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Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero ( = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.

For an exact form α, α = for some differential form β of one-lesser degree than α. The form β is called a "potential form" or "primitive" for α. Since d2 = 0, β is not unique, but can be modified by the addition of the differential of a two-step-lower-order form.

Because d2 = 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 domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, which allows one to obtain purely topological information using differential methods.

Contents

ExamplesEdit

 
Vector field corresponding to dθ.

A simple example of a form which is closed but not exact is the 1-form   given by the derivative of argument on the punctured plane  .[note 1] Since   is not actually a function (see the next paragraph)   is not an exact form. Still,   has vanishing derivative and is therefore closed.

Note that the argument   is only defined up to an integer multiple of   since a single point   can be assigned different arguments  ,  , etc. We can assign arguments in a locally consistent manner around  , but not in a globally consistent manner. This is because if we trace a loop from   counterclockwise around the origin and back to  , the argument increases by  . Generally, the argument   changes by

 

over a counter-clockwise oriented loop  .

Even though the argument   is not technically a function, the different local definitions of   at a point   differ from one another by constants. Since the derivative at   only uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative " ".[note 2]

The upshot is that   is a one-form on   that is not actually the derivative of any well-defined function  . We say that   is not exact. Explicitly,   is given as:

 ,

which by inspection has derivative zero. Because   has vanishing derivative, we say that it is closed.

This form generates the de Rham cohomology group   meaning that any closed form   is the sum of an exact form   and a multiple of     where   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.

Examples in low dimensionsEdit

Differential forms in R2 and R3 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 dxdy, so that it is the 1-forms

 

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

 

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

 

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

 

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

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, if the integral around any smooth closed curve is zero.

Vector field analogiesEdit

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.

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 (smooth scalar field), 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.

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).

The concepts of conservative and incompressible vector fields generalize to n dimensions, because gradient and divergence generalize to n dimensions; curl is defined only in three dimensions, thus the concept of irrotational vector field does not generalize in this way.

Poincaré lemmaEdit

The Poincaré lemma states that if B is an open ball in Rn, any smooth closed p-form ω defined on B is exact, for any integer p with 1 ≤ pn.[1]

Translating if necessary, it can be assumed that the ball B has centre 0. Let αs be the flow on Rn defined by αsx = esx. For s ≥ 0 it carries B into itself and induces an action on functions and differential forms. The derivative of the flow is the vector field X defined on functions f by Xf = dsf)/ds|s = 0: it is the radial vector field -r∂/∂r = ∑ -xi ∂/∂xi. The derivative of the flow on forms defines the Lie derivative with respect to X given by LX ω = dsω) /ds|s=0. In particular

 

Now define

 

By the fundamental theorem of calculus we have that

 

With   being the interior multiplication or contraction by the vector field X, Cartan's formula writes:

 

Using the fact that d commutes with h (as αs does with d) we get:

 .

It now follows that if ω is closed, i. e. dω = 0, then d(h ιXω) = ω, so that ω is exact and the Poincaré lemma is proved.

(In the language of homological algebra, h ∘ ιX. is a "contracting homotopy".)

The same method applies to any open set in Rn that is star-shaped about 0, i.e. any open set containing 0 and invariant under αt for <  .

Example. In two dimensions the Poincaré lemma can be proved directly for closed 1-forms and 2-forms as follows.[2]

If ω = p dx + q dy is a closed 1-form on (a,b) × (c,d), then py = qx. If ω = df then p = fx and q = fy. Set

 

so that gx = p. Then h = fg must satisfy hx = 0 and hy = qgy. The right hand side here is independent of x since its partial derivative with respect to x is 0. So

 

and hence

 

Similarly if Ω = r dxdy then Ω = d(a dx + b dy) with bxay = r. Thus a solution is given by a = 0 and

 

Formulation as cohomologyEdit

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

 

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.

Using contracting homotopies similar to the one used in the proof of the Poincaré lemma, it can be shown that de Rham cohomology is homotopy-invariant. In general, non-contractible differentiable manifolds have non-trivial de Rham cohomology. For instance, on the circle S1, parametrized by t in [0, 1], the closed 1-form dt is not exact.

Application in electrodynamicsEdit

In electrodynamics, the case of the magnetic field   produced by a stationary electrical current is important. There one deals with the vector potential   of this field. This case corresponds to k=2, and the defining region is the full   The current-density vector is   It corresponds to the current two-form

 

For the magnetic field   one has analogous results: it corresponds to the induction two-form   and can be derived from the vector potential  , or the corresponding one-form  ,

 

Thereby the vector potential   corresponds to the potential one-form

 

The closedness of the magnetic-induction two-form corresponds to the property of the magnetic field that it is source-free:     i.e. there are no magnetic monopoles.

In a special gauge,  , this implies for i = 1, 2, 3

 

(Here   is a constant, the magnetic vacuum permeability.)

This equation is remarkable, because it corresponds completely to a well-known formula for the electrical field  , namely for the electrostatic Coulomb potential   of a charge density  . At this place one can already guess that

  •   and  
  •   and  
  •   and  

can be unified to quantities with six rsp. four nontrivial components, which is the basis of the relativistic invariance of the Maxwell equations.

If the condition of stationarity is left, on the l.h.s. of the above-mentioned equation one must add, in the equations for   to the three space coordinates, as a fourth variable also the time t, whereas on the r.h.s., in   the so-called "retarded time",     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 c is the vacuum velocity of light.)

NotesEdit

  1. ^ This is an abuse of notation. The argument   is not a well-defined function, and   is not the differential any zero-form. The discussion that follows elaborates on this.
  2. ^ The article covering spaces has more information on the mathematics of functions that are only locally well-defined.

FootnotesEdit

  1. ^ Warner 1983, pp. 155-156
  2. ^ Napier & Ramachandran 2011, pp. 443-444

ReferencesEdit

  • Flanders, Harley (1989), Differential forms with applications to the physical sciences, New York: Dover Publications, ISBN 978-0-486-66169-8 .
  • Warner, Frank W. (1983), Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer, ISBN 0-387-90894-3 
  • Napier, Terrence; Ramachandran, Mohan (2011), An introduction to Riemann surfaces, Birkhäuser, ISBN 978-0-8176-4693-6