Elbow it!!!, Really fun :) Options

[Hell-Rell]

aight you play this game by typin a word or sentence with your elbows! the user above give you


I'll start the word off


cookies

[*iNyCxShoRT*]

dfoioiiiue3wq - oh yes COOKIES

[xxxSiERRAxxx]

cookiesx DANGGIT I WAS CLOSE!!!!!!!!

oh yeah...next word...
supercalifragalisticexpeealidotious

muahahahahahahaha

[Hell-Rell]

sxuporevc zal.iufgvtrgsxbbhgsx,kwqazartolicx....


Belly

[krispy_kreme333]

belly
yes i did it!!!!!!!!!!!

milky way

[much2muse]

mkiilllk 3wazyh

girl scout

[xxxSiERRAxxx]

girl sckooooooooout


haha...

supercalifragalisticexpealidotius

[VintagexCore]

wsfhjkauoykdfsjkhghf. ughh.

Ducky.

[misskentucky]

ducky.

I have small elbows.

Christine!

[xxxSiERRAxxx]

Cnhristine

stewardesses

[*lil_chubby_cheeks2*]

dsxgytdeewardsxznb esaz

what the hell was that?

supercalifragilisticexpialidocious

biatchhhhhhh

[xxxSiERRAxxx]

supercalifrfagalistcexpealidotious

howmuchwoodcouldawoodchuckchuckifawoodchuckcouldchuckwood


bah

[CHiiCKENBUTT]

bgazhyu

player

[SarahxJoy]

plzaayer

---

Dignified.

[sw33t_rouge]

dignifirdefd

chicken

[SarahxJoy]

chickken

---

Telephone.

[*RiC3xBoy*]

uyjhfdrel.;dfrcxe nbjhpkoesw3

_
F uck, its so hard typing it with your elbow.

[SarahxJoy]

Hm..

f**k, it's so hard typing it with your elboqw.,.

(I messed up at the very end.. *Sigh*)

---

Triumphant.

[*RiC3xBoy*]

ghyttr65opiui8,mjnhmsdxnm jyu
-
How do you know people type so welll...@!?#@!#@!#?!@

[*salcha*]

....eff that

[*wind&fire*]

GTRIUMPHANT

PWNAGE

[*salcha*]

Acetone, Propylene Carbonate, Dimethyl Glutarate, Dimethyl Succinate, Dimethyl Adipate, Tocopheryl Acetate, Panthenol

1. Acetone:

Acetone is a colorless mobile flammable liquid with melting point at -95.4 °C and boiling point at 56.53 °C. It has a relative density of 0.819 (at 0 °C). It is readily soluble in water, ethanol, ether, etc., and itself serves as an important solvent. The most familiar household use of acetone is as the active ingredient in nail polish remover. Acetone is also used to make plastic, fibers, drugs, and other chemicals.

Source: http://en.wikipedia.org/wiki/Acetone

Acetone is a colorless, flammable, and volatile liquid with a characteristic odor that can be detected at very low concentrations. It is used in consumer goods such as nail polish remover, model airplane glue, lacquers, and paints. Industrially, it is used mainly as a solvent and an ingredient to make other chemicals.
Acetone is the common name for the simplest of the ketones. The formula of acetone is CH3 CO CH3.
The International Union of Pure and Applied Chemistry's (IUPAC) systematic name for acetone is 2-propanone; it is also called dimethyl ketone. The molecular weight is 58.08. Its boiling point is 133°F (56°C) and the melting point is -139.63°F (-95.4°C). The specific gravity is .7899.
Acetone is the simplest and most important of the ketones. It is a polar organic solvent and therefore dissolves a wide variety of substances. It has low chemical reactivity. These traits, and its relatively low cost, make it the solvent of choice for many processes. About 25% of the acetone produced is used directly as a solvent.
About 20% is used in the manufacture of methyl methacrylate to make plastics such as acrylic plastic, which can be used in place of glass. Another 20% is used to manufacture methyl isobutyl ketone, which serves as a solvent in surface coatings. Acetone is important in the manufacture of artificial fibers,explosives, and polycarbonate resins.
Because of its importance as a solvent and as a starting material for so many chemical processes, acetone is produced in the United States in great quantities. In 1999, the worldwide acetone market reached 9.4 billion lb (4.27 billion kg) at a steady growth rate of 2-3% per year. Acetone is prepared by several routes, from petrochemical sources. The methods of its synthesis include oxidation of 2-propanol (isopropyl alcohol), the hydration of propene, and as a co-product (with phenol) of the O2-oxidation of cumene.
Acetone is normally present in low concentrations in human blood and urine. Diabetic patients produce it in larger amounts. Sometimes "acetone breath" is detected on the breath of diabetics by others and wrongly attributed to the drinking of liquor. If acetone is splashed in the eyes, irritation or damage to the cornea will result. Excessive breathing of fumes causes headache, weariness, and irritation of the nose and throat. Drying results from contact with the skin.
Source: http://www.bookrags.com/sciences/chemistry/acetone-woc.html

I am not sure what you mean by “desired properties”. So what I will do is
describe the molecular structure of acetone and relate that to the most
common
usage of acetone.

For those who are not familiar with the molecular geometry and chemical
structure of acetone, imagine a threebladed airplane propeller. At the
center
of this propeller there is a carbon atom. The three blades of the propeller
are the bonds from the carbon atom. At the tip of one of the blades we
find an
oxygen atom. At the other two blades’ tips we find a CH3 group (for our
purposes, we can just imagine these as carbon atoms as well).

Since two of the propeller blades are made up of carbontocarbon bonds (a
carbon is at the tip of the blade and bonded to the central carbon), these
two
blades are essentially nonpolar. The third blade, however, is representative
of a carbontooxygen bond (it is actually a doublebond, but that is not
important right now). We know that oxygen is far more electronegative than
carbon, that is, oxygen is better at attracting the pairs of electrons that
form bonds between the central carbon and the oxygen. We can therefore
imagine
that the electrons spend a lot more time near the oxygen than near the
carbon.
Since electrons are negatively charged, the oxygen becomes partially negative
relative to the central carbon. We say that bond is polar, and, since
there is
no other polar bond within the acetone structure that could cancel out this
polarity, we say that the molecule has a netdipole. Just like a magnet, it
can be very good at attracting other molecules with a netdipole.

A lot of the chemical and physical properties of acetone can be traced
back to
the presence of this netdipole within its geometry. For example, acetone has
a higher boiling point than other molecules with similar masses to acetone
but
which do not have the netdipole. It dissolves very well in water because
water also acts like a polar molecule. It is a good solvent for a wide
range of
organic compounds, because most of these compounds have some polarity as well.

The chemical reactions of acetone are also dependent on this carbontooxygen
bond. Most compounds that react with acetone will either have a positive
functionality that tends to be attracted to the negative site found in the
oxygen, or a negative functionality that tends to be attracted to the
positive
site found in the carbon.

I hope that helps.
R. Gregorious
====================================================
I found searching Google for "flame temperatures" I found a nice little
table:
http://www.engineeringtoolbox.com/9_422.html .

They are more like 1900C for fuel pre-mixed with air.
Pre-mixed is what a Bunsen burner does, and your gas stove, too.

There is not really very much variation, they are all around1950C.
Hydrogen about 2050C.
Acetylene with 2400C is exceptional. (Isn't an alkane, admittedly.)
I have not yet guessed how you got your 1300C number.

More scatter, but lots of references, at:
http://hypertextbook.com/facts/1998/JamesDanyluk.shtml

It is likely a diffusion-flame can have a lower temperature than a
pre-mixed flame.
But it would not really be a constant of the gas,
it would be an effect of a particular geometry of flame,
loosing heat in all directions about as fast as it can make that heat.
A larger diffusion flame, perhaps partly occupied by hot crumbs of rock to
obstruct heat-radiation,
would burn hotter inside.
Eventually you have re-invented some ancient-style pottery furnace,
and it might approach the temperature of a pre-mixed flame.

The alternatives are "pre-mixed" or "diffusion-flame"
(in addition to choosing between air and pure oxygen, of course).

A diffusion flame is like a candle flame. Fuel gas on the inside, air on
the outside,
diffusing through each other as fast as they can so they can react.
Not really very fast. A diffusion flame "drifts" upwards by
gravity/convection.

A pre-mixed flame is a jet of dangerously mixed cool gas going one way,
and a flame front racing the other way (upstream) as fast as it can.
There is a narrow point in the nozzle where the gas flows faster than the
flame can propagate,
then the flow spreads out and slows down.
That is why the flame front can stand still before your eyes.
It is the round bottom of the little blue flame in your gas stove top burner.
The gasses burn near-instantly as they cross the "flame front".
Suddenly the gasses reach their ideal maximum temperature, then they start
cooling down as they flow away.
You therefore have a small place where the torch-flame has not yet cooled
down by any significant amount.

If the distinction is not clear to you, think about the three components
required for fire -
fuel, oxidizer, and heat.
If you mix fuel and oxidizer first, then add heat, that is pre-mixed.
If you heat fuel and/or air first, then mix, it will be a diffusion-flame.
When the fuel and oxidizer enter the hot zone by different paths, it must
be a diffusion-flame.

hope that helps-
Jim Swenson


Source: http://www.newton.dep.anl.gov/askasci/chem03/chem03356.htm









You better type the whole thing.

[*chaneun*]

Acetone, Propylene Carbonate, Dimethyl Glutarate, Dimethyl Succinate, Dimethyl Adipate, Tocopheryl Acetate, Panthenol

1. Acetone:

Acetone is a colorless mobile flammable liquid with melting point at -95.4 °C and boiling point at 56.53 °C. It has a relative density of 0.819 (at 0 °C). It is readily soluble in water, ethanol, ether, etc., and itself serves as an important solvent. The most familiar household use of acetone is as the active ingredient in nail polish remover. Acetone is also used to make plastic, fibers, drugs, and other chemicals.

Source: http://en.wikipedia.org/wiki/Acetone

Acetone is a colorless, flammable, and volatile liquid with a characteristic odor that can be detected at very low concentrations. It is used in consumer goods such as nail polish remover, model airplane glue, lacquers, and paints. Industrially, it is used mainly as a solvent and an ingredient to make other chemicals.
Acetone is the common name for the simplest of the ketones. The formula of acetone is CH3 CO CH3.
The International Union of Pure and Applied Chemistry's (IUPAC) systematic name for acetone is 2-propanone; it is also called dimethyl ketone. The molecular weight is 58.08. Its boiling point is 133°F (56°C) and the melting point is -139.63°F (-95.4°C). The specific gravity is .7899.
Acetone is the simplest and most important of the ketones. It is a polar organic solvent and therefore dissolves a wide variety of substances. It has low chemical reactivity. These traits, and its relatively low cost, make it the solvent of choice for many processes. About 25% of the acetone produced is used directly as a solvent.
About 20% is used in the manufacture of methyl methacrylate to make plastics such as acrylic plastic, which can be used in place of glass. Another 20% is used to manufacture methyl isobutyl ketone, which serves as a solvent in surface coatings. Acetone is important in the manufacture of artificial fibers,explosives, and polycarbonate resins.
Because of its importance as a solvent and as a starting material for so many chemical processes, acetone is produced in the United States in great quantities. In 1999, the worldwide acetone market reached 9.4 billion lb (4.27 billion kg) at a steady growth rate of 2-3% per year. Acetone is prepared by several routes, from petrochemical sources. The methods of its synthesis include oxidation of 2-propanol (isopropyl alcohol), the hydration of propene, and as a co-product (with phenol) of the O2-oxidation of cumene.
Acetone is normally present in low concentrations in human blood and urine. Diabetic patients produce it in larger amounts. Sometimes "acetone breath" is detected on the breath of diabetics by others and wrongly attributed to the drinking of liquor. If acetone is splashed in the eyes, irritation or damage to the cornea will result. Excessive breathing of fumes causes headache, weariness, and irritation of the nose and throat. Drying results from contact with the skin.
Source: http://www.bookrags.com/sciences/chemistry/acetone-woc.html

I am not sure what you mean by “desired properties”. So what I will do is
describe the molecular structure of acetone and relate that to the most
common
usage of acetone.

For those who are not familiar with the molecular geometry and chemical
structure of acetone, imagine a threebladed airplane propeller. At the
center
of this propeller there is a carbon atom. The three blades of the propeller
are the bonds from the carbon atom. At the tip of one of the blades we
find an
oxygen atom. At the other two blades’ tips we find a CH3 group (for our
purposes, we can just imagine these as carbon atoms as well).

Since two of the propeller blades are made up of carbontocarbon bonds (a
carbon is at the tip of the blade and bonded to the central carbon), these
two
blades are essentially nonpolar. The third blade, however, is representative
of a carbontooxygen bond (it is actually a doublebond, but that is not
important right now). We know that oxygen is far more electronegative than
carbon, that is, oxygen is better at attracting the pairs of electrons that
form bonds between the central carbon and the oxygen. We can therefore
imagine
that the electrons spend a lot more time near the oxygen than near the
carbon.
Since electrons are negatively charged, the oxygen becomes partially negative
relative to the central carbon. We say that bond is polar, and, since
there is
no other polar bond within the acetone structure that could cancel out this
polarity, we say that the molecule has a netdipole. Just like a magnet, it
can be very good at attracting other molecules with a netdipole.

A lot of the chemical and physical properties of acetone can be traced
back to
the presence of this netdipole within its geometry. For example, acetone has
a higher boiling point than other molecules with similar masses to acetone
but
which do not have the netdipole. It dissolves very well in water because
water also acts like a polar molecule. It is a good solvent for a wide
range of
organic compounds, because most of these compounds have some polarity as well.

The chemical reactions of acetone are also dependent on this carbontooxygen
bond. Most compounds that react with acetone will either have a positive
functionality that tends to be attracted to the negative site found in the
oxygen, or a negative functionality that tends to be attracted to the
positive
site found in the carbon.

I hope that helps.
R. Gregorious
====================================================
I found searching Google for "flame temperatures" I found a nice little
table:
http://www.engineeringtoolbox.com/9_422.html .

They are more like 1900C for fuel pre-mixed with air.
Pre-mixed is what a Bunsen burner does, and your gas stove, too.

There is not really very much variation, they are all around1950C.
Hydrogen about 2050C.
Acetylene with 2400C is exceptional. (Isn't an alkane, admittedly.)
I have not yet guessed how you got your 1300C number.

More scatter, but lots of references, at:
http://hypertextbook.com/facts/1998/JamesDanyluk.shtml

It is likely a diffusion-flame can have a lower temperature than a
pre-mixed flame.
But it would not really be a constant of the gas,
it would be an effect of a particular geometry of flame,
loosing heat in all directions about as fast as it can make that heat.
A larger diffusion flame, perhaps partly occupied by hot crumbs of rock to
obstruct heat-radiation,
would burn hotter inside.
Eventually you have re-invented some ancient-style pottery furnace,
and it might approach the temperature of a pre-mixed flame.

The alternatives are "pre-mixed" or "diffusion-flame"
(in addition to choosing between air and pure oxygen, of course).

A diffusion flame is like a candle flame. Fuel gas on the inside, air on
the outside,
diffusing through each other as fast as they can so they can react.
Not really very fast. A diffusion flame "drifts" upwards by
gravity/convection.

A pre-mixed flame is a jet of dangerously mixed cool gas going one way,
and a flame front racing the other way (upstream) as fast as it can.
There is a narrow point in the nozzle where the gas flows faster than the
flame can propagate,
then the flow spreads out and slows down.
That is why the flame front can stand still before your eyes.
It is the round bottom of the little blue flame in your gas stove top burner.
The gasses burn near-instantly as they cross the "flame front".
Suddenly the gasses reach their ideal maximum temperature, then they start
cooling down as they flow away.
You therefore have a small place where the torch-flame has not yet cooled
down by any significant amount.

If the distinction is not clear to you, think about the three components
required for fire -
fuel, oxidizer, and heat.
If you mix fuel and oxidizer first, then add heat, that is pre-mixed.
If you heat fuel and/or air first, then mix, it will be a diffusion-flame.
When the fuel and oxidizer enter the hot zone by different paths, it must
be a diffusion-flame.

hope that helps-
Jim Swenson
Source: http://www.newton.dep.anl.gov/askasci/chem03/chem03356.htm
You better type the whole thing.

They are more like 1900C for fuel pre-mixed with air.
Pre-mixed is what a Bunsen burner does, and your gas stove, too.

There is not really very much variation, they are all around1950C.
Hydrogen about 2050C.
Acetylene with 2400C is exceptional. (Isn't an alkane, admittedly.)
I have not yet guessed how you got your 1300C number.

More scatter, but lots of references, at:
http://hypertextbook.com/facts/1998/JamesDanyluk.shtml

It is likely a diffusion-flame can have a lower temperature than a
pre-mixed flame.
But it would not really be a constant of the gas,
it would be an effect of a particular geometry of flame,
loosing heat in all directions about as fast as it can make that heat.
A larger diffusion flame, perhaps partly occupied by hot crumbs of rock to
obstruct heat-radiation,
would burn hotter inside.
Eventually you have re-invented some ancient-style pottery furnace,
and it might approach the temperature of a pre-mixed flame.

The alternatives are "pre-mixed" or "diffusion-flame"
(in addition to choosing between air and pure oxygen, of course).

A diffusion flame is like a candle flame. Fuel gas on the inside, air on
the outside,
diffusing through each other as fast as they can so they can react.
Not really very fast. A diffusion flame "drifts" upwards by
gravity/convection.

A pre-mixed flame is a jet of dangerously mixed cool gas going one way,
and a flame front racing the other way (upstream) as fast as it can.
There is a narrow point in the nozzle where the gas flows faster than the
flame can propagate,
then the flow spreads out and slows down.
That is why the flame front can stand still before your eyes.
It is the round bottom of the little blue flame in your gas stove top burner.
The gasses burn near-instantly as they cross the "flame front".
Suddenly the gasses reach their ideal maximum temperature, then they start
cooling down as they flow away.
You therefore have a small place where the torch-flame has not yet cooled
down by any significant amount.

If the distinction is not clear to you, think about the three components
required for fire -
fuel, oxidizer, and heat.
If you mix fuel and oxidizer first, then add heat, that is pre-mixed.
If you heat fuel and/or air first, then mix, it will be a diffusion-flame.
When the fuel and oxidizer enter the hot zone by different paths, it must
be a diffusion-flame.

I COPIED AND PASTED WITH MY ELBOWS. SO HA!




sally sucks at everything. no, she pwns.

[*RiC3xBoy*]

Quantum mechanics is a more fundamental theory than Newtonian mechanics and classical electromagnetism, in the sense that it provides accurate and precise descriptions for many phenomena that these "classical" theories simply cannot explain on the atomic and subatomic level. It is necessary to use quantum mechanics to understand the behavior of systems at atomic length scales and smaller. For example, if Newtonian mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus. However, in the natural world the electron normally remains in a stable orbit around a nucleus -- seemingly defying classical electromagnetism.

Quantum mechanics was initially developed to explain the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom developed as an explanation for the electron's staying in its orbital, which could not be explained by Newton's laws of motion and by classical electromagnetism.

Quantum mechanics uses complex number wave functions (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space to explain such effects. These are related to classical physics largely through probability. Probability in the context of quantum mechanics has to do with the likelihood of finding a system in a particular state at a certain time, for example, finding an electron, in a particular region around the nucleus at a particular time. Therefore, electrons cannot be pictured as localized particles in space but rather should be thought of as "clouds" of negative charge spread out over the entire orbit. These clouds represent the regions around the nucleus where the probability of "finding" an electron is the largest. This probability cloud obeys a quantum mechanical principle called Heisenberg's Uncertainty Principle, which states that there is an uncertainty in the classical position of any subatomic particle, including the electron; so instead of describing where an electron or other particle is, the entire range of possible values is used, describing a probability distribution. So in normal atoms with electrons in stationary states, the probability of the electron being within the nucleus (or somewhere else in atom within similarly small volume) is nearly zero according to the Uncertainty Principle (it is nearly zero as the nucleus has a volume and is not a point). Therefore, quantum mechanics, translated to Newton's equally deterministic description, leads to a probabilistic description of nature.

The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. This led to a theory of unity between subatomic particles and electromagnetic waves called wave-particle duality in which particles and waves were neither one nor the other, but had certain properties of both. While quantum mechanics describes the world of the very small, it also is needed to explain certain "macroscopic quantum systems" such as superconductors and superfluids.

Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (i) the quantization (discretization) of certain physical quantities, (ii) wave-particle duality, (iii) the uncertainty principle, and (iv) quantum entanglement. Each of these phenomena will be described in greater detail in subsequent sections.

Most physicists believe that quantum mechanics provides a correct description for the physical world under almost all circumstances. However, the effects of quantum mechanics are generally not significant when considering the observable Universe as a whole. This is because although atoms and subatomic particles are the building blocks of matter, when analyzing the universe on large scales one finds that the dominant force becomes gravity -- which is described using Einstein's general theory of relativity. In some cases, both general relativity and quantum mechanics converge. As an example, general relativity is unable to explain what will happen if a subatomic particle hits the singularity of a black hole which is a phenomenon predicted by general relativity and involves gravity in the macro world. Only quantum mechanics can provide the answer: the particle's position will have an uncertainty that follows the Heisenberg Uncertainty Principle, such that it might not really reach the singularity and thus escape the possible collapse to infinite density.

It is believed that the theories of general relativity and quantum mechanics, the two great achievements of physics in the 20th century, contradict one another for two main reasons. One is that the former is an essentially deterministic theory and the latter is essentially indeterministic. Secondly, general relativity relies mainly on the force of gravity while quantum mechanics relies mainly on the other three fundamental forces, those being the strong, the weak, and the electromagnetic. The question of how to resolve this contradiction remains an area of active research (see, for example, quantum gravity).

In certain situations, the laws of classical physics approximate the laws of quantum mechanics to a high degree of precision. This is often expressed by saying that in case of large quantum numbers quantum mechanics "reduces" to classical mechanics and classical electromagnetism . This situation is called the correspondence, or classical limit.

Quantum mechanics can be formulated in either a relativistic or non-relativistic manner. Relativistic quantum mechanics (quantum field theory) provides the framework for some of the most accurate physical theories known. Still, non-relativistic quantum mechanics is also used due to its simplicity and when relativistic effects are relatively small. We will use the terms quantum mechanics, quantum physics, and quantum theory synonymously, to refer to both relativistic and non-relativistic quantum mechanics. It should be noted, however, that certain authors refer to "quantum mechanics" in the more restricted sense of non-relativistic quantum mechanics. Also, in quantum mechanics, the use of the term particle refers to an elementary or subatomic particle.

[*wind&fire*]

type string theory bitch
Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figured out the first known string physics -- the harmonic relationship. Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notes (i.e. middle C and high C) if the ratio of the lengths of the two strings were a whole number.
Pythagoras discovered this by looking and listening. Today that information is more precisely encoded into mathematics, namely the wave equation for a string with a tension T and a mass per unit length m. If the string is described in coordinates as in the drawing below, where x is the distance along the string and y is the height of the string, as the string oscillates in time t,

then the equation of motion is the one-dimensional wave equation

Nonrelativistic string equation

where vw is the wave velocity along the string.
When solving the equations of motion, we need to know the "boundary conditions" of the string. Let's suppose that the string is fixed at each end and has an unstretched length L. The general solution to this equation can be written as a sum of "normal modes", here labeled by the integer n, such that

sum of string normal modes

The condition for a normal mode is that the wavelength be some integral fraction of twice the string length, or

The frequency of the normal mode is then

The normal modes are what we hear as notes. Notice that the string wave velocity vw increases as the tension of the string is increased, and so the normal frequency of the string increases as well. This is why a guitar string makes a higher note when it is tightened.
But that's for a nonrelativistic string, one with a wave velocity much smaller than the speed of light. How do we write the equation for a relativistic string?
According to Einstein's theory, a relativistic equation has to use coordinates that have the proper Lorentz transformation properties. But then we have a problem, because a string oscillates in space and time, and as it oscillates, it sweeps out a two-dimensional surface in spacetime that we call a world sheet (compared with the world line of a particle).
In the nonrelativistic string, there was a clear difference between the space coordinate along the string, and the time coordinate. But in a relativistic string theory, we wind up having to consider the world sheet of the string as a two-dimensional spacetime of its own, where the division between space and time depends upon the observer.
The classical equation can be written as

Relativistic string eqaution

where s and t are coordinates on the string world sheet representing space and time along the string, and the parameter c2 is the ratio of the string tension to the string mass per unit length.
These equations of motion can be derived from Euler-Lagrange equations from an action based on the string world sheet

Bosonic string action

The spacetime coordinates Xm of the string in this picture are also fields Xm in a two-dimension field theory defined on the surface that a string sweeps out as it travels in space. The partial derivatives are with respect to the coordinates s and t on the world sheet and hmn is the two-dimensional metric defined on the string world sheet.
The general solution to the relativistic string equations of motion looks very similar to the classical nonrelativistic case above. The transverse space coordinates can be expanded in normal modes as

The string solution above is unlike a guitar string in that it isn't tied down at either end and so travels freely through spacetime as it oscillates. The string above is an open string, with ends that are floppy.
For a closed string, the boundary conditions are periodic, and the resulting oscillating solution looks like two open string oscillations moving in the opposite direction around the string. These two types of closed string modes are called right-movers and left-movers, and this difference will be important later in the supersymmetric heterotic string theory.
This is classical string. When we add quantum mechanics by making the string momentum and position obey quantum commutation relations, the oscillator mode coefficients have the commutation relations

The quantized string oscillator modes wind up giving representations of the Poincaré group, through which quantum states of mass and spin are classified in a relativistic quantum field theory.
So this is where the elementary particle arise in string theory. Particles in a string theory are like the harmonic notes played on a string with a fixed tension

String tension

The parameter a' is called the string parameter and the square root of this number represents the approximate distance scale at which string effects should become observable.
In the generic quantum string theory, there are quantum states with negative norm, also known as ghosts. This happens because of the minus sign in the spacetime metric, which implies that

So there ends up being extra unphysical states in the string spectrum.
In 26 spacetime dimensions, these extra unphysical states wind up disappearing from the spectrum. Therefore. bosonic string quantum mechanics is only consistent if the dimension of spacetime is 26.
By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that the quantum modes of the string look just like the particles we see in spacetime, with mass that depends on the spin according to the formula

Regge formula

Remember that boundary conditions are important for string behavior. Strings can be open, with ends that travel at the speed of light, or closed, with their ends joined in a ring.
One of the particle states of a closed string has zero mass and two units of spin, the same mass and spin as a graviton, the particle that is supposed to be the carrier of the gravitational force.

[SarahxJoy]

Wudeva.

[*wind&fire*]

Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figured out the first known string physics -- the harmonic relationship. Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notes (i.e. middle C and high C) if the ratio of the lengths of the two strings were a whole number.
Pythagoras discovered this by looking and listening. Today that information is more precisely encoded into mathematics, namely the wave equation for a string with a tension T and a mass per unit length m. If the string is described in coordinates as in the drawing below, where x is the distance along the string and y is the height of the string, as the string oscillates in time t,

then the equation of motion is the one-dimensional wave equation

Nonrelativistic string equation

where vw is the wave velocity along the string.
When solving the equations of motion, we need to know the "boundary conditions" of the string. Let's suppose that the string is fixed at each end and has an unstretched length L. The general solution to this equation can be written as a sum of "normal modes", here labeled by the integer n, such that

sum of string normal modes

The condition for a normal mode is that the wavelength be some integral fraction of twice the string length, or

The frequency of the normal mode is then

The normal modes are what we hear as notes. Notice that the string wave velocity vw increases as the tension of the string is increased, and so the normal frequency of the string increases as well. This is why a guitar string makes a higher note when it is tightened.
But that's for a nonrelativistic string, one with a wave velocity much smaller than the speed of light. How do we write the equation for a relativistic string?
According to Einstein's theory, a relativistic equation has to use coordinates that have the proper Lorentz transformation properties. But then we have a problem, because a string oscillates in space and time, and as it oscillates, it sweeps out a two-dimensional surface in spacetime that we call a world sheet (compared with the world line of a particle).
In the nonrelativistic string, there was a clear difference between the space coordinate along the string, and the time coordinate. But in a relativistic string theory, we wind up having to consider the world sheet of the string as a two-dimensional spacetime of its own, where the division between space and time depends upon the observer.
The classical equation can be written as

Relativistic string eqaution

where s and t are coordinates on the string world sheet representing space and time along the string, and the parameter c2 is the ratio of the string tension to the string mass per unit length.
These equations of motion can be derived from Euler-Lagrange equations from an action based on the string world sheet

Bosonic string action

The spacetime coordinates Xm of the string in this picture are also fields Xm in a two-dimension field theory defined on the surface that a string sweeps out as it travels in space. The partial derivatives are with respect to the coordinates s and t on the world sheet and hmn is the two-dimensional metric defined on the string world sheet.
The general solution to the relativistic string equations of motion looks very similar to the classical nonrelativistic case above. The transverse space coordinates can be expanded in normal modes as

The string solution above is unlike a guitar string in that it isn't tied down at either end and so travels freely through spacetime as it oscillates. The string above is an open string, with ends that are floppy.
For a closed string, the boundary conditions are periodic, and the resulting oscillating solution looks like two open string oscillations moving in the opposite direction around the string. These two types of closed string modes are called right-movers and left-movers, and this difference will be important later in the supersymmetric heterotic string theory.
This is classical string. When we add quantum mechanics by making the string momentum and position obey quantum commutation relations, the oscillator mode coefficients have the commutation relations

The quantized string oscillator modes wind up giving representations of the Poincaré group, through which quantum states of mass and spin are classified in a relativistic quantum field theory.
So this is where the elementary particle arise in string theory. Particles in a string theory are like the harmonic notes played on a string with a fixed tension

String tension

The parameter a' is called the string parameter and the square root of this number represents the approximate distance scale at which string effects should become observable.
In the generic quantum string theory, there are quantum states with negative norm, also known as ghosts. This happens because of the minus sign in the spacetime metric, which implies that

So there ends up being extra unphysical states in the string spectrum.
In 26 spacetime dimensions, these extra unphysical states wind up disappearing from the spectrum. Therefore. bosonic string quantum mechanics is only consistent if the dimension of spacetime is 26.
By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that the quantum modes of the string look just like the particles we see in spacetime, with mass that depends on the spin according to the formula

Regge formula

Remember that boundary conditions are important for string behavior. Strings can be open, with ends that travel at the speed of light, or closed, with their ends joined in a ring.
One of the particle states of a closed string has zero mass and two units of spin, the same mass and spin as a graviton, the particle that is supposed to be the carrier of the gravitational force.

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The multidimensional continua that Riemann was concerned with are essentially instances of what is now known as a real n-dimensional smooth manifold. For brevity, call them ‘n-manifolds’. (For example, a sphere, a Möbius strip and the surface of a Henry Moore sculpture may be regarded as 2-manifolds; the spacetimes models of current cosmology are 4-manifolds.) Conditions (a) -- © provide a full characterization of n-manifolds.

(a) An n-manifold M is a set of points that can be pieced together from partially overlapping patches, such that every point of M lies in at least one patch.

(b) M is endowed with a neighborhood structure (a topology) such that, if U is a patch of M, there is a continuous one-one mapping f of U onto some region of Rn, with continuous inverse f-1. (Rn denotes here the collection of all real number n-tuples, with the standard topology generated by the open balls.) f is a coordinate system or chart of M; the k-th number in the n-tuple assigned by a chart f to a point P in f's patch is called the k-th coordinate of P by f; the k-th coordinate function of chart f is the real-valued function that assigns to each point of the patch its k-th coordinate by f.

© There is a collection A of charts of M, which contains at least one chart defined on each patch of M and is such that, if g and h belong to A, the composite mappings h g-1 and g h-1 --- known as coordinate transformations --- are differentiable to every order wherever they are well defined. (Denote the real number n-tuple <a1, ... , an> by a. The mapping h g-1 is well defined at a if a is the valued assigned by g to some point P of M to which h also assigns a value. Suppose that the latter value h(P) = <b1, ... ,bn> = b; then, b = h g-1(a). Since h g-1 maps a region of Rn into Rn, it makes sense to say that h g-1 is differentiable.) Such a collection A is called an atlas.[*]

It is the pair <M, A> that, strictly speaking, is an n-manifold, in the sense defined above. If <M1,A1> and <M2,A2> are an n-manifold and an m-manifold, respectively, it makes good sense to say that a mapping f of M1 into M2 is differentiable at a point P of M1 if, for a chart h defined at P and a chart g defined at f(P), the composite mapping g f h-1 is differentiable at h(P). (Condition © implies that the fulfillment of this requirement does not depend on the choice of h and g.) f is differentiable if it is differentiable at every point of M1.

Let <M,A> be an n-manifold. To each point P of M one associates a vector space, which is known as the tangent space at P and is denoted by TPM. The idea is based on the intuitive notion of a plane tangent to a surface at a given point. It can be constructed as follows. Let be a one-one differentiable mapping of a real open interval I into M. We can think of the successive values of as forming the path of a point that moves through M during a time interval represented by I. We call a curve in M (parametrized by u I). Put (t0) = P for a fixed number t0 in I. Consider the collection F(P) of all differentiable real-valued functions defined on some neighborhood of P. With the ordinary operations of function addition and multiplication by a constant, F(P) has the structure of a vector space. Each function f in F(P) varies smoothly with u, along the path of , in some neighborhood of P. Its rate of variation at P = (t0) is properly expressed by the derivative d(f)/du at u = t0. As f ranges over F(P), the value of d(f)/du at u = t0 is apt to vary in R. So we have here a mapping of F(P) into R, which we denote by (u). It is in fact a linear function and therefore a vector in the dual space F*(P) of real-valued linear functions on F(P). Call it the tangent to at P. The tangents at P to all the curves whose paths go through P span an n-dimensional subspace of F*(P). This subspace is, by definition, the tangent space TPM. The tangent spaces at all points of an n-manifold M can be bundled together in a natural way into a 2n-manifold TM. The projection mapping of TM onto M assigns to each tangent vector v in TPM the point (v) at which v is tangent to M. The structure <TM,M,> is the tangent bundle over M. A vector field on M is a section of TM, i.e., a differentiable mapping f of M into TM such that f sends each point P of M to itself; such a mapping obviously assigns to P a vector in TPM.

Any vector space V is automatically associated with other vector spaces, such as the dual space V* of linear functions on V, and the diverse spaces of multilinear functions on V, on V*, and on any possible combination of V and V*. This holds, of course, for each tangent space of an n-manifold M. The dual of TPM is known as the cotangent space at P. There is a natural way of bundling together the cotangent spaces of M into a 2n-manifold, the cotangent bundle. Generally speaking, all the vector spaces of a definite type associated with the tangent and cotangent spaces of M can be naturally bundled together into a k-manifolds (for suitable integers k, depending on the nature of the bundled items). A section of any of these bundles is a tensor field on M (of rank r, if the bundled objects are r-linear functions).

A Riemannian metric g on the n-manifold <M,A> is a tensor field of rank 2 on M. Thus, g assigns to each P in M a bilinear function gP on TPM. For any P in M and any vectors v, w, in TPM, gP must meet these requirements:

(i) gP(v,w) = gP(w,v) (symmetry)

(ii) gP(v,w) = 0 for all vectors w in TPM if and only if v is the 0-vector (non-degeneracy)

(iii) gP(v,v) > 0 unless v is the 0-vector (positive definiteness).

It is worth noting that the so-called Lorentzian metrics defined by relativity theory on its spacetime models meet requirements (i) and (ii), but not (iii), and are therefore usually said to be semi-Riemannian.

The length (v) of a vector v in TPM is defined by |(v)|2 = gP(v,v). Let be a curve in M. Let (u) be the tangent to at the point (u). The length of 's path from (a) to (b) is measured by the integral

((u))du

Thus, in Riemannian geometry, the length of the tangent vector (u) bears witness to the advance of curve g as it passes through the point (u). The definition of the length of a curve leads at once to the notion of a geodesic (or straightest) curve, which is characterized by the fact that its length is extremal; in other words, a geodesic is either the greatest or the shortest among all the curves that trace out neighboring paths between the same two points.

In his study of curved surfaces, Gauss introduced a real-valued function, the Gaussian curvature, which measures a surface's local deviation from flatness in terms of the surface's intrinsic geometry. Riemann extended this concept of curvature to Riemannian n-manifolds. He observed that each geodesic through a point in such a manifold is fully determined by its tangent vector at that point. Consider a point P in a Riemannian n-manifold <M,A,g> and two linearly independent vectors v and w in TPM. The geodesics determined by all linear combinations of v and w form a 2-manifold about P, with a definite Gaussian curvature KP(v,w) at P. The real number KP(v,w) measures the curvature of M at P in the ‘surface direction’ (Riemann 1854, p. 145) fixed by v and w. Riemann (1861) thought up a global mapping, depending on the metric g, that yields the said values KP(v,w) on appropriate arguments P, v and w. Nowadays this object is conceived as a tensor field of rank 4, which assigns to each point P in a Riemannian n-manifold <M,A,g> a 4-linear function on the tangent space TPM. It is therefore known as the Riemann tensor. Given the above definition of KP(v,w) it is clear that, if n = 2, the Riemann tensor reduces to the Gaussian curvature function.

[SarahxJoy]

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Calculus and Analysis > Calculus > Differential Calculus v
MathWorld Contributors > Derwent v

Derivative

COMMENT On this Page EXPLORE THIS TOPIC IN the MathWorld ClassroomDOWNLOAD Mathematica Notebook

The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have. The "simple" derivative of a function f with respect to a variable x is denoted either f^'(x) or
(df)/(dx), (1)

often written in-line as df/dx. When derivatives are taken with respect to time, they are often denoted using Newton's Eric Weisstein's World of Biography overdot notation for fluxions,
(dx)/(dt)==x^.. (2)

When a derivative is taken n times, the notation f^((n))(x) or
(d^nf)/(dx^n) (3)

is used, with
x^.,x^..,x^..., (4)

etc., the corresponding fluxion notation. When a function f(x,y,...) depends on more than one variable, a partial derivative
(partialf)/(partialx),(partial^2f)/(partialxpartialy), etc. (5)

can be used to specify the derivative with respect to one or more variables.

The derivative of a function f(x) with respect to the variable x is defined as
f^'(x)=lim_(h->0)(f(x+h)-f(x))/h, (6)

but may also be calculated more symmetrically as
f^'(x)==lim_(h->0)(f(x+h)-f(x-h))/(2h), (7)

provided the derivative is known to exist.

It should be noted that the above definitions refer to "real" derivatives, i.e., derivatives which are restricted to directions along the real axis. However, this restriction is artificial, and derivatives are most naturally defined in the complex plane, where they are sometimes explicitly referred to as complex derivatives. In order for complex derivatives to exist, the same result must be obtained for derivatives taken in any direction in the complex plane. Somewhat surprisingly, almost all of the important functions in mathematics satisfy this property, which is equivalent to saying that they satisfy the Cauchy-Riemann equations.

These considerations can lead to confusion for students because elementary calculus texts commonly consider only "real" derivatives, never alluding the the existence of complex derivatives, variables, or functions. For example, textbook examples to the contrary, the "derivative" (read: complex derivative) d|z|/dz of the absolute value function |z| does not exist because at every point in the complex plane, the value of the derivative depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than x==0 as
(d|x|)/(dx)=={-1 for x<0; undefined for x==0; 1 for x>0. (8)

As a result of the fact that computer algebra programs such as Mathematica generically deal with complex variables (i.e., the definition of derivative always means complex derivative), d|x|/dx correctly returns unevaluated by such software.

If the first derivative exists, the second derivative may be defined as
f^('')(x)=lim_(h->0)(f^'(x+h)-f^'(x))/h (9)

and calculated more symmetrically as
f^('')(x)==lim_(h->0)(f(x+2h)-2f(x+h)+f(x))/(h^2), (10)

again provided the second derivative is known to exist.

Note that in order for the limit to exist, both lim_(h->0^+) and lim_(h->0^-) must exist and be equal, so the function must be continuous. However, continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite Eric Weisstein's World of Biography wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

A three-dimensional generalization of the derivative to an arbitrary direction is known as the directional derivative. In general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into "tangent maps."

Performing numerical differentiation is in many ways more difficult than numerical integration. This is because while numerical integration requires only good continuity properties of the function being integrated, numerical differentiation requires more complicated properties such as Lipschitz classes.

Simple derivatives of some simple functions follow.
d/(dx)x^n = nx^(n-1) (11)
d/(dx)lnx = 1/x (12)
d/(dx)sinx = cosx (13)
d/(dx)cosx = -sinx (14)
d/(dx)tanx = sec^2x (15)
d/(dx)cscx = -cscxcotx (16)
d/(dx)secx = secxtanx (17)
d/(dx)cotx = -csc^2x (18)
d/(dx)e^x = e^x (19)
d/(dx)a^x = (lna)a^x (20)
d/(dx)sin^(-1)x = 1/(sqrt(1-x^2)) (21)
d/(dx)cos^(-1)x = -1/(sqrt(1-x^2)) (22)
d/(dx)tan^(-1)x = 1/(1+x^2) (23)
d/(dx)cot^(-1)x = -1/(1+x^2) (24)
d/(dx)sec^(-1)x = 1/(xsqrt(x^2-1)) (25)
d/(dx)csc^(-1)x = -1/(xsqrt(x^2-1)) (26)
d/(dx)sinhx = coshx (27)
d/(dx)coshx = sinhx (28)
d/(dx)tanhx = sech^2x (29)
d/(dx)cothx = -csch^2x (30)
d/(dx)sechx = -sechxtanhx (31)
d/(dx)cschx = -cschxcothx (32)
d/(dx)snx = cnxdnx (33)
d/(dx)cnx = -snxdnx (34)
d/(dx)dnx = -k^2snxcnx. (35)

where sn(x)=sn(x,k), cn(x)=cn(x,k), etc. are Jacobi elliptic functions, and the product rule and quotient rule have been used extensively to expand the derivatives.

There are a number of important rules for computing derivatives of certain combinations of functions. Derivatives of sums are equal to the sum of derivatives so that
[f(x)+...+h(x)]^'==f^'(x)+...+h^'(x). (36)

In addition, if c is a constant,
d/(dx)[cf(x)]==cf^'(x). (37)

The product rule for differentiation states
d/(dx)[f(x)g(x)]==f(x)g^'(x)+f^'(x)g(x), (38)

where f^' denotes the derivative of f with respect to x. This derivative rule can be applied iteratively to yield derivative rules for products of three or more functions, for example,
[fgh]^' = (fg)h^'+(fg)^'h==fgh^'+(fg^'+f^'g)h (39)
= f^'gh+fg^'h+fgh^'. (40)

The quotient rule for derivatives states that
d/(dx)[(f(x))/(g(x))]==(g(x)f^'(x)-f(x)g^'(x))/([g(x)]^2) (41)

while the power rule gives
d/(dx)(x^n)==nx^(n-1). (42)

Other very important rule for computing derivatives is the chain rule, which states that for y==y(u),
(dy)/(dx)==(dy)/(du).(du)/(dx), (43)

or more generally, for z==z(x(t),y(t))
(dz)/(dt)==(partialz)/(partialx)(dx)/(dt)+(partialz)/(partialy)(dy)/(dt), (44)

where partialz/partialx denotes a partial derivative.

Miscellaneous other derivative identities include
(dy)/(dx)==((dy)/(dt))/((dx)/(dt)) (45)
(dy)/(dx)==1/((dx)/(dy)). (46)

If F(x,y)==C, where C is a constant, then
dF==(partialF)/(partialy)dy+(partialF)/(partialx)dx==0, (47)

so
(dy)/(dx)==-((partialF)/(partialx))/((partialF)/(partialy)). (48)

Derivative identities of inverse functions include
(dx)/(dy) = 1/((dy)/(dx)) (49)
(d^2x)/(dy^2) = -(d^2y)/(dx^2)((dy)/(dx))^(-3) (50)
(d^3x)/(dy^3) = [3((d^2y)/(dx^2))^2-(d^3y)/(dx^3)(dy)/(dx)]((dy)/(dx))^(-5). (51)

A vector derivative of a vector function
X(t)=[x_1(t); x_2(t); |; x_k(t)] (52)

can be defined by
(dX)/(dt)==[(dx_1)/(dt); (dx_2)/(dt); |; (dx_k)/(dt)]. (53)

The nth derivatives of x^nf(x) for n==1, 2, ... are
d/(dx)[xf(x)] = f(x)+xf^'(x) (54)
(d^2)/(dx^2)[x^2f(x)] = 2f(x)+4xf^'(x)+x^2f^('')(x) (55)
(d^3)/(dx^3)[x^3f(x)] = 6f(x)+18xf^'(x)+9x^2f^('')(x)+x^3f^(''')(x). (56)

The nth row of the triangle of coefficients 1; 1, 1; 2, 4, 1; 6, 18, 9, 1; ... (Sloane's A021009) is given by the absolute values of the coefficients of the Laguerre polynomial L_n(x).

Faá di Bruno's formula gives an explicit formula for the nth derivative of the composition f(g(t)).

SEE ALSO: Blancmange Function, Calculus, Carathéodory Derivative, Cauchy-Riemann Equations, Chain Rule, Comma Derivative, Complex Derivative, Complex Differentiable, Convective Derivative, Covariant Derivative, Definite Integral, Differentiable, Differential Calculus, Differentiation, Directional Derivative, Euler-Lagrange Derivative, Faá di Bruno's Formula, Finite Difference, Fluxion, Fractional Calculus, Fréchet Derivative, Functional Derivative, Implicit Differentiation, Indefinite Integral, Integral, Lagrangian Derivative, Lie Derivative, Logarithmic Derivative, Numerical Differentiation, Pincherle Derivative, Power Rule, Product Rule, q-Derivative, Quotient Rule, Schwarzian Derivative, Semicolon Derivative, Total Derivative, Weierstrass Function. [Pages Linking Here]


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