Backpropagation - Revision history

en>Wfwhitney: Move the Algorithm section tag to not include the heading.

2014-12-08T16:28:45Z

Move the Algorithm section tag to not include the heading.

← Older revision		Revision as of 18:28, 8 December 2014
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	~~{{thermodynamics\|cTopic=[[Thermodynamic system\|Systems]]}}~~		<br><br>To get started Sign up for Hostgator. Plus, your MySpace page is apt to show up in search. No Spam!: Hostgator comes with a tool called Spam assassin! A reseller package is usually the best thing for a novice to choose because it gives the beginner room to grow. They are recognized worldwide as one of the best worldwide web hosting providers with over 7,000,000 hosting domains.Why decide on HostGator?It's the one of the cheapest webhosting [http://Data.gov.uk/data/search?q=services services] out there today, why pay a lot for something you can have at a better cost?. I will advise you about the best way to get a high level of support at NameCheap. And when it comes to this, there's nothing better than web hosting reviews to keep yourself informed - but of course, you'll need to know how to make proper use of them in order to get the full benefits that they offer. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. [http://search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=Hostgator%27s&Submit=Go Hostgator's] the best in my opinion, like I said before. Web users love to share blog posts - this is the foundation which can build your brand, business or community.<br><br><br><br>More or less, your website might experience more website downtimes rather than uptimes. Their webhosting packages also includes all around support when needed, that's correct whenever you need help with your website they will be there 24/7 to assist you at any time. If you have any questions concerning where by and how to use [http://iaco.ajou.ac.kr/?document_srl=514632 Cheap Hosting from Hostgator], you can speak to us at our web site. And trust me, I did my research before signing up for Hostgator and they were the obvious choice. You can set up unlimited websites under your own brand name, with your own packages , features plus prices. After that you'll want to scroll down until you see a happy face icon with the word 'Fantastico' below it. Plus, Wordpress blogs can be monetized very, very easily given that search engines love them and that there is a plethora of plugins that can immediately transform your pages into moneymaking digital machines. Next, create categories and quick edit each post to assign a category to it. Depending on your publisher, you may have assistance in promotion, or you may be on your own.
	~~In [[fluid mechanics]] and [[thermodynamics]]~~, ~~a '''control volume'''~~ is ~~a mathematical abstraction employed~~ in ~~the process of creating [[mathematical model]]s of physical processes~~. ~~In an [[inertial frame of reference]], it~~ is a ~~volume fixed in space or moving with constant velocity through which the [[fluid]] ([[gas]] or [[liquid]]) flows. The surface enclosing~~ the ~~control volume is referred~~ to ~~as the '''control surface'''~~.~~<ref>G.J. Van Wylen and R.E. Sonntag (1985), ''Fundamentals of Classical Thermodynamics'', Section 2.1 (3rd edition), John Wiley & Sons, Inc., New York ISBN 0-471-82933-1</ref>~~

	~~At [[steady state]], a control volume can be thought of~~ as ~~an arbitrary volume in which the mass~~ of the ~~fluid remains constant. As fluid moves through the control volume, the mass entering the control volume is equal to the mass leaving the control volume. At [[steady state]]~~, ~~and in the absence of work and heat transfer~~, ~~the energy within the control volume remains constant~~. It ~~is analogous to~~ the ~~classical mechanics concept~~ of the [~~[free body diagram]]~~.

	=~~=Overview==~~
	~~Typically~~, ~~to understand how~~ a ~~given [[physical law]] applies to the system under consideration, one first begins by considering how it applies to~~ a ~~small, control volume, or "representative volume"~~. ~~There is nothing special~~ about a ~~particular control volume, it simply represents a small part~~ of ~~the system to which physical laws can be easily applied~~. ~~This gives rise~~ to ~~what is termed a volumetric~~, ~~or volume~~-~~wise formulation~~ of the ~~mathematical model.~~

	~~One can then argue~~ that ~~since the [[physical law]]s behave in a certain way on a particular control volume,~~ they ~~behave the same way on all such volumes, since that particular control volume was not special in any way~~. ~~In this~~ way, ~~the corresponding point-wise formulation of the [[mathematical model]] can be developed so it can describe the physical behaviour of an entire (and maybe~~ more ~~complex) system~~.

	In [~~[fluid mechanics]] the [[Conservation laws\|conservation equations]] (for instance, the [[Navier-Stokes equations]]) are in integral form~~. ~~They therefore apply on volumes~~. ~~Finding forms of the equation that are <em>independent<~~/~~em> of the control volumes allows simplification of the integral signs.~~

	== ~~Substantive derivative~~ ==
	~~{{Main\|Material derivative}}~~
	~~Computations~~ in ~~[[fluid mechanics]] often require that~~ the ~~regular time [[Derivative\|derivation]] operator~~
	<~~math~~>~~d/dt\;~~<~~/math~~>
	~~is replaced by the [[substantive derivative]] operator~~
	<~~math~~>~~D/Dt~~<~~/math~~>.
	~~This can be seen as follows.~~

	~~Consider a bug that is moving through a volume where there is some [[scalar field\|scalar]]~~,
	e.~~g. [[pressure]]~~, that ~~varies~~ with time and ~~position~~:
	~~<math>p=p(t,x,y,z)\;<~~/~~math>~~.

	~~If the bug during the time interval from~~
	~~<math>t\;<~~/~~math>~~
	to
	~~<math>t+dt\;</math>~~
	~~moves~~ from
	~~<math>(x~~,~~y,z)\;</math>~~
	to
	~~<math>(x+dx~~, ~~y+dy, z+dz),\;</math>~~
	~~then~~ the ~~bug experiences a change <math>dp\;</math> in the scalar value~~,
	~~:<math>dp = \frac{\partial p}{\partial t}dt~~
	~~+ \frac{\partial p}{\partial x}dx~~
	~~+ \frac{\partial p}{\partial y}dy~~
	~~+ \frac{\partial p}{\partial z}dz</math>~~
	~~(the [[total derivative\|total differential]]). If the bug is moving~~ with ~~velocity~~
	~~<math>\mathbf v = (v_x, v_y, v_z)~~,~~</math>~~
	~~the change in position is~~
	~~<math>\mathbf vdt = (v_xdt, v_ydt, v_zdt),</math>~~
	~~and we may write~~
	~~:<math>\begin{alignat}{2}~~
	~~dp &~~
	~~= \frac{\partial p}{\partial t}dt~~
	~~+ \frac{\partial p}{\partial x}v_xdt~~
	~~+ \frac{\partial p}{\partial y}v_ydt~~
	~~+ \frac{\partial p}{\partial z}v_zdt \\ &~~
	~~= \left(~~
	~~\frac{\partial p}{\partial t}~~
	~~+ \frac{\partial p}{\partial x}v_x~~
	~~+ \frac{\partial p}{\partial y}v_y~~
	~~+ \frac{\partial p}{\partial z}v_z~~
	~~\right)dt \\ &~~
	~~= \left(~~
	~~\frac{\partial p}{\partial t}~~
	~~+ \mathbf v\cdot\nabla p~~
	~~\right)dt~~. \\
	~~\end{alignat}</math>~~
	~~where <math>\nabla p</math> is the [[gradient]] of the scalar field~~ '~~'p''. If the bug is just~~ a ~~fluid particle moving~~ with the ~~fluid~~'~~s velocity field~~, ~~the same formula applies~~, ~~but now the velocity vector is~~ that ~~of the fluid.~~
	~~The last parenthesized expression~~ is ~~the substantive derivative~~ of ~~the scalar pressure~~.
	~~Since the pressure p in this computation is an arbitrary scalar field~~, ~~we may abstract~~ it ~~and write the substantive derivative operator as~~
	~~:<math>\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf v\cdot\nabla~~.~~</math>~~

	~~== See also ==~~
	*[[Navier-Stokes equations]]
	*[[Special relativity]]
	*[[Substantive derivative]]
	*[[Fluid mechanics]]

	~~==References==~~
	*James R. Welty, ~~Charles E. Wicks~~, ~~Robert E~~. ~~Wilson & Gregory Rorrer ''Fundamentals of Momentum, Heat, and Mass Transfer'' ISBN 0-471-38149-7~~

	~~===Notes===~~
	~~{{reflist}}~~

	~~== External links ==~~
	* [http://s6.aeromech.usyd.edu.au/aero/cvanalysis/integral_approach.pdf Integral Approach to the Control Volume analysis of Fluid Flow]

	~~[[Category:Fluid mechanics]]~~
	~~[[Category:Thermodynamics]]~~

117.202.64.20: /* Derivation */

2014-01-15T10:21:54Z

Derivation

New page

{{thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}
In [[fluid mechanics]] and [[thermodynamics]], a '''control volume''' is a mathematical abstraction employed in the process of creating [[mathematical model]]s of physical processes. In an [[inertial frame of reference]], it is a volume fixed in space or moving with constant velocity through which the [[fluid]] ([[gas]] or [[liquid]]) flows. The surface enclosing the control volume is referred to as the '''control surface'''.<ref>G.J. Van Wylen and R.E. Sonntag (1985), ''Fundamentals of Classical Thermodynamics'', Section 2.1 (3rd edition), John Wiley & Sons, Inc., New York ISBN 0-471-82933-1</ref>

At [[steady state]], a control volume can be thought of as an arbitrary volume in which the mass of the fluid remains constant. As fluid moves through the control volume, the mass entering the control volume is equal to the mass leaving the control volume. At [[steady state]], and in the absence of work and heat transfer, the energy within the control volume remains constant. It is analogous to the classical mechanics concept of the [[free body diagram]].

==Overview==
Typically, to understand how a given [[physical law]] applies to the system under consideration, one first begins by considering how it applies to a small, control volume, or "representative volume". There is nothing special about a particular control volume, it simply represents a small part of the system to which physical laws can be easily applied. This gives rise to what is termed a volumetric, or volume-wise formulation of the mathematical model.

One can then argue that since the [[physical law]]s behave in a certain way on a particular control volume, they behave the same way on all such volumes, since that particular control volume was not special in any way. In this way, the corresponding point-wise formulation of the [[mathematical model]] can be developed so it can describe the physical behaviour of an entire (and maybe more complex) system.

In [[fluid mechanics]] the [[Conservation laws|conservation equations]] (for instance, the [[Navier-Stokes equations]]) are in integral form. They therefore apply on volumes. Finding forms of the equation that are <em>independent</em> of the control volumes allows simplification of the integral signs.

== Substantive derivative ==
{{Main|Material derivative}}
Computations in [[fluid mechanics]] often require that the regular time [[Derivative|derivation]] operator
<math>d/dt\;</math>
is replaced by the [[substantive derivative]] operator
<math>D/Dt</math>.
This can be seen as follows.

Consider a bug that is moving through a volume where there is some [[scalar field|scalar]],
e.g. [[pressure]], that varies with time and position:
<math>p=p(t,x,y,z)\;</math>.

If the bug during the time interval from
<math>t\;</math>
to
<math>t+dt\;</math>
moves from
<math>(x,y,z)\;</math>
to
<math>(x+dx, y+dy, z+dz),\;</math>
then the bug experiences a change <math>dp\;</math> in the scalar value,
:<math>dp = \frac{\partial p}{\partial t}dt
+ \frac{\partial p}{\partial x}dx
+ \frac{\partial p}{\partial y}dy
+ \frac{\partial p}{\partial z}dz</math>
(the [[total derivative|total differential]]). If the bug is moving with velocity
<math>\mathbf v = (v_x, v_y, v_z),</math>
the change in position is
<math>\mathbf vdt = (v_xdt, v_ydt, v_zdt),</math>
and we may write
:<math>\begin{alignat}{2}
dp &
= \frac{\partial p}{\partial t}dt
+ \frac{\partial p}{\partial x}v_xdt
+ \frac{\partial p}{\partial y}v_ydt
+ \frac{\partial p}{\partial z}v_zdt \\ &
= \left(
\frac{\partial p}{\partial t}
+ \frac{\partial p}{\partial x}v_x
+ \frac{\partial p}{\partial y}v_y
+ \frac{\partial p}{\partial z}v_z
\right)dt \\ &
= \left(
\frac{\partial p}{\partial t}
+ \mathbf v\cdot\nabla p
\right)dt. \\
\end{alignat}</math>
where <math>\nabla p</math> is the [[gradient]] of the scalar field ''p''. If the bug is just a fluid particle moving with the fluid's velocity field, the same formula applies, but now the velocity vector is that of the fluid.
The last parenthesized expression is the substantive derivative of the scalar pressure.
Since the pressure p in this computation is an arbitrary scalar field, we may abstract it and write the substantive derivative operator as
:<math>\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf v\cdot\nabla.</math>

== See also ==
*[[Navier-Stokes equations]]
*[[Special relativity]]
*[[Substantive derivative]]
*[[Fluid mechanics]]

==References==
*James R. Welty, Charles E. Wicks, Robert E. Wilson & Gregory Rorrer ''Fundamentals of Momentum, Heat, and Mass Transfer'' ISBN 0-471-38149-7

===Notes===
{{reflist}}

== External links ==
* [http://s6.aeromech.usyd.edu.au/aero/cvanalysis/integral_approach.pdf Integral Approach to the Control Volume analysis of Fluid Flow]

[[Category:Fluid mechanics]]
[[Category:Thermodynamics]]