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.
A simple example of a form that is closed but not exact is the 1-form
d\theta
\theta
d\theta
d\theta
Note that the argument
\theta
2\pi
p
p
\theta
\oint | |
S1 |
d\theta
Even though the argument
\theta
\theta
p
p
The upshot is that
d\theta
R2\smallsetminus\{0\}
d\theta
d\theta
d\theta=
-ydx+xdy | |
x2+y2 |
,
d\theta
This form generates the de Rham cohomology group
1 | |
H | |
dR |
(R2\smallsetminus\{0\})\congR,
\omega
df
Differential forms in
\R2
\R3
dx\wedgedy
\alpha=f(x,y)dx+g(x,y)dy
d
d\alpha=(gx-fy)dx\wedgedy
where the subscripts denote partial derivatives. Therefore the condition for
\alpha
fy=gx.
In this case if
h(x,y)
dh=hxdx+hydy.
The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to
x
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.
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.
The Poincaré lemma states that if B is an open ball in Rn, any closed p-form ω defined on B is exact, for any integer p with .
More generally, the lemma states that on a contractible open subset of a manifold (e.g.,
Rn
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-η=d\beta
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 electrodynamics, the case of the magnetic field
\vecB(r)
\vecA(r)
\R3
I:=j1(x1,x2,x3){\rmd}x2\wedge{\rmd}x3+j2(x1,x2,x3){\rmd}x3\wedge{\rmd}x1+j3(x1,x2,x3){\rmd}x1\wedge{\rmd}x2.
For the magnetic field
\vecB
\vecA
A
\vecB=\operatorname{curl}\vecA=\left\{
\partialA3 | - | |
\partialx2 |
\partialA2 | |
\partialx3 |
,
\partialA1 | - | |
\partialx3 |
\partialA3 | , | |
\partialx1 |
\partialA2 | - | |
\partialx1 |
\partialA1 | |
\partialx2 |
\right\}, or \PhiB={\rmd}A.
Thereby the vector potential
\vecA
A:=A1{\rmd}x1+A2{\rmd}x2+A3{\rmd}x3.
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{div}\vecA{~\stackrel{!}{=}~}0
Ai(\vecr)=\int
\mu0ji\left(\vecr'\right)dx1'dx2'dx3' | |
4\pi|\vecr-\vecr'| |
.
(Here
\mu0
This equation is remarkable, because it corresponds completely to a well-known formula for the electrical field
\vecE
\varphi(x1,x2,x3)
\rho(x1,x2,x3)
\vecE
\vecB,
\rho
\vecj,
\varphi
\vecA
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.)
\theta
d\theta