exact differential form

In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), 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'', for some differential form ''β'' of degree one less than that of ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since the exterior derivative of a closed form is zero, ''β'' is not unique, but can be modified by the addition of any closed form of degree one less than that of ''α''.

Because , every exact form is necessarily 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.

Examples

A simple example of a form which is closed but not exact is the 1-form $d\theta$This is an abuse of notation. The argument $\theta$ is not a well-defined function, and $d\theta$ is not the differential of any zero-form. The discussion that follows elaborates on this. given by the derivative of argument on the punctured plane Since $\theta$ is not actually a function (see the next paragraph) $d\theta$ is not an exact form. Still, $d\theta$ has vanishing derivative and is therefore closed.

Note that the argument $\theta$ is only defined up to an integer multiple of $2\pi$ since a single point $p$ 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 $p$ counterclockwise around the origin and back to the argument increases by Generally, the argument $\theta$ changes by
:$\oint_ d\theta$
over a counter-clockwise oriented loop

Even though the argument $\theta$ is not technically a function, the different ''local'' definitions of $\theta$ at a point $p$ differ from one another by constants. Since the derivative at $p$ only uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative The article covering spaces has more information on the mathematics of functions that are only locally well-defined.

The upshot is that $d\theta$ is a one-form on $\mathbf^2\setminus\$ that is not actually the derivative of any well-defined function We say that $d\theta$ is not ''exact''. Explicitly, $d\theta$ is given as:
:$d\theta = \frac ,$
which by inspection has derivative zero. Because $d\theta$ has vanishing derivative, we say that it is ''closed''.

This form generates the de Rham cohomology group $H^1_(\mathbf^2\setminus\) \cong \mathbf,$ meaning that any closed form $\omega$ is the sum of an exact form $df$ and a multiple of where $k = \frac\oint_ \omega$ 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 dimensions

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 , so that it is the 1-forms

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

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

:$d \alpha = (g_x-f_y) \, dx\wedge dy$

where the subscripts denote partial derivatives. Therefore the condition for $\alpha$ to be ''closed'' is

:$f_y=g_x.$

In this case if is a function then

:$dh = h_x \, dx + h_y \, dy.$

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

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 analogies

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 term incompressible is used because a non-zero divergence corresponds to the presence of sources and sinks in analogy with a fluid.

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 lemma

The Poincaré lemma states that if ''B'' is an open ball in R^''n'', any smooth closed ''p''-form ''ω'' defined on ''B'' is exact, for any integer ''p'' with .

Translating if necessary, it can be assumed that the ball ''B'' has centre 0. Let ''α''_''s'' be the flow on R^''n'' defined by . For 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 : it is the ''radial vector field'' . The derivative of the flow on forms defines the Lie derivative with respect to ''X'' given by In particular

:$\frac \alpha_s \omega = \alpha_s L_X \omega,$

Now define

:$h\,\omega =-\int_0^\infty \alpha_t\omega \, dt.$

By the fundamental theorem of calculus we have that

:L_X h \, \omega = -\int_0^\infty \alpha_t L_X \omega \, dt=-\int_0^\infty (\alpha_t \omega) \, dt= - alpha _t \omega0^\infty= \omega.

With $\iota_X$ being the interior multiplication or contraction by the vector field ''X'', Cartan's formula states that

:$L_X = d\iota_X + \iota_X d.$

Using the fact that ''d'' commutes with ''L''_''X'', $\alpha_s$ and ''h'', we get:

:$\omega = L_X h\, \omega = (d\iota_X + \iota_X d) h \omega = d (\iota_X h \omega) + \iota_X h d \omega.$

Setting

:$g(\omega)= \iota_X h (\omega),$

leads to the identity

:$(dg + gd)\, \omega = \omega.$

It now follows that if ''ω'' is closed, i. e. , then , so that ''ω'' is exact and the Poincaré lemma is proved.

(In the language of homological algebra, ''g'' is a "contracting homotopy".)

The same method applies to any open set in R^''n'' that is star-shaped about 0, i.e. any open set containing 0 and invariant under ''α''_''t'' for $1 < t < \infty$.

Another standard proof of the Poincaré lemma uses the homotopy invariance formula and can be found in , , and . The local form of the homotopy operator is described in and the connection of the lemma with the Maurer-Cartan form is explained in .

This formulation can be phrased in terms of homotopies between open domains ''U'' in ''R''^''m'' and ''V'' in ''R''^''n''. If ''F''(''t'',''x'') is a homotopy from ,1× ''U'' to ''V'', set ''F''_''t''(''x'') = ''F''(''t'',''x''). For $\omega$ a ''p''-form on ''V'', define
:$g(\omega)=\int_0^1 \iota_ (F_t^*(\omega)) \, dt$

Then

:$(dg+gd)\,\omega= \int_0^1 (d\iota_ + \iota_ d)F_t^*(\omega)\, dt =\int_0^1 L_ F_t^*(\omega) \, dt= \int_0^1 \partial_t F_t^*(\omega)\, dt = F_1^*(\omega) - F_0^*(\omega).$

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

If is a closed 1-form on , then . If then and . Set

:$g(x,y)=\int_a^x p(t,y)\, dt,$

so that . Then must satisfy and . The right hand side here is independent of ''x'' since its partial derivative with respect to ''x'' is 0. So

:$h(x,y)=\int_c^y q(a,s)\, ds - g(a,y)=\int_c^y q(a,s)\, ds,$

and hence

:$f(x,y)=\int_a^x p(t,y)\, dt + \int_c^y q(a,s)\, ds.$

Similarly, if then with . Thus a solution is given by and

:$b(x,y)=\int_a^x r(t,y) \, dt.$

Alternative notions

Related notions to the Poincaré lemma can be proven in other contexts.

On complex manifolds, the use of the Dolbeault operators $\partial$ and $\bar \partial$ for complex differential forms, which refine the exterior derivative by the formula $d=\partial + \bar \partial$, lead to the notion of $\bar \partial$-closed and $\bar \partial$-exact differential forms. The local exactness result for such closed forms is known as the Dolbeault–Grothendieck lemma (or $\bar \partial$-Poincaré lemma). Importantly, the geometry of the domain on which a $\bar \partial$-closed differential form is $\bar \partial$-exact is more restricted than for the Poincaré lemma, since the proof of the Dolbeault–Grothendieck lemma holds on a polydisk (a product of disks in the complex plane, on which the multidimensional Cauchy's integral formula may be applied) and there exist counterexamples to the lemma even on contractible domains.For counterexamples on contractible domains which have non-vanishing first Dolbeault cohomology, see the post https://mathoverflow.net/a/59554. The $\bar \partial$-Poincaré lemma holds in more generality for pseudoconvex domains.

Using both the Poincaré lemma and the $\bar \partial$-Poincaré lemma, a refined local $\partial \bar \partial$-Poincaré lemma can be proven, which is valid on domains upon which both the aforementioned lemmas are applicable. This lemma states that $d$-closed complex differential forms are actually locally $\partial \bar \partial$-exact (rather than just $d$ or $\bar \partial$-exact, as implied by the above lemmas).

Basic derivation for a 1-form

Let $G \subset \mathbf^n \, (n\ge2)\,$ be an open star domain and let $\mathbf=(a_1,...,a_n)\,$ be a vantage point of $G$, so for all $\mathbf=(x_1,...,x_n)\in G\,$ the line segment mathbf,\mathbf/math> lies fully in $G$.
And let $\,g_k: G \to \mathbf^m\,(k=1,...,n)\,$ be continuously differentiable functions. (Possibly some of them are zero.)

Now $\omega = \sum_^n g_j(\mathbf)dx_j\,$ is a (differential) 1-form and its exterior derivative is $d\omega = \sum_^n \sum_^n \fracdx_k\wedge dx_j\,$

Since $dx_j\wedge dx_j = 0,\, dx_j\wedge dx_k = -dx_k\wedge dx_j$ it follows $d\omega = \sum_^ \sum_^n (\frac dx_k\wedge dx_j + \frac dx_j\wedge dx_k) = \sum_^ \sum_^n (\frac - \frac)dx_j\wedge dx_k$

So $d\omega =0\,$ (i.e. $\omega$ is closed) implies:

For all $j,k \in \$ it should hold: $\,\,\frac(\mathbf) = \frac(\mathbf)$

Then \,f(\mathbf):=\int_0^1\sum_^n \left g_j\left(\mathbf+t\left(\mathbf-\mathbf\right)\right)\cdot(x_j-a_j) \rightdt\, defines a continuously differentiable function $\,G \to \mathbf^m\,$ with $\, \frac(\mathbf) = g_k(\mathbf) \,$ for all $k \in \$.

The integral is well-defined for all $\mathbf \in G\,$ because $\mathbf+t\left(\mathbf-\mathbf\right) \in G\,$ for all \, t \in ;1, and the integrand is a finite sum of continuous functions $t\mapsto g_j\left(\mathbf+t\left(\mathbf-\mathbf\right)\right)\cdot\left(\mathbf-\mathbf\right)$ over the closed real interval ;1/math>.

Now \,\frac(\mathbf) = \int_0^1 \sum_^n \frac \left g_j\left(\mathbf+t\left(\mathbf-\mathbf\right)\right)\cdot(x_j-a_j) \rightdt\,=\,
\int_0^1 \sum_^n \left frac \left(\mathbf+t\left(\mathbf-\mathbf\right)\right)\cdot t \cdot(x_j-a_j) \,+\ \rightdt\, = \,
\int_0^1 \left(g_k\left(\mathbf+t\left(\mathbf-\mathbf\right)\right) + \sum_^n \left frac \left(\mathbf+t\left(\mathbf-\mathbf\right)\right)\cdot t \cdot(x_j-a_j)\right\right) dt\, = \,
\int_0^1 \frac\left \cdot g_k(\mathbf+t(\mathbf-\mathbf))\rightdt = g_k(\mathbf). (Source found at math.stackexchange.)

Formulation as cohomology

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

:$\zeta - \eta = d\beta$

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.

Application in electrodynamics

In electrodynamics, the case of the magnetic field $\vec B(\mathbf r)$ produced by a stationary electrical current is important. There one deals with the vector potential $\vec A(\mathbf r )$ of this field. This case corresponds to , and the defining region is the full $\R^3$. The current-density vector is It corresponds to the current two-form

:$\mathbf I :=j_1(x_1,x_2, x_3) \, x_2\wedge x_3+j_2(x_1,x_2, x_3) \, x_3\wedge x_1+j_3(x_1,x_2, x_3) \, x_1\wedge x_2.$

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

:$\vec B =\operatorname\vec A =\left\, \text \Phi_B=\mathbf A.$

Thereby the vector potential $\vec A$ corresponds to the potential one-form

:$\mathbf A:=A_1 \, x_1+A_2 \, x_2+A_3 \, x_3.$

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

In a special gauge, $\operatorname\vec A0$, this implies

:$A_i(\vec r) = \int \frac \, .$

(Here $\mu_0$ is a constant, the magnetic vacuum permeability.)

This equation is remarkable, because it corresponds completely to a well-known formula for the ''electrical'' field $\vec E$, namely for the ''electrostatic Coulomb potential'' $\phi (x_1,x_2, x_3)$ of a ''charge density'' $\rho (x_1,x_2,x_3)$. At this place one can already guess that

*$\vec E$ and $\vec B ,$
*$\rho$ and $\vec j ,$
*$\phi$ and $\vec A$

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 left-hand side 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 right-hand side, 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.)

Notes

Footnotes

References

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{{Manifolds

Differential forms
Lemmas in analysis