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| In [[mathematics]], especially [[vector calculus]] and [[differential topology]], a '''closed form''' is a [[differential form]] ''α'' whose [[exterior derivative]] is zero (''dα'' = 0), and an '''exact form''' is a differential form that is the exterior derivative of another differential form ''β''. Thus, an '''exact''' form is in the ''[[image (mathematics)|image]]'' of ''d'', and a '''closed''' form is in the ''[[kernel (algebra)|kernel]]'' of ''d''.
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| For an exact form ''α'', {{nowrap|1=''α'' = ''dβ''}} for some differential form ''β'' of one-lesser degree than ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since {{nowrap|1=''d''<sup>2</sup> = 0}}, ''β'' is not unique, but can be modified by the addition of the differential of a two-step-lower-order form.
| | In the case when as a parent you might be [http://En.Wiktionary.org/wiki/concerned concerned] with movie on the internet game content, control what online mods are put globe sport. These online mods are usually written by players, perhaps not often the gaming businesses, therefore there's no ranking system. You actually thought was a realistically un-risky game can revert a lot worse considering any of these mods.<br><br>There is a patch game button that you will want to click after entering that this desired values. when you check back concerning the game after their late twenties seconds to a minute, you will already have the items. Recently there is nothing wrong by making use of secret sauce. To hack was the best way into enjoy clash of clans cheats. Make use of usually the Resources that you have, and take advantage of this 2013 Clash for Clans download! Reason why pay for coins potentially gems when you could well get the needed gadgets with this tool! Hurry and get one's own very own Clash because of Clans hack tool right now. The needed valuables are just a couple clicks away.<br><br>There are no consequences in the least to be attacking other players in addition , losing, so just tackle and savor it. Win or lose, anybody may lose the a great deal troops you have within the attack since these types of people are only beneficial in order to one mission, nevertheless, an individual can steal more guides with the enemy vill than it cost in order to really make the troops. And you just have more troops within your primary barracks. It''s a new good idea to grab them queued up until now you decide to tackle and that means the person are rebuilding your militia through the battle.<br><br>Web site not only provides end tools, there is usually Clash of Clans compromise no survey by virtually anyone. Strict anti ban system probable users to utilize pounds and play without type of hindrance. If manufacturers are interested in finding the right program, they are just required to visit this wonderful site and obtain the hack tool trainer at this instant. The name of the internet sites is Amazing Cheats. A number of site have different types related to software by which many can get past harsh stages in the task.<br><br>Make sure that shoppers build and buy a handful new laboratory so you are able to research improved barbarians. Should you loved this short article and you would want to receive more details with regards to [http://prometeu.net how to hack clash of clans] kindly visit our web site. Eventually, in proceedings you take part their game for most months, you might finally obtain the nirvana of five-star barbarians.<br><br>A lot of them are not really cheats, they are excuses. The odds are quality that unless you can be found dating a certain as a professional golfer or a diamond star along the method by which this is not likely to happen to you. In John 4:23 and 24 Jesus tells usa we are to praise God "in spirit and in truth. Once entered, the Ruzzle cheat should then show a list from all of the possible words that can be accomplished. Using a PSP Market Emulator is a straightforward way to hack your individual PSP and open together new worlds of fun. s these university students played Poker online and other casino activities simply for fun. |
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| Because {{nowrap|1=''d''<sup>2</sup> = 0}}, any exact form is automatically closed. The question of whether ''every'' closed form is exact depends on the [[topology]] of the domain of interest. On a [[Contractible space|contractible]] domain, every closed form is exact by the [[Closed and exact differential forms#Poincaré_lemma|Poincaré lemma]]. More general questions of this kind on an arbitrary [[differentiable manifold]] are the subject of [[de Rham cohomology]], that allows one to obtain purely [[algebraic topology|topological]] information using differential methods.
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| == Examples ==
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| {{details|Winding number}}
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| [[File:Irrotationalfield.svg|thumb|Vector field corresponding to "d''θ''".]]
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| The simplest example of a form which is closed but not exact is the 1-form "d''θ''" (quotes because it is not the derivative of a globally defined function), defined on the [[punctured plane]] <math>\mathbf{R}^2\setminus\{0\},</math> which is locally given as the derivative of the [[Argument (complex analysis)|argument]] - note that argument is locally but not globally defined, since a loop around the origin increases (or decreases, depending on direction) the argument by 2''π'', which corresponds to the integral:
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| :<math>\oint_{S^1} d\theta = 2\pi,</math>
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| and for general paths is known as the [[winding number]]. The ''differential'' of the argument is however globally defined (except at the origin), since differentiation only requires local data and different values of the argument differ by a constant, so the derivatives of different local definitions are equal; this line of thought is generalized in the notion of [[covering space]]s.
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| Explicitly, the form is given as:
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| :<math>d\theta = \frac{1}{x^2+y^2}\left(-y\,dx + x\,dy\right),</math>
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| which is not defined at the origin. This can be computed from a [[Argument_(complex_analysis)#Computation|formula for the argument]], most simply via arctan(''y''/''x'') (''y''/''x'' is the slope of the line passing through (''x'',''y''), and arctan converts slope to angle), recognizing 1/(''x''<sup>2</sup>+''y''<sup>2</sup>) as corresponding to the derivative of arctan, which is 1/(''x''<sup>2</sup>+1) (these agree on the line ''y''=1). While the differential is correctly computed by symbolically differentiating this expression, this formula is only strictly correct on the halfplane ''x''>0, and properly one must use a correct formula for the argument.
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| This form generates the de Rham cohomology group <math>H^1_{dR}(\mathbf{R}^2\setminus\{0\}) \cong \mathbf{R},</math> meaning that any closed form <math>\omega</math> is the sum of an exact form <math>df</math> and a multiple of <math>d\theta:</math> <math>\omega = df + k\cdot d\theta,</math> where <math>\textstyle{k = \frac{1}{2\pi}\oint_{S^1} \omega}</math> accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a [[potential function]]) being the derivative of a globally defined function.
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| ==Examples in low dimensions==
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| Differential forms in '''R'''<sup>2</sup> and '''R'''<sup>3</sup> were well known in the [[mathematical physics]] of the nineteenth century. In the plane, 0-forms are just functions, and 2-forms are functions times the basic area element ''dx''∧''dy'', so that it is the 1-forms
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| :<math> \alpha = f(x,y) \, dx + g(x,y) \, dy</math>
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| that are of real interest. The formula for the [[exterior derivative]] ''d'' here is
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| :<math> d \alpha = (g_x-f_y) \, dx\wedge dy\,</math>
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| where the subscripts denote [[partial derivative]]s. Therefore the condition for ''α'' to be ''closed'' is
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| :<math> f_y=g_x.\,</math>
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| In this case if ''h''(''x'',''y'') is a function then
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| :<math> dh = h_x \, dx + h_y \, dy.\,</math>
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| The implication from 'exact' to 'closed' is then a consequence of the [[symmetry of second derivatives]], with respect to ''x'' and ''y''.
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| The [[gradient theorem]] asserts that a 1-form is exact if and only if the line integral of the form depends only on the endpoints of the curve, or equivalently,
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| if the integral around any smooth closed curve is zero.
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| ===Vector field analogies===
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| On a [[Riemannian manifold]], or more generally a [[pseudo-Riemannian manifold]], ''k''-forms correspond to ''k''-vector fields (by duality via the metric), so there is a notion of a vector field corresponding to a closed or exact form.
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| In 3 dimensions, an exact vector field (thought of as a 1-form) is called a [[conservative vector field]], meaning that it is the derivative ([[gradient]]) of a 0-form (function), called the [[scalar potential]]. A closed vector field (thought of as a 1-form) is one whose derivative (curl) vanishes, and is called an [[irrotational vector field]].
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| Thinking of a vector field as a 2-form instead, a closed vector field is one whose derivative ([[divergence]]) vanishes, and is called an [[incompressible flow]] (sometimes [[solenoidal vector field]]).
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| Conservative and incompressible vector fields generalize to ''n''-dimensions (gradient and divergence generalize to ''n'' dimensions); curl and hence irrotational does not generalize in this way.
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| ==Poincaré lemma==
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| The '''Poincaré lemma''' states that if ''X'' is a [[Contractible space|contractible]] open subset of '''R'''<sup>''n''</sup>, any smooth closed ''p''-form ''α'' defined on ''X'' is exact, for any integer ''p'' > 0 (this has content only when ''p'' ≤ ''n'').
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| Contractibility means that there is a [[homotopy]] ''F''<sub>''t''</sub> : ''X''×[0,1] → ''X'' that continuously deforms ''X'' to a point. Thus every cycle ''c'' in ''X'' is the boundary of some "cone"; one may take the cone to be the image of ''c'' under the homotopy. A dual version of this gives the Poincaré lemma.
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| More specifically, we associate to ''X'' the cylinder ''X''×[0,1]. Identify the top and bottom of the cylinder with the maps ''j''<sub>1</sub>(''x'') = (''x'', 1) and ''j''<sub>0</sub>(''x'') = (''x'', 0) respectively. On the differential forms, the induced maps ''j''<sub>1</sub>* and ''j''<sub>0</sub>* are related by a [[chain homotopy|cochain homotopy]] ''K'':
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| :<math>K d + d K = j_1^* - j_0 ^*.</math>
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| Let Ω<sup>''p''</sup>(''X'') denote the ''p''-forms on ''X''. The map ''K'': Ω<sup>''p'' + 1</sup>( ''X''×[0,1] ) → Ω<sup>''p''</sup>(''X'') is the dual of the cylinder map and defined by
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| :<math>a(x,t) \, d x^{p+1} \mapsto 0, \; a(x,t) \, dt \, dx^p \mapsto \left(\int_0 ^1 a(x,t) \, dt\right) dx^p,</math>
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| where ''dx''<sup>''p''</sup> is a monomial ''p''-form with no ''dt'' in it. So if ''F'' is a homotopy deforming ''X'' to a point ''Q'', then
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| :<math>F \circ j_1 = \mathrm{id}, \; F \circ j_0 = Q.</math>
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| On forms,
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| :<math>j_1 ^* \circ F^* = \mathrm{id}, \; j_0^* \circ F^* = 0.</math>
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| Inserting these two equations into the cochain homotopy equation proves the Poincaré lemma.
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| ==Formulation as cohomology==
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| When the difference of two closed forms is an exact form, they are said to be '''cohomologous''' to each other. That is, if ζ and η are closed forms, and one can find some β such that
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| :<math>\zeta - \eta = d\beta\,</math>
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| then one says that ζ and η are cohomologous to each other. Exact forms are sometimes said to be '''cohomologous to zero'''. The set of all forms cohomologous to a given form (and thus to each other) is called a [[de Rham cohomology]] class; the general study of such classes is known as [[cohomology]]. It makes no real sense to ask whether a 0-form (smooth function) is exact, since ''d'' increases degree by 1; but the clues from topology suggest that only the zero function should be called "exact". The cohomology classes are identified with [[locally constant]] functions.
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| A corollary of the Poincaré lemma is that de Rham cohomology is homotopy-invariant. Non-contractible spaces need not have trivial de Rham cohomology. For instance, on the circle ''S''<sup>1</sup>, parametrized by ''t'' in [0, 1], the closed 1-form ''dt'' is not exact. | |
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| ==Application in electrodynamics==
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| In electrodynamics, the case of the magnetic field <math>\vec B(\mathbf r)</math> produced by a stationary electrical current is important. There one deals with the [[vector potential]] <math>\vec A(\mathbf r )</math> of this field. This case corresponds to k=2, and the defining region is the full <math>\mathbb R^3\,.</math> The current-density vector is <math>\vec j\,.</math> It corresponds to the current two-form
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| :<math>\mathbf I :=j_1(x,y,z) \, {\rm d}x_2\wedge {\rm d}x_3+j_2(x,y,z) \, {\rm d}x_3\wedge {\rm d}x_1+j_3(x,y,z) \, {\rm d}x_1\wedge {\rm d}x_2.</math>
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| For the magnetic field <math>\vec B</math> one has analogous results: it corresponds to the induction two-form <math>\Phi_B:=B_1{\rm d}x_2\wedge {\rm d}x_3 +\cdots ,</math> and can be derived from the vector potential <math>\vec A</math>, or the corresponding one-form <math>\mathbf A</math>,
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| :<math> \vec B ={\rm curl\,\,}\vec A =\left\{ \frac{\partial A_3}{\partial x_2}-\frac{\partial A_2}{\partial x_3} , \frac{\partial A_1}{\partial x_3}-\frac{\partial A_3}{\partial x_1} ,\frac{\partial A_2}{\partial x_1}-\frac{\partial A_1}{\partial x_2}\right\},\text{ or }\Phi_B={\rm d}\mathbf A.</math>
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|
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| Thereby the vector potential <math>\vec A</math> corresponds to the potential one-form
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| :<math>\mathbf A:=A_1 \, {\rm d}x_1+A_2 \, {\rm d}x_2+A_3 \, {\rm d}x_3.</math>
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| The closedness of the magnetic-induction two-form corresponds to the property of the magnetic field that it is source-free: <math> {\rm div\,\,}\vec B \equiv 0,</math> i.e. there are no magnetic [[Magnetic monopole|monopole]]s.
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| In a special gauge, <math>{\rm div\,\,}\vec A\stackrel{!}{=}0</math>, this implies for ''i'' = 1, 2, 3
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| :<math>A_i(\vec r)
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| =\int \frac{\mu_0 j_i(\vec r^{\,'})\,\, dx_1'dx_2'dx_3'}{4\pi |\vec r -\vec r^{\,'}|}\,.</math>
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| (Here <math>\mu_0</math> is a constant, the magnetic vacuum permeability.)
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| This equation is remarkable, because it corresponds completely to a well-known formula for the ''electrical'' field <math>\vec E</math>, namely for the ''electrostatic Coulomb potential'' <math>\,\phi (x_1,x_2, x_3)</math> of a ''charge density'' <math>\rho (x_1,x_2,x_3)</math>. At this place one can already guess that
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| *<math>\vec E</math> and <math>\vec B ,</math>
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| *<math>\rho </math> and <math>\vec j ,</math>
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| *<math>\,\phi</math> and <math>\vec A</math>
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| can be ''unified'' to quantities with six rsp. four nontrivial components, which is the basis of the [[relativistic invariance]] of the [[Maxwell equations]]. | |
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| If the condition of stationarity is left, on the ''l.h.s.'' of the above-mentioned equation one must add, in the equations for <math>A_i\,,</math> to the three space coordinates, as a fourth variable also the time ''t'', whereas on the ''r.h.s.'', in <math>j_i' \,,</math> the so-called "retarded time", <math>t':=t-\frac{|\vec r -\vec r^{\,'}|}{c}\,,</math> must be used, i.e. it is added to the argument of the current-density. Finally, as before, one integrates over the three primed space coordinates. (As usual
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| ''c'' is the vacuum velocity of light.)
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| ==References==
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| * {{Citation | last1=Flanders | first1=Harley | title=Differential forms with applications to the physical sciences | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-66169-8 | year=1989}}.
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| [[Category:Differential forms]]
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| [[Category:Lemmas]]
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