Closed and exact differential forms Totally Explained

Everything about Closed Differential Form totally explained

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose differential is zero (dα = 0), and an exact form is a differential form that's the differential of another differential form (α = dβ for some differential form β, known as a primitive for α). Since d² = 0, to be exact is a sufficient condition to be closed. The main interest of this pair of definitions, thus, is that asking whether this is also a necessary condition is a way of detecting topological information, by differential conditions. It makes no real sense to ask whether a 0-form is exact, since d increases degree by 1.
When the difference of two closed forms is an exact form, they're said to be cohomologous to each other. That is, if ζ and η are closed forms, and one can find some β such that ť

zeta - eta = deta,

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 each other form an element of a de Rham cohomology class; the general study of such classes is known as cohomology.
The cases of differential forms in R² and R³ were already 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's the 1-forms ť

alpha = f(x,y)dx + g(x,y)dy,

that are of real interest. The formula for the exterior derivative d here is ť

d alpha = (g_x - f_y)dxwedge dy,

where the subscripts denote partial derivatives. Therefore the condition for α to be closed is ť

f_y=g_x.,

In this case if h(x,y) is a function then ť

dh = h_x dx + h_y dy.,

The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y.

Poincaré lemma

The Poincaré lemma states that if X is a contractible open subset of Rⁿ, any smooth closed p-form α defined on X is exact, for any integer p > 0 (this has content only when p ≤ n).
Contractability means that there's a homotopy F_t : 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 Poincaré lemma.
More specificly, we associate to X the cylinder X×[0,1]. Identify the top and bottom of the cylinder with the maps j₁(x) = (x, 1) and j₀(x) = (x, 0) respectively. On the differential forms, the induced maps j₁* and j₀* are related by a cochain homotopy K:
ť

K d + d K = j_1^* - j_0 ^*.

Let Ω^p(X) denote the p-forms on X. The map K: Ω^{p + 1}(X×[0,1] ) → Ω^p(X) is the dual of the cylinder map and defined by ť

a(x,t) d x^ mapsto 0, ; a(x,t) dt dx^p  mapsto (int_0 ^1 a(x,t) dt) dx^p,

where dx^p is a monomial p-form with no dt in it. So if F is a homotopy deforming X to a point Q, then ť

F circ j_1 = id, ; F circ j_0 = Q.

On forms, ť

j_1 ^* circ F^* = id, ; j_0^* circ F^* = 0.

Inserting these two equations into the cochain homotopy equation proves Poincaré lemma.
   A corollary of the lemma is that de Rham cohomology is homotopy invariant.
   Non-contractible spaces need not have trivial de Rham cohomology. For instance, on the circle S¹, parametrized by t in [0,1], the closed 1-form dt isn't exact.

Further Information

Get more info on 'Closed Differential Form'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://closed_and_exact_differential_forms.totallyexplained.com">Closed and exact differential forms Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.