Closed and exact differential forms Totally Explained

Everything about Exact 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

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