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43-573: (Redirected from Dr ) [REDACTED] Look up Dr , dr , or Dr. in Wiktionary, the free dictionary. DR , Dr , dr , may refer to: Doctor (title) , a person who has obtained a doctoral degree, or a courtesy title for a medical or dental practitioner Arts and media [ edit ] Dalereckoning , a fictional numbering of years in Dungeons & Dragons Danganronpa ,

86-487: A homotopy operator . Since it is also nilpotent , it forms a dual chain complex with the arrows reversed compared to the de Rham complex. This is the situation described in the Poincaré lemma . The idea behind de Rham cohomology is to define equivalence classes of closed forms on a manifold. One classifies two closed forms α , β ∈ Ω ( M ) as cohomologous if they differ by an exact form, that is, if α − β

129-492: A Chinese airline Places [ edit ] Dominican Republic , a country on the eastern portion of the Caribbean island of Hispaniola Dadar railway station , Mumbai, India (Central railway station code) Science and technology [ edit ] Data Room , a space used for housing data, usually of a secure or privileged nature Dead reckoning , a process of estimating global position Demand response ,

172-525: A given equivalence class of closed forms can be written as where α {\displaystyle \alpha } is exact and γ {\displaystyle \gamma } is harmonic: Δ γ = 0 {\displaystyle \Delta \gamma =0} . Any harmonic function on a compact connected Riemannian manifold is a constant. Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on

215-480: A long exact sequence in cohomology. Since the sheaf Ω 0 {\textstyle \Omega ^{0}} of C ∞ {\textstyle C^{\infty }} functions on M admits partitions of unity , any Ω 0 {\textstyle \Omega ^{0}} -module is a fine sheaf ; in particular, the sheaves Ω k {\textstyle \Omega ^{k}} are all fine. Therefore,

258-516: A manufacturer of guitar strings DR (broadcaster) , a Danish government-owned radio and television public broadcasting company D/R or Design Research, a retail lifestyle store chain (1953–1978) DR Motor Company , an Italian automobile company Deutsche Reichsbahn (East Germany) , former German railway company Digital Research , a defunct software company Duane Reade , a New York pharmacy chain Ruili Airlines (IATA code DR),

301-500: A method of managing consumer consumption of electricity Design rationale , documentation of reasons behind decisions made during technical design Designated Router, a concept used in the OSPF routing protocol Digital radiography , a form of x-ray imaging using digital sensors Disaster recovery , reestablishing systems following a disaster Discrepancy reporting, in software project management Dose–response relationship ,

344-412: A shorthand for "Debit", a bookkeeping concept Death row , a prison or section of a prison that houses prisoners awaiting execution Democratic Republic , designating a country that is both a democracy and a republic Depositary receipt , negotiable financial instrument issued by a bank to represent a foreign company's publicly traded securities Derealization , an alteration in the perception of

387-678: A smooth manifold M , this map is in fact an isomorphism . More precisely, consider the map defined as follows: for any [ ω ] ∈ H d R p ( M ) {\displaystyle [\omega ]\in H_{\mathrm {dR} }^{p}(M)} , let I ( ω ) be the element of Hom ( H p ( M ) , R ) ≃ H p ( M ; R ) {\displaystyle {\text{Hom}}(H_{p}(M),\mathbb {R} )\simeq H^{p}(M;\mathbb {R} )} that acts as follows: The theorem of de Rham asserts that this

430-496: A video game series and anime Deltarune , a game by Toby Fox Diário da República , the official gazette of the government of Portugal Douay–Rheims Bible , a translation of the Christian Bible Drag Race , a drag reality competition tv show franchise Dress rehearsal (disambiguation) , a full-scale rehearsal of a public performance Businesses [ edit ] DR Handmade Strings ,

473-430: Is G {\displaystyle G} -invariant if given any diffeomorphism induced by G {\displaystyle G} , ⋅ g : X → X {\displaystyle \cdot g:X\to X} we have ( ⋅ g ) ∗ ( ω ) = ω {\displaystyle (\cdot g)^{*}(\omega )=\omega } . In particular,

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516-531: Is compact and oriented , the dimension of the kernel of the Laplacian acting upon the space of k -forms is then equal (by Hodge theory ) to that of the de Rham cohomology group in degree k {\displaystyle k} : the Laplacian picks out a unique harmonic form in each cohomology class of closed forms . In particular, the space of all harmonic k {\displaystyle k} -forms on M {\displaystyle M}

559-420: Is connected , we have that This follows from the fact that any smooth function on M with zero derivative everywhere is separately constant on each of the connected components of M . One may often find the general de Rham cohomologies of a manifold using the above fact about the zero cohomology and a Mayer–Vietoris sequence . Another useful fact is that the de Rham cohomology is a homotopy invariant. While

602-467: Is an isomorphism between de Rham cohomology and singular cohomology. The exterior product endows the direct sum of these groups with a ring structure. A further result of the theorem is that the two cohomology rings are isomorphic (as graded rings ), where the analogous product on singular cohomology is the cup product . For any smooth manifold M , let R _ {\textstyle {\underline {\mathbb {R} }}} be

645-465: Is different from Wikidata All article disambiguation pages All disambiguation pages Dr">Dr The requested page title contains unsupported characters : ">". Return to Main Page . De Rham cohomology On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in

688-473: Is exact. This classification induces an equivalence relation on the space of closed forms in Ω ( M ) . One then defines the k -th de Rham cohomology group H d R k ( M ) {\displaystyle H_{\mathrm {dR} }^{k}(M)} to be the set of equivalence classes, that is, the set of closed forms in Ω ( M ) modulo the exact forms. Note that, for any manifold M composed of m disconnected components, each of which

731-534: Is harmonic if the Laplacian is zero, Δ γ = 0 {\displaystyle \Delta \gamma =0} . This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is defined with respect to the L inner product on Ω k ( M ) {\displaystyle \Omega ^{k}(M)} : By use of Sobolev spaces or distributions ,

774-524: Is isomorphic to H k ( M ; R ) . {\displaystyle H^{k}(M;\mathbb {R} ).} The dimension of each such space is finite, and is given by the k {\displaystyle k} -th Betti number . Let M {\displaystyle M} be a compact oriented Riemannian manifold . The Hodge decomposition states that any k {\displaystyle k} -form on M {\displaystyle M} uniquely splits into

817-473: Is not an invariant 0 {\displaystyle 0} -form. This with injectivity implies that Since the cohomology ring of a torus is generated by H 1 {\displaystyle H^{1}} , taking the exterior products of these forms gives all of the explicit representatives for the de Rham cohomology of a torus. Punctured Euclidean space is simply R n {\displaystyle \mathbb {R} ^{n}} with

860-437: Is paracompact Hausdorff we have that sheaf cohomology is isomorphic to the Čech cohomology H ˇ ∗ ( U , R _ ) {\textstyle {\check {H}}^{*}({\mathcal {U}},{\underline {\mathbb {R} }})} for any good cover U {\textstyle {\mathcal {U}}} of M .) The standard proof proceeds by showing that

903-510: Is the Cartesian product: T n = S 1 × ⋯ × S 1 ⏟ n {\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}} . Similarly, allowing n ≥ 1 {\displaystyle n\geq 1} here, we obtain We can also find explicit generators for

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946-475: Is the sheaf cohomology of R _ {\textstyle {\underline {\mathbb {R} }}} and at the other lies the de Rham cohomology. The de Rham cohomology has inspired many mathematical ideas, including Dolbeault cohomology , Hodge theory , and the Atiyah–Singer index theorem . However, even in more classical contexts, the theorem has inspired a number of developments. Firstly,

989-409: The 1 -form corresponding to the derivative of angle from a reference point at its centre, typically written as dθ (described at Closed and exact differential forms ). There is no function θ defined on the whole circle such that dθ is its derivative; the increase of 2 π in going once around the circle in the positive direction implies a multivalued function θ . Removing one point of

1032-724: The Hodge theory proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of harmonic forms and of the Hodge theorem. For further details see Hodge theory . If M is a compact Riemannian manifold , then each equivalence class in H d R k ( M ) {\displaystyle H_{\mathrm {dR} }^{k}(M)} contains exactly one harmonic form . That is, every member ω {\displaystyle \omega } of

1075-620: The constant sheaf on M associated to the abelian group R {\textstyle \mathbb {R} } ; in other words, R _ {\textstyle {\underline {\mathbb {R} }}} is the sheaf of locally constant real-valued functions on M. Then we have a natural isomorphism between the de Rham cohomology and the sheaf cohomology of R _ {\textstyle {\underline {\mathbb {R} }}} . (Note that this shows that de Rham cohomology may also be computed in terms of Čech cohomology ; indeed, since every smooth manifold

1118-405: The exterior derivative and δ {\displaystyle \delta } the codifferential . The Laplacian is a homogeneous (in grading ) linear differential operator acting upon the exterior algebra of differential forms : we can look at its action on each component of degree k {\displaystyle k} separately. If M {\displaystyle M}

1161-500: The change in effect on an organism caused by differing levels of exposure Dram (unit) , a unit of mass and volume Dreieckrechner , a German flight computer manufactured in the 1930s and 1940s Dynamic range , the ratio between the largest and smallest possible values of a quantity, such as sound and light H dR k {\displaystyle H_{\text{dR}}^{k}} , notation for de Rham cohomology groups Other uses [ edit ] Dr. or DR ,

1204-403: The circle obviates this, at the same time changing the topology of the manifold. One prominent example when all closed forms are exact is when the underlying space is contractible to a point or, more generally, if it is simply connected (no-holes condition). In this case the exterior derivative d {\displaystyle d} restricted to closed forms has a local inverse called

1247-418: The computation is not given, the following are the computed de Rham cohomologies for some common topological objects: For the n -sphere , S n {\displaystyle S^{n}} , and also when taken together with a product of open intervals, we have the following. Let n > 0, m ≥ 0 , and I be an open real interval. Then The n {\displaystyle n} -torus

1290-503: The de Rham cohomology group for the n {\displaystyle n} -torus is thus n {\displaystyle n} choose k {\displaystyle k} . More precisely, for a differential manifold M , one may equip it with some auxiliary Riemannian metric . Then the Laplacian Δ {\displaystyle \Delta } is defined by with d {\displaystyle d}

1333-428: The de Rham cohomology of the torus directly using differential forms. Given a quotient manifold π : X → X / G {\displaystyle \pi :X\to X/G} and a differential form ω ∈ Ω k ( X ) {\displaystyle \omega \in \Omega ^{k}(X)} we can say that ω {\displaystyle \omega }

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1376-474: The de Rham complex, when viewed as a complex of sheaves, is an acyclic resolution of R _ {\textstyle {\underline {\mathbb {R} }}} . In more detail, let m be the dimension of M and let Ω k {\textstyle \Omega ^{k}} denote the sheaf of germs of k {\displaystyle k} -forms on M (with Ω 0 {\textstyle \Omega ^{0}}

1419-425: The external world such that it seems unreal Diminishing returns See also [ edit ] [REDACTED] Search for "dr" on Misplaced Pages. Digital recorder (disambiguation) Doctor (disambiguation) All pages with titles beginning with dr All pages with titles beginning with DR All pages with titles containing DR RD (disambiguation) Topics referred to by

1462-404: The image of other forms under the exterior derivative , plus the constant 0 function in Ω ( M ) , are called exact and forms whose exterior derivative is 0 are called closed (see Closed and exact differential forms ); the relationship d = 0 then says that exact forms are closed. In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and

1505-441: The manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. The de Rham complex is the cochain complex of differential forms on some smooth manifold M , with the exterior derivative as the differential: where Ω ( M ) is the space of smooth functions on M , Ω ( M ) is the space of 1 -forms , and so forth. Forms that are

1548-551: The manifold. For example, on a 2 - torus , one may envision a constant 1 -form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations. In particular, this implies that the 1st Betti number of a 2 -torus is two. More generally, on an n {\displaystyle n} -dimensional torus T n {\displaystyle T^{n}} , one can consider

1591-780: The origin removed. We may deduce from the fact that the Möbius strip , M , can be deformation retracted to the 1 -sphere (i.e. the real unit circle), that: Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains . It says that the pairing of differential forms and chains, via integration, gives a homomorphism from de Rham cohomology H d R k ( M ) {\displaystyle H_{\mathrm {dR} }^{k}(M)} to singular cohomology groups H k ( M ; R ) . {\displaystyle H^{k}(M;\mathbb {R} ).} de Rham's theorem, proved by Georges de Rham in 1931, states that for

1634-854: The pullback of any form on X / G {\displaystyle X/G} is G {\displaystyle G} -invariant. Also, the pullback is an injective morphism. In our case of R n / Z n {\displaystyle \mathbb {R} ^{n}/\mathbb {Z} ^{n}} the differential forms d x i {\displaystyle dx_{i}} are Z n {\displaystyle \mathbb {Z} ^{n}} -invariant since d ( x i + k ) = d x i {\displaystyle d(x_{i}+k)=dx_{i}} . But, notice that x i + α {\displaystyle x_{i}+\alpha } for α ∈ R {\displaystyle \alpha \in \mathbb {R} }

1677-401: The same term [REDACTED] This disambiguation page lists articles associated with the title DR . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=DR&oldid=1259642322 " Category : Disambiguation pages Hidden categories: Short description

1720-402: The sheaf cohomology groups H i ( M , Ω k ) {\textstyle H^{i}(M,\Omega ^{k})} vanish for i > 0 {\textstyle i>0} since all fine sheaves on paracompact spaces are acyclic. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain

1763-563: The sheaf of C ∞ {\textstyle C^{\infty }} functions on M ). By the Poincaré lemma , the following sequence of sheaves is exact (in the abelian category of sheaves): This long exact sequence now breaks up into short exact sequences of sheaves where by exactness we have isomorphisms i m d k − 1 ≅ k e r d k {\textstyle \mathrm {im} \,d_{k-1}\cong \mathrm {ker} \,d_{k}} for all k . Each of these induces

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1806-706: The sum of three L components: where α {\displaystyle \alpha } is exact, β {\displaystyle \beta } is co-exact, and γ {\displaystyle \gamma } is harmonic. One says that a form β {\displaystyle \beta } is co-closed if δ β = 0 {\displaystyle \delta \beta =0} and co-exact if β = δ η {\displaystyle \beta =\delta \eta } for some form η {\displaystyle \eta } , and that γ {\displaystyle \gamma }

1849-450: The various combings of k {\displaystyle k} -forms on the torus. There are n {\displaystyle n} choose k {\displaystyle k} such combings that can be used to form the basis vectors for H dR k ( T n ) {\displaystyle H_{\text{dR}}^{k}(T^{n})} ; the k {\displaystyle k} -th Betti number for

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