,$$
 * $$3=C_1 \,$$
 * $$\int f'(x) \,dx = \int 2x+3 \,dx$$
 * $$f(x)=x^2+3x+C_2 \,$$
 * $$f(0)=0^2+3(0)+C_2 \,$$
 * $$2=C_2 \,$$
 * $$f(x)=x^2+3x+2 \,$$

Without initial conditions, the answer we obtain is known as the general solution or the solution to the family of equations. With them, our solution is known as a specific solution.

Basic first order DEs
In this section we will consider four main types of differential equations:
 * separable
 * homogeneous
 * linear
 * exact

There are many other forms of differential equation, however, and these will be dealt with in the next section

Separable equations
A separable equation is in the form (using dy/dx notation which will serve us greatly here)
 * $${dy \over dx} = f(x)/g(y)$$

Previously we have only dealt with simple differential equations with g(y)=1. How do we solve such a separable equation as above?

We group x and dx terms together, and y and dy terms together as well.
 * $$g(y)\ dy = f(x)\ dx$$

Integrating both sides with respect to y on the left hand side and x on the right hand side:
 * $$\int g(y)\,dy=\int f(x)\,dx+C$$

we will obtain the solution.

Worked example
Here is a worked example illustrating the process.

We are asked to solve
 * $${dy \over dx} = 3x^2y$$

Separating
 * $${dy \over y} = (3x^2)\,dx$$

Integrating
 * $$\int {dy \over y} = \int 3x^2\,dx$$
 * $$\ln{y}=x^3+C \,\!$$
 * $$y=e^{x^3+C}$$

Letting $$k = e^C$$ where k is a constant we obtain
 * $$y=ke^{x^3}$$

which is the general solution.

Verification
This step does not need to be part of your work, but if you want to check your solution, you can verify your answer by differentiation.

We obtained
 * $$y=ke^{x^3}$$

as the solution to
 * $${dy \over dx} = 3x^2y$$

Differentiating our solution with respect to x,
 * $${dy \over dx} = 3kx^2e^{x^3}$$

And since $$y=ke^{x^3}$$, we can write
 * $${dy \over dx} = 3x^2y$$

We see that we obtain our original differential equation, thus our work must be correct.

Homogeneous equations
A homogeneous equation is in the form
 * $${dy \over dx} = f(y/x)$$

This looks difficult as it stands, however we can utilize the substitution
 * $$v = {y \over x}$$

so that we are now dealing with F(v) rather than F(y/x).

Now we can express y in terms of v, as y=xv and use the product rule.

The equation above then becomes, using the product rule
 * $${dy \over dx} = v+x{dv \over dx}$$

Then
 * $$v+x{dv \over dx} = f(v)$$
 * $$x{dv \over dx} = f(v)-v$$
 * $${dv \over dx} = {f(v)-v \over x}$$

which is a separable equation and can be solved as above.

However let's look at a worked equation to see how homogeneous equations are solved.

Worked example
We have the equation
 * $$ {dy \over dx} = {y^2 + x^2 \over yx}$$

This does not appear to be immediately separable, but let us expand to get
 * $$ {dy \over dx} = {y^2 \over yx} + {x^2 \over yx}$$
 * $$ {dy \over dx} = {x \over y} + {y \over x}$$

Substituting y=xv which is the same as substituting v=y/x:
 * $$ {dy \over dx} = 1/v + v$$

Now
 * $$ v+x{dv \over dx} = 1/v + v$$

Canceling v from both sides
 * $$ x{dv \over dx} = 1/v$$

Separating
 * $$ v\, dv = dx/x$$

Integrating both sides
 * $$ {1 \over 2}v^2+C= \ln(x) \,$$
 * $$ {1 \over 2}\left({y \over x}\right)^2= \ln(x)-C$$
 * $$ y^2 = 2x^2 \ln(x) - 2Cx^2 \,$$
 * $$ y = x\sqrt{2 \ln(x) - 2C}$$

which is our desired solution.

Linear equations
A linear first order differential equation is a differential equation in the form
 * $$ a(x){dy \over dx} + b(x)y=c(x)$$

Multiplying or dividing this equation by any non-zero function of x makes no difference to its solutions so we could always divide by a(x) to make the coefficient of the differential 1, but writing the equation in this more general form may offer insights.

At first glance, it is not possible to integrate the left hand side, but there is one special case. If b happens to be the differential of a then we can write

 * $$a(x){dy \over dx} + b(x)y = a(x){dy \over dx} + y{da \over dx}

= {d \over dx}a(x)y $$

and integration is now straightforward.

Since we can freely multiply by any function, lets see if we can use this freedom to write the left hand side in this special form.

We multiply the entire equation by an arbitrary, I(x), getting

 * $$ aI{dy \over dx} + bIy=cI$$

then impose the condition

 * $$ \frac{d}{dx}aI = bI $$

If this is satisfied the new left hand side will have the special form. Note that multiplying I by any constant will leave this condition still satisfied.

Rearranging this condition gives

 * $$ \frac{1}{I}\frac{dI}{dx} = \frac{b-\frac{da}{dx}}{a} $$

We can integrate this to get

 * $$ \ln I(x) = \int \frac{b(z)}{a(z)}dz - \ln a(x) + c \quad

I(x)=\frac{k}{a(x)}e^{\int \frac{b(z)}{a(z)}dz}$$

We can set the constant k to be 1, since this makes no difference.

Next we use I on the original differential equation, getting

 * $$ e^{\int \frac{b(z)}{a(z)}dz}{dy \over dx} +

e^{\int \frac{b(z)}{a(z)}dz} \frac{b(x)}{a(x)}y =e^{\int \frac{b(z)}{a(z)}dz}\frac{c(x)}{a(x)}$$

Because we've chosen I to put the left hand side in the special form we can rewrite this as

 * $$ {d \over dx}(ye^{\int \frac{b(z)}{a(z)}dz}) =

e^{\int \frac{b(z)}{a(z)}dz}\frac{c(x)}{a(x)}$$

Integrating both sides and dividing by $$e^I$$ we obtain the final result

 * $$ y = e^{-\int \frac{b(z)}{a(z)}dz}

\left(\int e^{\int \frac{b(z)}{a(z)}dz}\frac{c(x)}{a(x)}dx + C\right) $$

We call I an integrating factor. Similar techniques can be used on some other calculus problems.

Example
Consider

 * $$\frac{dy}{dx} + y \tan x = 1 \quad y(0)=0$$

First we calculate the integrating factor.

 * $$I=e^{\int \tan x dx} = e^ {\ln \sec x} = \sec x $$

Multiplying the equation by this gives

 * $$\sec x \frac{dy}{dx} + y \sec x \tan x = \sec x $$

or

 * $$ \frac{d}{dx} y\sec x = \sec x $$

We can now integrate

 * $$ y = \cos x \int_0^x \sec z \, dz = \cos x \ln (\sec x + \tan x) $$

Exact equations
An exact equation is in the form
 * f(x, y) dx + g(x, y) dy = 0

and, has the property that
 * Dx f = Dy g

(If the differential equation does not have this property then we can't proceed any further).

As a result of this, if we have an exact equation then there exists a function h(x, y) such that
 * Dy h = f and Dx h = g

So then the solutions are in the form
 * h(x, y) = c

by using the fact of the total differential. We can find then h(x, y) by integration

Basic second and higher order ODE's
The generic solution of a nth order ODE will contain n constants of integration. To calculate them we need n more equations. Most often, we have either
 * boundary conditions, the values of y and its derivatives take for two different values of x

or
 * initial conditions, the values of y and its first n-1 derivatives take for one particular value of x.

Reducible ODE's
1. If the independent variable, x, does not occur in the differential equation then its order can be lowered by one. This will reduce a second order ODE to first order.

Consider the equation:
 * $$F\left(y,\frac{dy}{dx},\frac{d^2y}{dx^2}\right)=0$$

Define
 * $$u=\frac{dy}{dx}$$

Then
 * $$\frac{d^2y}{dx^2}=\frac{du}{dx}=\frac{du}{dy}\cdot\frac{dy}{dx}=\frac{du}{dy}\cdot u$$

Substitute these two expression into the equation and we get
 * $$F\left(y,u,\frac{du}{dy}\cdot u\right)$$=0

which is a first order ODE

Example
Solve
 * $$1+2y^2\operatorname{D}^2y=0$$

if at x=0, y=Dy=1

First, we make the substitution, getting
 * $$1+2y^2 u \frac{du}{dy}=0$$

This is a first order ODE. By rearranging terms we can separate the variables
 * $$udu=-\frac{dy}{2y^2}$$

Integrating this gives
 * $$u^2/2=c+1/2y$$

We know the values of y and u when x=0 so we can find c
 * $$c=u^2/2-1/2y=1^2/2-1/(2\cdot 1)=1/2-1/2=0$$

Next, we reverse the substitution
 * $$\frac{dy}{dx}^2=u^2=\frac{1}{y}$$

and take the square root
 * $$\frac{dy}{dx}=\pm \frac{1}{\sqrt{y}}$$

To find out which sign of the square root to keep, we use the initial condition, Dy=1 at x=0, again, and rule out the negative square root. We now have another separable first order ODE,
 * $$\frac{dy}{dx}=\frac{1}{\sqrt{y}}$$

Its solution is
 * $$\frac{2}{3}y^\frac{3}{2}= x+d$$

Since y=1 when x=0, d=2/3, and
 * $$y=\left(1 + \frac{3x}{2} \right)^\frac{2}{3}$$

2. If the dependent variable, y, does not occur in the differential equation then it may also be reduced to a first order equation.

Consider the equation:
 * $$F\left(x,\frac{dy}{dx},\frac{d^2y}{dx^2}\right)=0$$

Define
 * $$u=\frac{dy}{dx}$$

Then
 * $$\frac{d^2y}{dx^2}=\frac{du}{dx}$$

Substitute these two expressions into the first equation and we get
 * $$F\left(x,u,\frac{du}{dx}\right)$$=0

which is a first order ODE

Linear ODEs
An ODE of the form
 * $$\frac{d^ny}{dx^n}+a_1(x)\frac{d^{n-1}y}{dx^{n-1}}+ ... +a_n y=F(x)$$

is called linear. Such equations are much simpler to solve than typical non-linear ODEs. Though only a few special cases can be solved exactly in terms of elementary functions, there is much that can be said about the solution of a generic linear ODE. A full account would be beyond the scope of this book

If F(x)=0 for all x the ODE is called homogeneous

Two useful properties of generic linear equations are
 * 1) Any linear combination of solutions of an homogeneous linear equation is also a solution.
 * 2) If we have a solution of a nonhomogeneous linear equation and we add any solution of the corresponding homogenous linear equation we get another solution of the nonhomogeneous linear equation

Variation of constants
Suppose we have a linear ODE,
 * $$\frac{d^ny}{dx^n}+a_1(x)\frac{d^{n-1}y}{dx^{n-1}}+ ... +a_n y=0$$

and we know one solution, y=w(x)

The other solutions can always be written as y=wz. This substitution in the ODE will give us terms involving every differential of z upto the nth, no higher, so we'll end up with an nth order linear ODE for z.

We know that z is constant is one solution, so the ODE for z must not contain a z term, which means it will effectively be an n-1th order linear ODE. We will have reduced the order by one.

Lets see how this works in practice.

Example
Consider
 * $$\frac{d^2y}{dx^2}+\frac{2}{x}\frac{dy}{dx}-\frac{6}{x^2}y=0$$

One solution of this is y=x2, so substitute y=zx2 into this equation.

 * $$\left( x^2\frac{d^2z}{dx^2}+4x\frac{dz}{dx}+2z\right)

+\frac{2}{x} \left( x^2\frac{dz}{dx}+2xz \right) -\frac{6}{x^2}x^2 z=0$$

Rearrange and simplify.
 * $$x^2 D^2 z + 6xD z=0$$

This is first order for Dz. We can solve it to get
 * $$z=A x^{-5} \quad y=A x^{-3}$$

Since the equation is linear we can add this to any multiple of the other solution to get the general solution,

 * $$y=A x^{-3} + B x^2$$

Linear homogeneous ODE's with constant coefficients
Suppose we have a ODE
 * $$ (D^n+a_1 D^{n-1}+ ... + a_{n-1}D+a_0)y=0$$

we can take an inspired guess at a solution (motivate this)
 * $$y=e^{px}$$

For this function Dny=pny so the ODE becomes
 * $$ (p^n+a_1 p^{n-1}+ ... + a_{n-1}p+a_0)y=0$$

y=0 is a trivial solution of the ODE so we can discard it. We are then left with the equation
 * $$ p^n+a_1 p^{n-1}+ ... + a_{n-1}p+a_0=0$$

This is called the characteristic equation of the ODE.

It can have up to n roots, p1, p2 …  pn, each root giving us a different solution of the ODE.

Because the ODE is linear, we can add all those solution together in any linear combination to get a general solution
 * $$y=A_1 e^{p_1 x} +A_2 e^{p_2 x} + ... + A_n e^{p_n x}$$

To see how this works in practice we will look at the second order case. Solving equations like this of higher order uses exactly the same principles; only the algebra is more complex.

Second order
If the ODE is second order,
 * $$D^2 y + bDy+cy=0$$

then the characteristic equation is a quadratic,
 * $$p^2+bp+c=0$$

with roots
 * $$p_{\pm}=\frac{-b \pm \sqrt{b^2-4c}}{2}$$

What these roots are like depends on the sign of b2-4c, so we have three cases to consider.

1) b2 > 4c

In this case we have two different real roots, so we can write down the solution straight away.
 * $$ y=A_{+}e^{p_{+}}+A_{-}e^{p_{-}}$$

2) b2 < 4c

In this case, both roots are imaginary. We could just put them directly in the formula, but if we are interested in real solutions it is more useful to write them another way.

Defining k2=4c-b2, then the solution is
 * $$y=A_{+}e^{ikx-\frac{bx}{2}}+A_{-}e^{-ikx-\frac{bx}{2}}$$

For this to be real, the A ' s must be complex conjugates
 * $$A_{\pm}=A e^{\pm ia}$$

Make this substitution and we can write,
 * $$y=A e^{-bx/2}\cos (kx+a)$$

If b is positive, this is a damped oscillation.

3) b2 = 4c

In this case the characteristic equation only gives us one root, p=-b/2. We must use another method to find the other solution.

We'll use the method of variation of constants. The ODE we need to solve is,
 * $$D^2 y -2pDy+p^2y=0$$

rewriting b and c in terms of the root. From the characteristic equation we know one solution is $$y=e^{px}$$ so we make the substitution $$y=ze^{px}$$, giving
 * $$ (e^{px}D^2z+2pe^{px}Dz+p^2e^{px}z)-2p(e^{px}Dz+pe^{px}z)+p^2e^{px}z=0 $$

This simplifies to D2z=0, which is easily solved. We get
 * $$z=Ax+B \quad y=(Ax+B)e^{px}$$

so the second solution is the first multiplied by x.

Higher order linear constant coefficient ODE's behave similarly: an exponential for every real root of the characteristic and a exponent multiplied by a trig factor for every complex conjugate pair, both being multiplied by a polynomial if the root is repeated.

E.g., if the characteristic equation factors to
 * $$(p-1)^4(p-3)(p^2+1)^2=0$$

the general solution of the ODE will be
 * $$y=(A+Bx+Cx^2+Dx^3)e^x + Ee^{3x}+ F \cos (x+a) +Gx \cos(x+b)$$

The most difficult part is finding the roots of the characteristic equation.

Linear nonhomogeneous ODEs with constant coefficients
First, let's consider the ODE
 * $$Dy-y=x$$

a nonhomogeneous first order ODE which we know how to solve.

Using the integrating factor e-x we find
 * $$y=c e^{-x} +1 -x$$

This is the sum of a solution of the corresponding homogeneous equation, and a polynomial.

Nonhomogeneous ODE's of higher order behave similarly.

If we have a single solution, yp of the nonhomogeneous ODE, called a particular solution,
 * $$ (D^n+a_1 D^{n-1} + \cdots + a_n)y=F(x)$$

then the general solution is y=yp+yh, where yh is the general solution of the homogeneous ODE.

Find yp for an arbitrary F(x) requires methods beyond the scope of this chapter, but there are some special cases where finding yp is straightforward.

Remember that in the first order problem yp for a polynomial F(x) was itself a polynomial of the same order. We can extend this to higher orders.

Example:
 * $$D^2y+y=x^3-x+1$$

Consider a particular solution
 * $$y_p=b_0+b_1 x+b_2 x^2 + x^3$$

Substitute for y and collect coefficients
 * $$x^3 + b_2 x^2 +(6+b_1)x +(2b_2+b_0)=x^3-x+1$$

So b2=0, b1=-7, b0=1, and the general solution is
 * $$y=a \sin x + b \cos x + 1 -7x + x^3$$

This works because all the derivatives of a polynomial are themselves polynomials.

Two other special cases are
 * $$F(x)=P_n e^{kx} \quad y_p(x)=Q_n e^{kx}$$
 * $$F(x)=A_n \sin kx +B_n \cos kx \quad

y_p(x)=P_n \sin kx +Q_n \cos kx$$ where Pn,Qn,An, and Bn are all polynomials of degree n.

Making these substitutions will give a set of simultaneous linear equations for the coefficients of the polynomials.


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CACS/Glossary/Basic

Basic Language is a high level computer language that was intrroducd with the PC in the 1960s. BASIC is an acronym for Beginner's All-purpose Symbolic Instruction Code. Basic was originally an interpreted language, but compilers were soon introduced. While there is an ANSI Standard Basic, most vendors use proprietary extensions. There is a subset or similar language used in Web pages called Basic Script. Microsoft platform applications include another subset called Visual Basic for Applications (VBA). The .BAS file extension is frequently used for Basic source code.

title: CACS/Glossary/Basic


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CACS/Glossary/Binary

Binary is a number system in the base 2, which contains only the digits 0 and 1. Since computer memory is made up of off-on switches, all computer programs and data are ultimately encoded in binary.

Binary refers to a program that has been compiled into an executable module. When the module is stored, the .BIN file extension is used. Some development platform compilers produce a binary module that must still be linked with other modules to produce an executable program.

title: CACS/Glossary/Binary


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CACS/Glossary/Borland

Borland (Company) is a software developer and vendor specializing in development tools, language compilers, and data base platforms. It was founded in 1983, and is located in Scotts Valley, California. Their Web address is Borland.com.

title: CACS/Glossary/Borland


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CACS/Glossary/Computer

A Computer is programmable machine containing memory, input and output devices, and a central processor (CPU); and capable of executing a stored program. Each computer has a well defined instruction set that is used in programming it.

title: CACS/Glossary/Computer


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Spanish/Typing Spanish Characters

Language Settings (fastest)
The idea of changing the language settings is that you can then type characters quickly and easily (for example by pressing Alt+a for typing á).

Windows XP
This is a list that describes how to change the language settings for Windows XP.

 * 1) Click the Start button
 * 2) Click on Control Panel
 * 3) Click Regional and Language Options - a new window will pop up
 * 4) Click the Languages tab
 * 5) Click on the Details button
 * 6) Click the Add button - a new window will pop up
 * 7) Select "United States-International (keyboard layout menu)" or "United States-Dvorak (International)"
 * 8) Click OK
 * 9) Click Apply

Now you should see a keyboard icon at your task bar at the bottom. Click on this icon to switch to the United States International keyboard layout. This keyboard layout has a new key (AltGr) and 5 dead keys. The dead keys are explained below. An interactive diagram of this layout can be found in.

The US International keyboard has two different Alt keys. The left Alt key continues to be the regular Alt key, normally associated with Windows menus. The right Alt key becomes what is called AltGr (or graphic Alt) key. This key lets you type very quickly special characters in Spanish and other languages by using AltGr and then typing a character from the list below.

Note: + indicate typing one key after the other. - is typing two keys at the same time.

Lower-Case Characters
á -> AltGr+a é -> AltGr+e í -> AltGr+i ó -> AltGr+o ú -> AltGr+u ü -> AltGr+y ñ -> AltGr+n

ç -> AltGr+, å -> AltGr+w

Upper-Case Characters
Á -> AltGr+Shift-a É -> AltGr+Shift-e Í -> AltGr+Shift-i Ó -> AltGr+Shift-o Ú -> AltGr+Shift-u Ü -> AltGr+Shift-y Ñ -> AltGr+Shift-n

Ç -> AltGr+Shift-, Ð -> AltGr+Shift-d

Other Symbols
¿ -> AltGr+? ¡ -> AltGr+! « -> AltGr+[     Spaniards prefer «angular» quotes » -> AltGr+]     Latin Americans prefer the “curly” ones ° -> AltGr+:     Degree sign; ordinal sign, as in 4° año = cuarto año € -> AltGr+5 ¢ -> AltGr+Shift-c £ -> AltGr+$ ¥ -> AltGr+-

Dead Keys
The US International keyboard has five dead keys. They add the symbol they have marked at the top of the following letter.

'+a = á; '+e = é; ... "+a = ä; ...; "+u = ü ~+a = ã; ~+n = ñ ^+a = â; ... `+a = à; ... '+Shift-a = Á; '+Shift+e = É; ... "+Shift-u = Ü; ... ~+Shift-n = Ñ; ...

To enter what's written on a dead key you need to add a space. ' followed by space generates an actual apostrophe.

Other Operating Systems
Information on some other operation systems can be found here.

Alt Number Codes
You can type a special character by pressing and holding down the Alt button and then typing a number code on the number pad of your keyboard. The most frequently used characters have both a three-digit and a four-digit code. Less frequent characters (such as Á) have only a four-digit code. This page contains a good overview of special characters for different languages

Lower Case Characters
á -> Alt-160 or Alt-0225 ç -> Alt-135 or Alt-0231 é -> Alt-130 or Alt-0233 í -> Alt-161 or Alt-0237 ñ -> Alt-164 or Alt-0241 ó -> Alt-162 or Alt-0243 ú -> Alt-163 or Alt-0250 ü -> Alt-129 or Alt-0252

Upper Case Characters
Á ->           Alt-0193 Ç -> Alt-128 or Alt-0199 É -> Alt-144 or Alt-0201 Í ->           Alt-0205 Ñ -> Alt-165 or Alt-0209 Ó ->           Alt-0211 Ú ->           Alt-0218 Ü -> Alt-154 or Alt-0220

Punctuation Marks
¿ Alt-168 or Alt-0191 ¡ Alt-173 or Alt-0161

Copy & Paste
This method can be useful if you are just writing a short text (for example an e-mail) and don't have a computer where you can/want change language settings. Just try to pull up a web page or a document that contains the special characters and paste them into your text. For longer texts, however, this can become quite tedious.

Search & Replace
If you are working with a text editor you have the option to search for text and replace it with other text. This feature can be used to 'type' special characters. The idea is to mark a character for becoming a special character, for example typing ~a when you mean á. After you have written your text you replace marked characters (the ~a) with special characters (the á). Of course you have to either type in the Alt number code or paste the character, but the point is that you only have to do it once for the whole text and not for every single á that you want to type.

Automated Search & Replace
If you know a programming language that allows string processing you can automate the Search & Replace process by a computer program which automatically replaces all your marked characters with the appropriate special characters after you are done with typing your text.

Macintosh
Compared to Windows, typing Spanish characters on a Macintosh is relatively easy. So long as you are using a standard American or UK-style QWERTY keyboard, you may just use the following keyboard commands. (Note that you should release the Option (Opt) key before striking the second letter; for example, for á, hold down Option, strike E, release Option, strike A.)

One good way to practice typing Spanish characters on a Mac is to use the Key Caps program, which should be in the Utilities folder in the Applications folder. This simple program will show you what characters you can type next if you hold down the Option and/or Shift keys.

Lower-Case Characters
á -> Opt+E, A é -> Opt+E, E í -> Opt+E, I ó -> Opt+E, O ú -> Opt+E, U ü -> Opt+U, U ñ -> Opt+N, N

ç -> Opt+C å -> Opt+A

Upper-Case Characters
Á -> Opt+E, Shift+A É -> Opt+E, Shift+E Í -> Opt+E, Shift+I Ó -> Opt+E, Shift+O Ú -> Opt+E, Shift+U Ü -> Opt+U, Shift+U Ñ -> Opt+N, Shift+N

Ç -> Opt+Shift+C

Other Symbols
¿ -> Opt+Shift+/ (forward slash; same key as ?) ¡ -> Opt+1 « -> Opt+\ (back slash; under the Delete key) » -> Opt+Shift+\ *Note that Spaniards prefer «angle quotes,» whereas Latin Americans prefer “curly quotes” as in English. ° -> Opt+Shift+8     Degree sign; ordinal sign, as in 4° año = cuarto año € -> Opt+Shift+2 ¢ -> Opt+4 £ -> Opt+3 ¥ -> Opt+Y

KDE/GNOME
In KDE and GNOME you can choose the international US keyboard layout.

In KDE, go to Regional & Accessibility - Keyboard Layout in the KDE Control Center. Add the international US keyboard layout to your active layouts. With the flag icon in your taskbar you can now switch between different layouts.

In GNOME, find preferences in the system menu, and select keyboard from the preferences menu. Add the US English International layout to your layouts. Alt+Alt will cycle through the various layouts you have added. Alternatively, keyboard layout selection and management may be done via the GNOME keyboard indicator applet, which may be added to your panel.

With the appropriate keyboard layout selected, you can type á by typing ' and then a:

'a á 'e é 'i í 'o ó 'u ú "u ü ~n ñ

X Windows
In X Windows (the default windowing system for Linux which both KDE and Gnome run on top of) you can create a custom X keyboard layout with the special Spanish letters added; more information is here.

title: Spanish/Typing Spanish Characters


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Spanish/Web Resources

Leer en línea (read online)
Todos los enlaces llevan a páginas en español (all links go to Spanish language pages).
 * VeinteMundos Spanish E-Magazine (free online resource with vocab and podcasts)
 * TeleSurTV (South America's own CNN)
 * Yahoo
 * CNN
 * BBC Mundo BBC World
 * Lista de periódicos (list of newspapers)
 * Informacion sobre la salud Health information
 * La Biblia the Bible
 * Wikipedia

Estudiar el español (to study Spanish)

 * http://www.learning-spanish-online.org (Large collection of free learning Spanish resources)
 * http://www.lingolex.com/spanish.htm (Large collection of free learning Spanish resources)
 * http://www.learnspanishinspainwith.us (Learn Spanish in Spain)
 * http://www.wiracochaschool.org (Immersion Spanish courses)
 * http://www.spanishpod101.com (Podcasts for learning Spanish)
 * http://www.wiracochaschool.org (Learn spanish in Peru)
 * http://www.onlinespanishhelp.com
 * http://www.patagoniaschool.com (Spanish Courses in Argentina -Patagonia)
 * http://www.executivespanishimmersion.com/ (Spanish Immersion Programs)
 * http://www.rocketspanish.com/ (Learn Spanish)
 * http://www.english-spanish-translator.org (Spanish Language Community)
 * http://www.spanish-translator-services.com/ (Free Spanish Translator Resources)
 * http://www.jetpunk.com/tags/spanish-language-vocabulary.php (Spanish Vocabulary Quizzes)
 * http://www.elcastellano.org/
 * http://study-spanish.net/
 * http://spanish.about.com/
 * http://www.spanicity.com (grammar and audio lessons)
 * http://spanish.allinfo-about.com/
 * http://www.spanishstudyholidays.com/
 * http://www.ielanguages.com/spanish.html
 * http://www.rae.es/ (Spanish Royal Academy of Language, see http://www.wikipedia.org/wiki/Real_Academia_Espa%F1ola .)
 * http://www.bbc.co.uk/languages/spanish/
 * http://www.indiana.edu/~call/reglas.html (focusses on Spanish grammar)
 * http://www.notesfromspain.com/ (podcasts of everyday Spanish)
 * http://www.supremespanish.com/ (Conversational Spanish)
 * http://www.spanishlanguageimmersion.net/
 * http://www.perulanguage.com/
 * http://www.spanish-schools.com.ar/ (Learn Spanish)
 * http://www.spanishconversationclub.com/ (learn Spanish for free)
 * http://www.thespanishforum.co.uk/ (Spanish language forum)
 * http://www.gooverseas.com/study-abroad/peru (reviews of study abroad programs in Peru)

Software (software)
www.spanishconversationclub.com (Spansih as foreign language conversation club in London)
 * (links to sites where you can download software for learning Spanish)
 * Spanish Éxito The most effective Spanish learning program available.
 * Pythoñol Pythonol, an opensource Spanish learning program. Note: For Linux users
 * WordsGalore 9,100 Spanish and English Words. Flash cards. Make your own MP3s. Windows only.
 * Learn Spanish Quickly - Android app to learn Spanish

Diccionarios (dictionaries)

 * http://www.diccionarios.com/
 * http://www.spanish-translator-services.com/
 * http://education.yahoo.com/reference/dict_en_es/
 * http://www.spanishdict.com/
 * http://dictionary.reference.com/
 * http://www.wordreference.com
 * http://en.wiktionary.org/wiki/Main_Page
 * http://www.wordmagicsoft.com/dictionary/tools/index.php

Misceláneo (miscellaneous)

 * http://www.lenguaje.com/cgi-bin/motnumeng.exe (change numbers into letters)
 * http://www.english-spanish-translations.com (Spanish Resources in English)
 * Medicine Spanish.

title: Spanish/Web Resources


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Spanish/History

Here is a little history about the development of this Wikibook, detailing milestones since the beginning of the project.

1 Jun 2006

 * 109 pages under Spanish namespace
 * 29 pages outside of Spanish namespace
 * 31 pages linked to from main page
 * 1 entry page
 * 10 lessons
 * 3 vocabulary
 * 3 grammar
 * 4 verbs
 * 4 learning tools
 * 3 "and more" pages
 * 1 history of the project
 * 1 list of contributors
 * 1 requests page

 * 10 lessons complete (one more linked to but as of yet unedited)
 * 2 stubs
 * Index of sound files (but all links are broken)
 * 14 exercises
 * 8 external links
 * 8 users listed as main contributors

03 Oct 2003

 * 6 lessons complete with vocabulary, grammar, and exercises
 * about 10 exercises (all of which are "fill in the blank" type of exercises)

25 Aug 2003

 * Most titles now in both Spanish and English
 * Index of soundfiles
 * First exercise (regular verbs)

16 Aug 2003

 * Introduction
 * 8 Lessons (3 lesson with more than just mere vocabulary)
 * 4 Appendices (Characters, Slang, Verb Tenses, Web Resources)
 * A handful of images and sound files
 * A total of about 10 users have contributed to the book
 * Talk page contains a huge number of potential topics

09 Aug 2003

 * Introduction
 * 6 Lessons (1 lesson with more than just mere vocabulary)
 * 1 Appendix (Slang)

02 Aug 2003

 * Spanish took off as the first Wikibook on a foreign language

title: Spanish/History


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Talk:Spanish/History

Untitled
Wow this whole thing is pretty exciting dont you think ? --Karl Wick

Yep, when looking back it is quite amazing how much has been already done in two weeks. If the number of contributers keeps increasing steadily then the book might become pretty good ;) -- Thomas Strohmann

I've updated the history because now it's about two months since the Spanish Wikibook started. I'm really glad how far it has developed by now. But as it's always the case with a Wiki Website: there is always a lot that can be improved. So keep up the good work! Thomas

Periodic Updating of History
It looks like the history was being updated weekly when this Wikibook started. This only continued for five weeks. Then it was not updated again for two years and eight months! So, if we're going to update this regularly again, how often does everyone think it should be updated? --Think Fast 14:55, 26 June 2006 (UTC)
 * 1 month --Think Fast 14:19, 27 June 2006 (UTC)

title: Talk:Spanish/History


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Talk:Spanish/Verbs

Someone who knows, please rearrange this from alphabetical order into a logical order for learning, such as from simplest to most complex.

I think having an alphabetic order for this appendix is OK since it would be used more likely for quick look-ups (learning the tenses from simple to complex should be done with the lessons structure).

I'm going to add all the forms of the typical 1st, 2nd and 3rd conjugation verbs in Spanish: amar, temer and partir. There should also be a page for the irregular verbs. Sabbut 10:20, 19 Dec 2003 (UTC)

title: Talk:Spanish/Verbs


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User:WouterVH~enwikibooks

Projects I'm currently working on:

 * Open Cursus Initiatief
 * Vrijschrift
 * Open-Education.org

title: User:WouterVH~enwikibooks


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Biochemistry/Thermodynamics

<< Introduction to Biochemistry | Thermodynamics | Catalysis >>

Why Do Substances React?
Chemical (and thus, biochemical) reactions only occur to a significant extent if they are energetically favorable. If the products are more stable than the reactants, then in general the reaction will, over time, tend to go forward. Ashes are more stable than wood, so once the energy of activation is supplied (e.g., by a match), the wood will burn. There are plenty of exceptions to the rule, of course, but as a rule of thumb it's pretty safe to say that if the products of a reaction represent a more stable state, then that reaction will go in the forward direction.

There are two factors that determine whether or not reactions changing reactants into products are considered to be favorable: these two factors are simply called enthalpy and entropy.

Enthalpy
Simply put, enthalpy is the heat content of a substance (H). Most people have an intuitive understanding of what heat is... we learn as children not to touch the burners on the stove when they are glowing orange. Enthalpy is not the same as that kind of heat. Enthalpy is the sum of all the internal energy of a substance's matter plus its pressure times its volume. Enthalpy is therefore defined by the following equation:

 * $$H = U + PV \,$$

where (all units given in SI)
 * H is the enthalpy
 * U is the internal energy, (joules)
 * P is the pressure of the system, (Pascals)
 * and V is the volume, (cubic meters)

If the enthalpy of the reactants while being converted to products ends up decreasing (ΔH < 0), that means that the products have less enthalpy than the reactants and energy is released to the environment. This reaction type is termed exothermic. In the course of most biochemical processes there is little change in pressure or volume, so the change in enthalpy accompanying a reaction generally reflects the change in the internal energy of the system. Thus, exothermic reactions in biochemistry are processes in which the products are lower in energy than the starting materials.

As an example, consider the reaction of glucose with oxygen to give carbon dioxide and water. Strong bonds form in the products, reducing the internal energy of the system relative to the reactants. This is a highly exothermic reaction, releasing 2805 kJ of energy per mole of glucose that burns (ΔH = -2805 kJ/mol). That energy is given off as heat.

Entropy
Entropy (symbol S) is the measure of randomness in something. It represents the most likely of statistical possibilities of a system, so the concept has extremely broad applications. In chemistry of all types, entropy is generally considered important in determining whether or not a reaction goes forward based on the principle that a less-ordered system is more statistically probable than a more-ordered system.

What does that mean, really? Well, if the volcano Mt. Vesuvius erupted next to a Roman-Empire era Mediterranean city, would the volcano be more likely to destroy the city, or build a couple of skyscrapers there? It's pretty obvious what would happen (or, rather, what did happen) because it makes sense to us that natural occurrences favor randomness (destruction) over order (construction, or in this case, skyscrapers). Entropy is just a mathematical way of expressing these essential differences.

When it comes to chemistry, there are three major concepts based on the concept of entropy:
 * 1) Intramolecular states (Degrees of freedom)
 * 2) * The more degrees of freedom (how much the molecules can move in space) a molecule has, the greater the degree of randomness, and thus, the greater the entropy.
 * 3) * There are three ways molecules can move in space, and each has a name: rotation = movement around an axis, vibration = intramolecular movement of two bonded atoms in relation to each other, and translation = a molecule moving from place to place.
 * 4) Intermolecular structures
 * 5) * When molecules can interact with each other by forming non-covalent bonds a structure is often created.
 * 6) * This tends to reduce randomness (and thus entropy) since any such association between molecules stabilizes the motion of both and decreases the possibilities for a random distribution.
 * 7) Number of possibilities
 * 8) * The more molecules present, the more ways of distributing the molecules in space - which because of statistical probabilities means more potential for randomness.
 * 9) * Also, if there is more space available to distribute the molecules within, the randomness increases for precisely the same reason
 * 10) * solid matter (least entropy) << liquids << gases (most entropy)

Changes in entropy are denoted as ΔS. For the reasons stated above (in the volcano situation), the increase of entropy (ΔS > 0) is considered to be favorable as far as the Universe in general is concerned. A decrease in entropy is generally not considered favorable unless an energetic component in the reaction system can make up for the decrease in entropy (see free energy below).

Gibbs Free Energy
Changes of both enthalpy (ΔH) and entropy (ΔS) combined decide how favorable a reaction is. For instance, burning a piece of wood releases energy (exothermic, favorable) and results in a substance with less structure (CO2 and H2O gas, both of which are less 'ordered' than solid wood). Thus, one could predict that once a piece of wood was set on fire, it would continue to burn until it was gone. The fact that it does so is ascribed to the change in its Gibbs Free Energy.

The overall favorability of a reaction was first described by the prominent chemist Josiah Willard Gibbs, who defined the free energy of a reaction as
 * ΔG = ΔH - T ΔS

where T is the temperature on the Kelvin temperature scale. The formula above assumes that pressure and temperature are constant during the reaction, which is almost always the case for biochemical reactions, and so this book makes the same assumption throughout.

The unit of ΔG (for Gibbs) is the "joule" in SI systems, but the unit of "calorie" is also often used because of its convenient relation to the properties of water. This book will use both terms as convenient, but the preference should really be for the SI notation.

What Does ΔG Really Mean?
If ΔG < 0 then the reactants should convert into products (signifying a forward reaction)... eventually. (Gibbs free energy says nothing about a reaction's rate, only its probability.) Likewise, for a given reaction if ΔG > 0 then it is known that the reverse reaction is favored to take place. A state where ΔG = 0 is called equilibrium, and this is the state where the reaction in both the forward and reverse directions take place at the same rate, thus not changing the net effect on the system.

How is equilibrium best explained? Alright, as an example set yourself on the living room carpet with your most gullible younger relative (a little nephew, niece or cousin will work fine). Take out a set of Monopoly, take one ten dollar bill for yourself and give your little relative the rest. Now both of you give the other 5% of all that you have. Do this again, and again, and again-again-again until eventually... you both have the same amount of money. This is precisely what the equilibrium of a reaction means, though equilibrium only very rarely results in an even, 50-50% split of products and reactants.

ΔG naturally varies with the concentration of reactants and products. When ΔG reaches 0, the reaction rate in the forward direction and the reaction rate in the reverse direction are the same, and the concentration of reactants and products no longer appears to change; this state is called the point of chemical equilibrium. You and your gullible little relative have stopped gaining and losing Monopoly money, respectively; you both keep exchanging the same amount each turn. Note again that equilibrium is dynamic. Chemical reaction does not cease at equilibrium, but products are converted to reactants and reactants are converted to products at exactly the same rate.

A small ΔG (that is, a value of ΔG close to 0) indicates that a reaction is somewhat reversible; the reaction can actually run backwards, converting products back to reactants. A very large ΔG (that is, ΔG >> 0 or ΔG << 0) is precisely the opposite, because it indicates that a given reaction is irreversible, i.e., once the reactants become products there are very few molecules that go back to reactants.

Metabolic pathways
The food we consume is processed to become a part of our cells; DNA, proteins, etc. If the biochemical reactions involved in this process were reversible, we would convert our own DNA back to food molecules if we stop eating even for a short period of time. To prevent this from happening, our metabolism is organized in metabolic pathways. These pathways are a series of biochemical reactions which are, as a whole, irreversible. The reactions of a pathway occur in a row, with the products of the first reaction being the reactants of the second, and so on:
 * A ⇌ B ⇌ C ⇌ D ⇌ E

At least one of these reactions has to be irreversible, e.g.:
 * A ⇀ B ⇌ C ⇌ D ⇀ E

The control of the irreversible steps (e.g., A → B) enables the cell to control the whole pathway and, thus, the amount of reactants used, as well as the amount of products generated.

Some metabolic pathways do have a "way back", but it is not the same pathway backwards. Instead, while using the reversible steps of the existing pathway, at least one of the irreversible reactions is bypassed by another (irreversible) one on the way back from E to A:
 * E ⇀ X ⇌ C ⇌ B ⇀ A

This reaction is itself controlled, letting the cell choose the direction in which the pathway is running.

Free energy and equilibrium
For ΔG, the free energy of a reaction, standard conditions were defined: Under these standard conditions, ΔG0' is defined as the standard free energy change.
 * concentration of reactants and products at 1Mol/dm³
 * temperature at 25°C
 * acidity at pH 7.0

For a reaction
 * A + B ⇌ C + D

the ratio of products to reactants is given by keq' (=keq at pH 7.0):
 * $$k_{eq}^\prime = {\text{products} \over \text{reactants}} = {[C][D] \over [A][B]}$$

The relationship of ΔG0' and keq' is
 * ΔG0' = - R T ln keq' = - R T 2.30 log10 keq'

with
 * R = 8.315 [J mol-1 K-1] (molar gas constant)
 * T = temperature [K]
 * In = loge ("e" equals to 2.71828...)

In theory, we can now decide if a reaction is favorable (ΔG0' < 0). However, the reaction might need a catalyst to occur within a reasonable amount of time. In biochemistry, such a catalyst is called an enzyme.

<< Introduction to Biochemistry | Thermodynamics | Catalysis >>

The purpose of DNA melting or DNA denaturation is emphasizing and demonstrating the life cycles of all organisms and the origin of replication. The origin of replication specific structure varies from species to species. Furthermore, the particular sequence of the origin of replication is in a genome which is the human genes. Nevertheless, DNA replication is also part of origin of replication which examen in the living organism such as prokaryotes and eukaryotes.

Thermodynamically, there are two important contributions on the DNA denaturation. One of them is the breaking all of the hydrogen bonds between the bases in the double helix; the other one is to overcome the stacking stability/energy of bases on top of each other. There are several methods to denature DNA; heat is known as the most common one use in laboratory. We just have to heat the sample to reach above its melting point, the unstack ability of DNA can be then monitored. Melting point and denaturation of DNA depend on several factors: the length of DNA, base-composition of DNA, the condition of the DNA and also the composition of buffer. For instance, the longer DNA will contain more H-bonds and more intermolecular forces compared to the shorter one; therefore, denaturations of longer DNA requires more time and more heat. Base-composition of DNA can also play as a key factor because A:T requires two hydrogen bonds and G:C interaction requires three hydrogen bonds. The region of DNA which contains more A:T will melt/denature more rapidly compared to G:C. We can also see how the condition of DNA is important because condition of DNA is related to whether the DNA is relax, supercoiled, linear or heavily nicked. It is important because it allow us to examine how much intermolecular forces existing in the double helix. Finally, condition of buffer is also playing an essential role to study DNA denaturation because it allow us to control the amount of ions present in the solution during the entire process.

Biologically, DNA denaturation can happen inside the cell during DNA replication or translation. In both cases, DNA denaturation is an essential step and a beginning to start each of the process. Most of the time, denaturation happened because of binding of protein or enzymes to a specific region of DNA, the binding will likely lead to open or denature of the helix. However, the actual meaning of the DNA melting is the denaturation of DNA which changes the structure of DNA from double stranded into single stranded. The processes of DNA denaturation is unwinding the double stranded deoxyribonucleic acid and breaks it into two single stranded by breaking the hydrogen bonding between the bases. DNA denaturation is also known of DNA annealing because it is reservable. The main steps DNA annealing are double helical will go through the denaturation to become partially denatured DNA then it will separated the strands into two single strand of DNA in random coils.

title: Biochemistry/Thermodynamics


text:
Biochemistry/Authors

Authors of this book

 * Magnus Manske : Started this book
 * Cyclop : Began the work on Proteins section
 * Brandon : Lipids and membranes chapter
 * Justin : Filled in a few gaps

title: Biochemistry/Authors


text:
Cell Biology/Introduction

 * 1) What is interesting about cell biology
 * 2) What is living
 * 3) What is a cell
 * 4) Size of cells
 * 5) Chemical elements necessary for life

Клетъчна биология/Въведение

title: Cell Biology/Introduction


text:
Cell Biology/Cell types

Types of Cells

 * 1) /Prokaryotes/
 * 2) /Eukaryotes/
 * 3) /Plant cells/
 * 4) /Bacteria/

Клетъчна биология/Типове клетки

title: Cell Biology/Cell types


text:
Cell Biology/Parts of the cell

 * 1) Membranes
 * 2) Organs
 * 3) Genetic material
 * 4) Energy supply (chloroplasts and mitochondria)|Energy supply (chloroplasts and mitochondria)

Клетъчна биология/Части на клетката

title: Cell Biology/Parts of the cell


text:
Cell Biology/Cell division

 * 1) /Cell cycle/
 * 2) /Meiosis/
 * 3) /Mitosis/

Клетъчна биология/Клетъчно делене

title: Cell Biology/Cell division


text:
Cell Biology/Gene expression and translation

 * 1) Gene expression
 * 2) Gene translation

Клетъчна биология/Генна експресия и транслация