Conservapedia - User contributions [en]

Fluid mechanics

2009-07-14T01:08:36Z

RoyL: moved the definition of fluid to its article

'''Fluid mechanics''' is the study of the motion of [[fluid]]s and the forces that leads to these motions. It can be further classified into two categories: [[fluid statics]] (studies of fluids at rest) and [[fluid dynamics]] (studies of fluids in motion).

==See also==
*[[Fluid statics]]
*[[Fluid dynamics]]

[[Category:Fluid mechanics]]
[[Category:mechanics]]
[[Category:Physics]]

Fluid

2009-07-14T01:07:26Z

RoyL:

In the field of [[fluid mechanics]], '''fluid''' is defined<ref>James R. Welty, Charles E. Wicks, Robert E. Wilson, and Gregory L. Rorrer, Fundamentals of Momentum, Heat, and Mass Transfer, 4th Ed. Toronto: John Wiley & Sons Inc, 2001.</ref> to be "a substance which deforms continuously under the action of a shear stress", which includes both liquids and gases.

==References==
{{Reflist}}
==See also==
*[[Fluid mechanics]]

[[Category:Physics]]

Fluid

2009-07-14T01:07:16Z

RoyL: rewrite

Diminished Responsibility

2009-07-14T00:37:36Z

RoyL: Cat

'''Diminished responsibility''' or '''diminished capacity''' is a defense against criminal charges, which claims that the defendant should not be held criminally liable for their violations of the law, due to their diminished capacity for rational thought at the time the crime was committed.
[[Category:Law]]

Fluid mechanics

2009-07-14T00:25:57Z

RoyL: internal links

'''Fluid mechanics''' is the study of the motion of fluids and the forces that leads to these motions. It can be further classified into two categories: [[fluid statics]] (studies of fluids at rest) and [[fluid dynamics]] (studies of fluids in motion). The "fluid" in the term fluid mechanics is defined<ref>James R. Welty, Charles E. Wicks, Robert E. Wilson, and Gregory L. Rorrer, Fundamentals of Momentum, Heat, and Mass Transfer, 4th Ed. Toronto: John Wiley & Sons Inc, 2001.</ref> to be "a substance which deforms continuously under the action of a shear stress", which includes both liquids and gases.

==See also==
*[[Fluid statics]]
*[[Fluid dynamics]]

[[Category:Fluid mechanics]]
[[Category:mechanics]]
[[Category:Physics]]

Fluid mechanics

2009-07-14T00:25:22Z

RoyL: ref for definition of fluid

'''Fluid mechanics''' is the study of the motion of fluids and the forces that leads to these motions. It can be further classified into two categories: fluid statics (studies of fluids at rest) and fluid dynamics (studies of fluids in motion). The "fluid" in the term fluid mechanics is defined<ref>James R. Welty, Charles E. Wicks, Robert E. Wilson, and Gregory L. Rorrer, Fundamentals of Momentum, Heat, and Mass Transfer, 4th Ed. Toronto: John Wiley & Sons Inc, 2001.</ref> to be "a substance which deforms continuously under the action of a shear stress", which includes both liquids and gases.

==See also==
*[[Fluid statics]]
*[[Fluid dynamics]]

[[Category:Fluid mechanics]]
[[Category:mechanics]]
[[Category:Physics]]

Fluid mechanics

2009-07-14T00:22:37Z

RoyL: Created page with ''''Fluid mechanics''' is the study of the motion of fluids and the forces that leads to these motions. It can be further classified into two categories: fluid statics (studies ...'

'''Fluid mechanics''' is the study of the motion of fluids and the forces that leads to these motions. It can be further classified into two categories: fluid statics (studies of fluids at rest) and fluid dynamics (studies of fluids in motion). The "fluid" in the term fluid mechanics is defined to be "a substance which deforms continuously under the action of a shear stress", which includes both liquids and gases.

==See also==
*[[Fluid statics]]
*[[Fluid dynamics]]

[[Category:Fluid mechanics]]
[[Category:mechanics]]
[[Category:Physics]]

Fluid statics

2009-07-14T00:20:53Z

RoyL: It's a start.

'''Fluid statics''' is the branch of [[Fluid mechanics|fluid mechanics]] which studies fluids at rest.

==Equations==
===Pressure variation in a static Fluid===
Applying [[Isaac Newton|Newton]]'s [[Newton's Laws of Motion|laws of motions]] to a fluid at rest, it can be determined that the sum of forces must equal zero throughout the fluid, which means any arbitrary element of the fluid is subjected to forces that sums to zero. Since the fluid is not deforming while it is at rest, the only forces acting on the fluid are those due to gravity and pressure.

For a given fluid element, the pressure is given by the differential equation<ref>James R. Welty, Charles E. Wicks, Robert E. Wilson, and Gregory L. Rorrer, ''Fundamentals of Momentum, Heat, and Mass Transfer'', 4th Ed. Toronto: John Wiley & Sons Inc, 2001.</ref>
:<math>\rho\mathbf{g} =\nabla P</math>

where <math>\rho</math> is the density of the fluid, '''g''' is the gravitational acceleration (vector), and P is the pressure at the fluid element.

For small changes in altitude, '''g''' can be assumed to be constant (A more accurate description for large change in altitude can be found [[Gravitation|here]]). Simplifications can also be made for constant density, and reducing the analysis to only the y dimension:
:<math>P-P_0=\rho g\left( y-y_0\right)</math>

==Examples==

For system filled with an isothermal (constant temperature throughout) ideal gas, a relation between pressure altitude can be established using the following method:
For ideal gases, the density is given by
:<math>\rho =\frac{PM}{RT}</math>
Where M is the molar mass, R is the ideal gas constant and T is the [[temperature]] (Absolute temperature, in Kelvin or Rankine) of the gas.
we can then set up the equation as follows:
:<math>\frac{dP}{dy}=-\rho g=-\frac{PMg}{RT}</math>
Note that we are using the scalar value of gravity (g), so the minus sign is included due to gravity is downwards, in the negative direction of the y-axis.
using the method of [[separation of variables]], we can rearrange the equation so
:<math>\frac{dP}{P}=-\frac{Mg}{RT}dy</math>
Integrating both sides gives
:<math>\ln\frac{P}{P_0}=-\frac{Mg\left(y-y_0\right)}{RT}</math>
Where P0 is the reference pressure at point y0 (often taken at the point which P0 is the atmospheric pressure).

Rearranging gives
:<math>P=P_0 \exp{\left(-\frac{Mg\left(y-y_0\right)}{RT}\right)}</math>
==References==
{{Reflist}}
[[Category:Fluid mechanics]]
[[Category:mechanics]]
[[Category:Physics]]

Motion

2009-07-13T23:02:07Z

RoyL: added link to physical term

:For the application of the word "motion" in physics, see [[Newton's Laws of Motion]].
A '''motion''' is an application made to a [[court]] or judge which requests a ruling or order in favor of the applicant.

It is a request by a [[litigant]] to a judge for a decision on an issue relating to the case.

== Example ==

In [[New Jersey]] state court, Rule 1:6-3 establishes the requirements for filing and serving motions:<ref>This is as of Jan. 2008 and, like all court rules, is subject to change.</ref>

:(a) Motions Generally. Other than an [[ex parte]] motion and except as otherwise provided by R. 4:46-1 ([[summary judgment]]) and R. 5:5-4(c) (post judgment motions), a notice of motion shall be filed and served not later than 16 days before the specified return date unless otherwise provided by court order, which may be applied for ex parte. Thus, for example, if the return date of the motion is a Friday, the motion must be filed and served not later than the Wednesday, 16 days prior. If a motion is supported by [[affidavit]] or certification, the affidavit or certification shall be filed and served with the motion. Except as provided by R. 4:49-1(b) (motion for new trial), any opposing affidavits, certifications or objections filed pursuant to R. 1:6-2 shall be filed and served not later than 8 days before the return date unless the court relaxes that time. Thus, for example, if the return date is on a Friday, any response must be filed and served no later than Thursday of the prior week. Reply papers responding to opposing affidavits or certifications shall be filed and served not later than 4 days before the return date unless the court otherwise orders. Thus, for example, such papers must be filed and served on Monday for a return date of the following Friday. No other papers may be filed without leave of court. ...

== Reference ==

<references/>

== See also ==

*[[Claim]]
*[[Prejudicial]]
*[[Immunity]]

== External links ==

*[http://dictionary.law.com/ Law Dictionary]
*[http://research.lawyers.com/glossary/ Glossary of Legal Terms]

[[Category:Legal Terms]]

Counterexample

2009-07-13T22:32:38Z

RoyL: Reordering the examples

A '''counterexample''' is an example that is opposite to a theory or argument.

==In Science and Logic==

A '''counterexample''' cannot disprove correlation or causation. An example of flawed [[liberal logic]] is the use of a counterexample to argue that a theory is false, as in "I don't go to church and I'm not a bad person, and therefore church is not needed."

Another example is the well-proven theory that cigarettes cause [[lung cancer]]. An '''counterexample''' of the (rare) individual who might smoke his entire life and not develop lung cancer. The counterexample does not disprove the theory.

A theory of universality, such as the claim that ''all'' species [[theory of evolution|evolved]], can be disproved by just one '''counterexample''', such as a species (like the [[whale]]) that lacks a plausible evolutionary path. See [[Counterexamples to Evolution]].
[[category:logic]]

==In Mathematics==

The word counterexample originated in mathematics. It means an example that contradicts a general statement; therefore, the single example proves the statement false. Consider the statement

"Every number ending in the digit 7 is prime." 7, 17, 37, 47, 57, and 67 are all prime, but 27 is not, so 27 is a counterexample.[[Category:Mathematics]]

Talk:Evolution syndrome

2009-07-13T22:05:58Z

RoyL: /* Minor formatting request */ new section

Evolutionists have a compulsion to censor opposing or alternative views. This can be seen, among other places, on wikis. So, hypothetically, if there was a wiki whose pro-evolution was locked to prevent editing by anyone except for a single sysop, and where anyone who attempted to add alternative views was reverted or blocked, would that be an example of the type of censorship you have in mind? [[User:Eoinc|Eoinc]] 08:52, 17 September 2008 (EDT)
:"insist that their belief in evolution must somehow be true" Isn't "their belief" superfluous? "Believing" something is "holding it to be true", so someone must believe it's true to believe it at all. --[[User:Toffeeman|Toffeeman]] 09:06, 17 September 2008 (EDT)

:: The locking is due to the prevalence of [[evolution syndrome]]. If someone can propose a solution, I welcome it.

:: No, "their belief" is not superfluous. There are are different beliefs about evolution, but the person who has [[evolution syndrome]] insists that ''his'' belief is true to the exclusion of other beliefs. Most normal people do not take that attitude.--[[User:Aschlafly|Aschlafly]] 09:21, 17 September 2008 (EDT)

::: But there is a difference between asserting that the theory of evolution is true (I.E. that it is factual), and a person asserting that their ''belief'' in it is true (I.E. that they ''truly'' believe in it). The text currently suggests that evolution syndrome is an insistance that one has a true belief in evolution. Are you implying that sufferers of this 'syndrome' do not actually truly believe in evolution despite insisting that they do? [[User:Sideways|Sideways]] 15:15, 17 September 2008 (EDT)
::::Since there's been no response, I have taken the iniative & changed the wording. I am assuming that the intended meaning is that sufferers of this "syndrome" insist that evolution itself is true, not only that they truly believe in it. [[User:Sideways|Sideways]]

==Most atheists to many atheists change==

While most atheists certainly believe in evolution as the article states, it seems unlikely that most also contribute 90% edits about evolution. That only leaves 10% for other liberal atheist issues like promoting abortion, etc. [[User:Sulli|Sulli]] 13:22, 27 December 2008 (EST)

== Notability ==
Does CP have a notability guideline regarding articles? I've never heard this term used anywhere else and the fact that it has no citations seems to suggest it's a made-up term. I like it, but our job is to chronicle, not create, information.--[[User:DReynolds|DReynolds]] 12:31, 8 February 2009 (EST)

Since no one has responded, I've nominated this article for deletion. No sources, no support. It violates the commandments. Kill it --[[User:DReynolds|DReynolds]] 00:14, 11 February 2009 (EST).
:Your comments display a violence of language and closed-mindedness characteristic of ES-sufferers. ''Our job is to chronicle, not create, information'' - what better example of chronicling information could there be than to typologise atheist behaviours and to give a name to a previously nameless - but real - condition? [[User:RegalBruin|RegalBruin]] 09:48, 11 February 2009 (EST)

A "violence of language"? Where in his statement is there a "violence of language?" I see nothing malicious or offensive by any standard. If there were ad hominem attacks or, THAT would be violence of language. It seems like you instantly categorized him as an "ES-sufferer" without actually reading his comment. What he was trying to say was that because there are no sources for this article, and the term does not appear frequently elsewhere, that it is of questionable validity. And, like he said, it violates the basic principles of Conservapedia. The example you gave of typologizing atheist behaviors proves exactly what he was suggesting about creating information. YOU are creating a term for something, rather than defining a previously existing term. The point of an encyclopedia is not to come up with new terms and words, but rather to explain those that are used elsewhere in the world. Unless you can provide a valid source of this term, then it is original research. And ironically, you seemed to have missed the statement "I like it" in his comment, which suggests that he is not a crazed ES-sufferer, but rather trying to keep Conservapedia true to its own commandments, despite his general agreement with the information in the article. Personally, I also agree with the information, however I feel that unless it can be verified, it is an embarassingly baseless generalization of anti-creationist efforts. Certainly there are people that are like this article describes. There are radicals and extremists in every group. There are even extremists among the Creationists, which cannot seriously be denied (I am a Creationist by the way). To create a page about these nutjobs is not notable UNLESS you are defining a particular group of extremists (for instance, Al Queda among Islamic extremists), the generalization of the radical group is widely recognized (for instance: terrorists) and as such can be supported with reputable sources, or you are attempting to describe the entirety of the group as having these radical characteristics. In the case of Evolution Syndrome, none of these are present. If you were to find a source that supports that Evolution Syndrome is a legitimate term, let alone mental condition, then the article would be notable. Atleast provide evidence of the crimes that those with "ES" are accused of. If they censor Creationism in the classroom, provide concrete support of that accusation, or else it is baseless slander. Here's hoping that somebody has the time or ability to find support for all of the accustations in this article. This may very well be an important social trend of the "activities of atheists", like you've said, but until there is valid support for the term, it isn't. [[User:Br1|Br1]] 00:19, 15 June 2009 (EDT)

== Minor formatting request ==

Perhaps format the "See also" section with bullets/unenumerated list? The page is locked for editing so I can't make that change. Thanks. [[User:RoyL|RoyL]] 18:05, 13 July 2009 (EDT)

Curl

2009-07-11T02:20:12Z

RoyL:

{{math-h}}
The '''curl''' is a way of expressing a certain type of [[derivative]] of a [[vector field]]. It is defined for fields of 3-dimensional vectors on 3-dimensional space. The curl of a vector field is another vector field.

More precisely, it is defined<ref>[http://mathworld.wolfram.com/Curl.html Curl] at Wolfram Mathworld</ref> as the limiting value of rotation per unit area. Written explicitly,

:<math>(\nabla \times \mathbf{F}) \cdot \mathbf{\hat{n}} \equiv \lim_{A \to 0} \frac{\oint_{C} \mathbf{F} \cdot d\mathbf{s}}{A}</math>

where <math>\mathbf{\hat{n}}</math> is the unit [[normal vector]] of the area element A and C is the boundary of the area element (<math>C =\partial A</math>). The right side of the definition is a path integral along the boundary of the area element A, which is allow to shrink in the limiting process.

==Cartesian coordinates==
In [[Cartesian coordinates]], the curl is written as though it were the [[cross product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \times \vec V</math>. This is usually pronounced "curl V" or "del cross V".

In 3-D [[Cartesian coordinates]], the curl is calculated as:

:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ )</math>

or, using a notation like the [[determinant]] notation for cross product,
:<math>\nabla \times \vec V = \begin{vmatrix}
\hat x & \hat y & \hat z \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
V_x & V_y & V_z
\end{vmatrix}</math>
where <math>\hat x</math>, <math>\hat y</math>, and <math>\hat z</math> are the unit basis vectors.

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the cross product notation makes sense. This is also useful for remembering how to calculate a curl.

The curl is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.

The curl operation has an intrinsic "handedness" to it. Any physical phenomenon described by the curl operation (for example, magnetic fields), involves some kind of "right-hand rule".

The curl is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in [[Maxwell's Equations]].

Intuitively, the curl measures the degree to which the vector field rotates around a given point. If you were to measure the curl of the vector field of wind speed in the vicinity of a meteorological low pressure area, then, keeping in mind that the [[Coriolis force]] makes the wind move in a counterclockwise vortex in the Northern hemisphere, it would be a vector pointing upward. (The ''reason'' for the Coriolis force is not important here, we're just talking about the ''observation'' that the air moves in a counterclockwise vortex.) The way to see that the curl points upward is to visualize a giant right hand with the fingers curled counterclockwise—the thumb would point upward.

Vector fields with a curl of zero are called ''irrotational''.

==References==
{{reflist}}

[[Category:Calculus]]
[[Category:Physics]]

L'Hopital's rule

2009-07-11T01:50:57Z

RoyL: /* Example 3 */

'''L'Hôpital's Rule''' is a method in differential [[calculus]] for calculating the [[limit_(mathematics)|limit]] of a [[quotient]] of two [[function]]s wherein the entire expression approaches an [[indeterminate form]] (e.g. 0/0, [[infinity]]/infinity). In the event that this is the case, the limit is equal to the limit of the quotient of the first [[derivative]]s of the two functions (provided that limit exists). Should this also yield an indeterminate form, the process is repeated until a meaningful result is obtained.<ref>http://mathworld.wolfram.com/LHospitalsRule.html</ref>
{| style="border:1px solid #ccc;margin:auto" align="center"
|+ '''Math form'''
|
:<math>g'(x)\not\equiv 0</math>
:C is some number such that
::<math>f(c)=0</math>
::<math>g(c)=0</math>
:<math>\lim_{x\rightarrow c} \frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}</math>
|}

L'Hopital's Rule is not to be confused with the [[quotient rule]], which allows for the calculation of the derivative of a single function that contains a quotient.

== Examples ==
===Example 1===
A standard application of L'Hopital's rule is in evaluating the limit

::<math>\lim_{x \to 0} \frac{\sin x}{x}.</math>

In the preceding notation, this is the situation with <math>f(x) = \sin x</math> and <math>g(x) = x</math>. Both the numerator and the denominator tend to 0 as <math>x</math> tends to 0, i.e., <math>\lim_{x \to 0} \sin x = \lim_{x \to 0} x = 0</math>, and so L'Hôpital's rule implies that

::<math>
\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\frac{d}{dx} \sin x}{\frac{d}{dx} x} = \lim_{x \to 0} \frac{\cos x}{1} = \lim_{x \to 0} \cos x = \cos 0 = 1.
</math>

===Example 2===
L'Hopital's rule may also be used in the evaluation of the indeterminate form infinity/infinity. This version of the rule is useful in computing the horizontal [[asymptote|asymptotes]] of rational functions. For example, suppose we seek to compute

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7}.</math>

This is an indeterminate form <math>\infty/\infty</math>. Applying L'Hopital's rule once yields

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = \lim_{x \to \infty} \frac{\frac{d}{dx}(2x^2+3x+2)}{\frac{d}{dx}(x^2-5x+7)} = \lim_{x \to \infty} \frac{4x+3}{2x-5}.</math>

This is still an indeterminate form. To evaluate the limit, it is necessary to invoke L'Hopital's rule a second time:
::<math>\lim_{x \to \infty} \frac{4x+3}{2x-5} = \lim_{x \to \infty} \frac{\frac{d}{dx}(4x+3)}{\frac{d}{dx}(2x-5)} = \lim_{x \to \infty} \frac{4}{2} = 2.</math>

We conclude that

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = 2.</math>

An easy extension of this argument is useful for finding horizontal asymptotes of more general rational functions. Suppose that <math>f</math> and <math>g</math> are two polynomials of equal degree <math>n</math>. Applying L'Hopital's rule <math>n</math> times we may discover that

::<math>\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{f_n}{g_n},</math>

where <math>f_n</math> and <math>g_n</math> are the leading coefficients of <math>f</math> and <math>g</math> (i.e., the coefficients on the term <math>x^n</math> in these two polynomials). The example given is a case of this fact with <math>n=2</math> (since both <math>f</math> and <math>g</math> are quadratic), and with <math>f_n = 2</math> and <math>g_n = 1</math>.

===Example 3===
We can use L'Hôpital's rule to prove the following:

:<math>\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=0 \quad\forall n< \infty</math>

This is another example where the limit is in the form of <math>\infty/\infty</math>. 
'''Proof'''
:For Integer n:
::<math>\forall n< \infty\quad\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=
\lim_{x\rightarrow\infty} \frac{nx^{n-1}}{e^x}=
\lim_{x\rightarrow\infty} \frac{n\left(n-1\right)x^{n-2}}{e^x}= \cdots=
\lim_{x\rightarrow\infty} \frac{n!}{e^x}= 0</math>
:For non-integer n:
:::<math>\forall n< \infty\quad\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=
\lim_{x\rightarrow\infty} \frac{nx^{n-1}}{e^x}=
\lim_{x\rightarrow\infty} \frac{n\left(n-1\right)x^{n-2}}{e^x}= \cdots=
\lim_{x\rightarrow\infty} \frac{\left[n \left(n-1 \right)\left(n-2 \right)\cdots \left(n-\lfloor n \rfloor \right) \right]x^{n-\lceil n \rceil}}{e^x}</math>
:::(where <math>\lfloor n \rfloor</math> is the [[floor function]] of n and <math>\lceil n \rceil</math> is the [[ceiling function]] of n)
::Since <math>n-\lceil n \rceil <0</math>, <math>\lim_{x\rightarrow\infty}x^{n-\lceil n \rceil}=0</math> and therefore
:::<math>\lim_{x\rightarrow\infty} \frac{\left[n \left(n-1 \right)\left(n-2 \right)\cdots \left(n-\lfloor n \rfloor \right) \right]x^{n-\lceil n \rceil}}{e^x}=0</math>
::This completes the proof.

== Outside Links ==
*[http://books.google.com/books?id=H-0L9Dxor6sC&pg=PA144&lpg=PA144&dq=L%27H%C3%B4pital%27s+rule&source=bll&ots=3HfHZW3-DZ&sig=Ripjx5A7yw-OcrHN91sxEN8Gr5U&hl=en&ei=kg2WSePdKYKUsQOT_JiuBw&sa=X&oi=book_result&resnum=11&ct=result The Complete Idiot's Guide to Calculus]
== References ==
{{reflist}}

[[Category:Mathematics]]

L'Hopital's rule

2009-07-11T01:50:37Z

RoyL: /* Example 3 */

'''L'Hôpital's Rule''' is a method in differential [[calculus]] for calculating the [[limit_(mathematics)|limit]] of a [[quotient]] of two [[function]]s wherein the entire expression approaches an [[indeterminate form]] (e.g. 0/0, [[infinity]]/infinity). In the event that this is the case, the limit is equal to the limit of the quotient of the first [[derivative]]s of the two functions (provided that limit exists). Should this also yield an indeterminate form, the process is repeated until a meaningful result is obtained.<ref>http://mathworld.wolfram.com/LHospitalsRule.html</ref>
{| style="border:1px solid #ccc;margin:auto" align="center"
|+ '''Math form'''
|
:<math>g'(x)\not\equiv 0</math>
:C is some number such that
::<math>f(c)=0</math>
::<math>g(c)=0</math>
:<math>\lim_{x\rightarrow c} \frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}</math>
|}

L'Hopital's Rule is not to be confused with the [[quotient rule]], which allows for the calculation of the derivative of a single function that contains a quotient.

== Examples ==
===Example 1===
A standard application of L'Hopital's rule is in evaluating the limit

::<math>\lim_{x \to 0} \frac{\sin x}{x}.</math>

In the preceding notation, this is the situation with <math>f(x) = \sin x</math> and <math>g(x) = x</math>. Both the numerator and the denominator tend to 0 as <math>x</math> tends to 0, i.e., <math>\lim_{x \to 0} \sin x = \lim_{x \to 0} x = 0</math>, and so L'Hôpital's rule implies that

::<math>
\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\frac{d}{dx} \sin x}{\frac{d}{dx} x} = \lim_{x \to 0} \frac{\cos x}{1} = \lim_{x \to 0} \cos x = \cos 0 = 1.
</math>

===Example 2===
L'Hopital's rule may also be used in the evaluation of the indeterminate form infinity/infinity. This version of the rule is useful in computing the horizontal [[asymptote|asymptotes]] of rational functions. For example, suppose we seek to compute

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7}.</math>

This is an indeterminate form <math>\infty/\infty</math>. Applying L'Hopital's rule once yields

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = \lim_{x \to \infty} \frac{\frac{d}{dx}(2x^2+3x+2)}{\frac{d}{dx}(x^2-5x+7)} = \lim_{x \to \infty} \frac{4x+3}{2x-5}.</math>

This is still an indeterminate form. To evaluate the limit, it is necessary to invoke L'Hopital's rule a second time:
::<math>\lim_{x \to \infty} \frac{4x+3}{2x-5} = \lim_{x \to \infty} \frac{\frac{d}{dx}(4x+3)}{\frac{d}{dx}(2x-5)} = \lim_{x \to \infty} \frac{4}{2} = 2.</math>

We conclude that

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = 2.</math>

An easy extension of this argument is useful for finding horizontal asymptotes of more general rational functions. Suppose that <math>f</math> and <math>g</math> are two polynomials of equal degree <math>n</math>. Applying L'Hopital's rule <math>n</math> times we may discover that

::<math>\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{f_n}{g_n},</math>

where <math>f_n</math> and <math>g_n</math> are the leading coefficients of <math>f</math> and <math>g</math> (i.e., the coefficients on the term <math>x^n</math> in these two polynomials). The example given is a case of this fact with <math>n=2</math> (since both <math>f</math> and <math>g</math> are quadratic), and with <math>f_n = 2</math> and <math>g_n = 1</math>.

===Example 3===
We can use L'Hôpital's rule to prove the following:

:<math>\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=0 \quad\forall n< \infty</math>

This is another example where the limit is in the form of <math>\infty/\infty</math>. 
'''Proof'''
:For Integer n:
::<math>\forall n< \infty\quad\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=
\lim_{x\rightarrow\infty} \frac{nx^{n-1}}{e^x}=
\lim_{x\rightarrow\infty} \frac{n\left(n-1\right)x^{n-2}}{e^x}= \cdots
\lim_{x\rightarrow\infty} \frac{n!}{e^x}= 0</math>
:For non-integer n:
:::<math>\forall n< \infty\quad\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=
\lim_{x\rightarrow\infty} \frac{nx^{n-1}}{e^x}=
\lim_{x\rightarrow\infty} \frac{n\left(n-1\right)x^{n-2}}{e^x}= \cdots=
\lim_{x\rightarrow\infty} \frac{\left[n \left(n-1 \right)\left(n-2 \right)\cdots \left(n-\lfloor n \rfloor \right) \right]x^{n-\lceil n \rceil}}{e^x}</math>
:::(where <math>\lfloor n \rfloor</math> is the [[floor function]] of n and <math>\lceil n \rceil</math> is the [[ceiling function]] of n)
::Since <math>n-\lceil n \rceil <0</math>, <math>\lim_{x\rightarrow\infty}x^{n-\lceil n \rceil}=0</math> and therefore
:::<math>\lim_{x\rightarrow\infty} \frac{\left[n \left(n-1 \right)\left(n-2 \right)\cdots \left(n-\lfloor n \rfloor \right) \right]x^{n-\lceil n \rceil}}{e^x}=0</math>
::This completes the proof.

== Outside Links ==
*[http://books.google.com/books?id=H-0L9Dxor6sC&pg=PA144&lpg=PA144&dq=L%27H%C3%B4pital%27s+rule&source=bll&ots=3HfHZW3-DZ&sig=Ripjx5A7yw-OcrHN91sxEN8Gr5U&hl=en&ei=kg2WSePdKYKUsQOT_JiuBw&sa=X&oi=book_result&resnum=11&ct=result The Complete Idiot's Guide to Calculus]
== References ==
{{reflist}}

[[Category:Mathematics]]

L'Hopital's rule

2009-07-11T01:49:54Z

RoyL:

'''L'Hôpital's Rule''' is a method in differential [[calculus]] for calculating the [[limit_(mathematics)|limit]] of a [[quotient]] of two [[function]]s wherein the entire expression approaches an [[indeterminate form]] (e.g. 0/0, [[infinity]]/infinity). In the event that this is the case, the limit is equal to the limit of the quotient of the first [[derivative]]s of the two functions (provided that limit exists). Should this also yield an indeterminate form, the process is repeated until a meaningful result is obtained.<ref>http://mathworld.wolfram.com/LHospitalsRule.html</ref>
{| style="border:1px solid #ccc;margin:auto" align="center"
|+ '''Math form'''
|
:<math>g'(x)\not\equiv 0</math>
:C is some number such that
::<math>f(c)=0</math>
::<math>g(c)=0</math>
:<math>\lim_{x\rightarrow c} \frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}</math>
|}

L'Hopital's Rule is not to be confused with the [[quotient rule]], which allows for the calculation of the derivative of a single function that contains a quotient.

== Examples ==
===Example 1===
A standard application of L'Hopital's rule is in evaluating the limit

::<math>\lim_{x \to 0} \frac{\sin x}{x}.</math>

In the preceding notation, this is the situation with <math>f(x) = \sin x</math> and <math>g(x) = x</math>. Both the numerator and the denominator tend to 0 as <math>x</math> tends to 0, i.e., <math>\lim_{x \to 0} \sin x = \lim_{x \to 0} x = 0</math>, and so L'Hôpital's rule implies that

::<math>
\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\frac{d}{dx} \sin x}{\frac{d}{dx} x} = \lim_{x \to 0} \frac{\cos x}{1} = \lim_{x \to 0} \cos x = \cos 0 = 1.
</math>

===Example 2===
L'Hopital's rule may also be used in the evaluation of the indeterminate form infinity/infinity. This version of the rule is useful in computing the horizontal [[asymptote|asymptotes]] of rational functions. For example, suppose we seek to compute

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7}.</math>

This is an indeterminate form <math>\infty/\infty</math>. Applying L'Hopital's rule once yields

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = \lim_{x \to \infty} \frac{\frac{d}{dx}(2x^2+3x+2)}{\frac{d}{dx}(x^2-5x+7)} = \lim_{x \to \infty} \frac{4x+3}{2x-5}.</math>

This is still an indeterminate form. To evaluate the limit, it is necessary to invoke L'Hopital's rule a second time:
::<math>\lim_{x \to \infty} \frac{4x+3}{2x-5} = \lim_{x \to \infty} \frac{\frac{d}{dx}(4x+3)}{\frac{d}{dx}(2x-5)} = \lim_{x \to \infty} \frac{4}{2} = 2.</math>

We conclude that

::<math>\lim_{x \to \infty} \frac{2x^2+3x+2}{x^2-5x+7} = 2.</math>

An easy extension of this argument is useful for finding horizontal asymptotes of more general rational functions. Suppose that <math>f</math> and <math>g</math> are two polynomials of equal degree <math>n</math>. Applying L'Hopital's rule <math>n</math> times we may discover that

::<math>\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{f_n}{g_n},</math>

where <math>f_n</math> and <math>g_n</math> are the leading coefficients of <math>f</math> and <math>g</math> (i.e., the coefficients on the term <math>x^n</math> in these two polynomials). The example given is a case of this fact with <math>n=2</math> (since both <math>f</math> and <math>g</math> are quadratic), and with <math>f_n = 2</math> and <math>g_n = 1</math>.

===Example 3===
We can use L'Hôpital's rule to prove the following:

:<math>\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=0 \quad\forall n< \infty</math>

This is another example where the limit is in the form of <math>\infty/\infty</math>. 
'''Proof'''
:For Integer n:
::<math>\forall n< \infty\quad\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=
\lim_{x\rightarrow\infty} \frac{nx^{n-1}}{e^x}=
\lim_{x\rightarrow\infty} \frac{n\left(n-1\right)x^{n-2}}{e^x}= \cdots
\lim_{x\rightarrow\infty} \frac{n!}{e^x}= 0</math>
:For non-integer n:
:::<math>\forall n< \infty\quad\lim_{x\rightarrow\infty} \frac{x^n}{e^x}=
\lim_{x\rightarrow\infty} \frac{nx^{n-1}}{e^x}=
\lim_{x\rightarrow\infty} \frac{n\left(n-1\right)x^{n-2}}{e^x}= \cdots
\lim_{x\rightarrow\infty} \frac{\left[n \left(n-1 \right)\left(n-2 \right)\cdots \left(n-\lfloor n \rfloor \right) \right]x^{n-\lceil n \rceil}}{e^x}</math>
:::(where <math>\lfloor n \rfloor</math> is the [[floor function]] of n and <math>\lceil n \rceil</math> is the [[ceiling function]] of n)
::Since <math>n-\lceil n \rceil <0</math>, <math>\lim_{x\rightarrow\infty}x^{n-\lceil n \rceil}=0</math> and therefore
:::<math>\lim_{x\rightarrow\infty} \frac{\left[n \left(n-1 \right)\left(n-2 \right)\cdots \left(n-\lfloor n \rfloor \right) \right]x^{n-\lceil n \rceil}}{e^x}=0</math>
::This completes the proof.

== Outside Links ==
*[http://books.google.com/books?id=H-0L9Dxor6sC&pg=PA144&lpg=PA144&dq=L%27H%C3%B4pital%27s+rule&source=bll&ots=3HfHZW3-DZ&sig=Ripjx5A7yw-OcrHN91sxEN8Gr5U&hl=en&ei=kg2WSePdKYKUsQOT_JiuBw&sa=X&oi=book_result&resnum=11&ct=result The Complete Idiot's Guide to Calculus]
== References ==
{{reflist}}

[[Category:Mathematics]]

Divergence

2009-07-10T04:18:51Z

RoyL:

For divergence of a series, see [[Convergence]]

The '''divergence''' is a way of expressing a certain type of [[derivative]] of a [[vector field]]. It is typically defined for fields of 3-dimensional vectors on 3-dimensional space, but other dimensions are possible. The divergence of a vector field is a [[scalar field]], that is, just a number at each point in space. Vector fields with a divergence of zero are called ''divergenless'' or ''solenoidal''.

Strictly speaking, The divergence of a Vector Field '''F''' is defined as the limit of the surface integral as the volume shrinks to 0:

:<math>\operatorname{div}\,\mathbf{F} = \lim_{V \rightarrow 0} \frac{ \oint_\mathbf{S} \mathbf{F} \cdot d\mathbf{a} }{V}</math>

Where the integral is over the boundary surface <math>\mathbf{S}=\partial V</math> surrounding the volume element V, which is taken to be zero in the limit.

==Cartesian coordinates==
For [[Cartesian coordinates]], the divergence is written as though it were the [[dot product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \cdot \vec V</math>. This is usually pronounced "div V" or "del dot V".

In ordinary 3-dimensional [[Cartesian coordinates]], the divergence is calculated as:

:<math>\operatorname{div}\,\mathbf{F} = \nabla \cdot \vec V = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}</math>
or, using summation notation to extend it for n-dimension,
:<math>\nabla \cdot \vec V = \sum_{i=1}^n \frac{\partial V_i}{\partial x_i}</math>

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the dot product notation makes sense. This is also useful for remembering how to calculate a divergence.

The divergence is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.

The divergence is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in [[Maxwell's Equations]].

Intuitively, the divergence measures the degree to which the vector field is diverging from a given point. If you were to measure the divergence of the vector field of wind speed in the vicinity of a meteorological high pressure area, it would be positive, because the net motion of air is outward. If measured near a low pressure area, the divergence would be negative.
==References==
*[http://mathworld.wolfram.com/Divergence.html Divergence] from Wolfram Mathworld

[[Category:Calculus]]
[[Category:Physics]]

Gradient

2009-07-10T03:36:50Z

RoyL: I am sure some university teaches it in freshman/sophomore level

{{math-h}}

In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].

More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:

<math>
\nabla f = (\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})
</math>

consisting of the various [[partial derivative]]s of <math>f</math>. If <math>u</math> is a [[unit vector]] in <math>R^n</math>, then, by the [[chain rule]], the directional derivative of <math>f</math> in the direction of <math>u</math> is simply the [[dot product]]:

<math>
\nabla f \cdot u
</math>

Evidently by the [[Cauchy-Schwartz inequality]], the directional derivative in the direction <math>u</math> is maximal in the direction of the gradient, and equal to <math>||\nabla f||</math> for <math>u</math> a unit vector in the direction of the gradient.

==Properties of the Gradient==

If <math>f</math> is a differentiable function with smooth [[level set]]s <math>f^{-1}(c)</math>, then the gradient vector field <math>\nabla f</math> is [[perpendicular]] to the level sets of <math>f</math>. For fix a level set <math>S = f^{-1}(c)</math>, and let <math>v</math> be a vector [[tangent]] to <math>S</math> at <math>p</math>. Then we can find a [[curve]] <math>\gamma(t)</math> on <math>S</math> with <math>\gamma'(0) = v</math>. Now

<math>
f\circ\gamma(t) = c
</math>

since <math>S</math> is a level set. Taking derivatives of both sides and applying the chain rule, we get that

<math>
\nabla f\cdot \gamma'(0) = \nabla f\cdot v = 0
</math>

Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.

[[Category:calculus]]