In mathematics, especially vector calculus and differential topology, a **closed form** is a differential form *α* whose exterior derivative is zero (*dα* = 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*.

- Examples
- Examples in low dimensions
- Vector field analogies
- Poincaré lemma
- Formulation as cohomology
- Application in electrodynamics
- Notes
- Footnotes
- References

For an exact form *α*, *α* = *dβ* 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 *d*^{2} = 0, 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 which is closed but not exact is the 1-form ^{ [note 1] } given by the derivative of argument on the punctured plane . 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.

Differential forms in **R**^{2} and **R**^{3} 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 *dx* ∧ *dy*, 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.

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 **R**^{n}, any smooth closed *p*-form *ω* defined on *B* is exact, for any integer *p* with 1 ≤ *p* ≤ *n*.^{ [1] }

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 *α*_{s}**x** = *e*^{−s}**x**. 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* = *d*(*α*_{s}*f*)/*ds*|_{s = 0}: it is the *radial vector field*−*r*∂/∂*r* = −∑ *x*_{i}∂/∂*x*_{i}. The derivative of the flow on forms defines the Lie derivative with respect to *X* given by . 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 states that^{ [2] }

Using the fact that *d* commutes with *L*_{X}, and *h*, we get:

Setting

leads to the identity

It now follows that if *ω* is closed, i. e. *dω* = 0, then *d*(*g**ω*) = *ω*, 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 .

Another standard proof of the Poincaré lemma uses the homotopy invariance formula and can be found in Singer & Thorpe (1976 , pp. 128-132), Lee (2012), Tu (2011) and Bott & Tu (1982).^{ [3] }^{ [4] }^{ [5] } The local form of the homotopy operator is described in Edelen (2005) and the connection of the lemma with the Maurer-Cartan form is explained in Sharpe (1997).^{ [6] }^{ [7] }

This formulation can be phrased in terms of homotopies between open domains *U* in *R*^{m} and *V* in *R*^{n}.^{ [8] } If *F*(*t*,*x*) is a homotopy from [0,1] × *U* to *V*, set *F*_{t}(*x*) = *F*(*t*,*x*). For a *p*-form on *V*, define

Then

**Example**: In two dimensions the Poincaré lemma can be proved directly for closed 1-forms and 2-forms as follows.^{ [9] }

If *ω* = *p**dx* + *q**dy* is a closed 1-form on (*a*, *b*) × (*c*, *d*), then *p*_{y} = *q*_{x}. If *ω* = *df* then *p* = *f*_{x} and *q* = *f*_{y}. Set

so that *g*_{x} = *p*. Then *h* = *f* − *g* must satisfy *h*_{x} = 0 and *h*_{y} = *q* − *g*_{y}. 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**dx* ∧ *dy* then Ω = *d*(*a**dx* + *b**dy*) with *b*_{x} − *a*_{y} = *r*. Thus a solution is given by *a* = 0 and

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.^{ [10] }

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., that 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 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.)

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

- ↑ Warner 1983 , pp. 155-156
- ↑ Warner 1983 , pp. 69-72
- ↑ Lee, John M. (2012).
*Introduction to smooth manifolds*(2nd ed.). New York: Springer. ISBN 978-1-4419-9982-5. OCLC 808682771. - ↑ Tu, Loring W. (2011).
*An introduction to manifolds*(2nd ed.). New York: Springer. ISBN 978-1-4419-7400-6. OCLC 682907530. - ↑ Bott, Raoul; Tu, Loring W. (1982).
*Differential Forms in Algebraic Topology*. Graduate Texts in Mathematics.**82**. New York, NY: Springer New York. doi:10.1007/978-1-4757-3951-0. ISBN 978-1-4419-2815-3. - ↑ Edelen, Dominic G. B. (2005).
*Applied exterior calculus*(Rev ed.). Mineola, N.Y.: Dover Publications. ISBN 0-486-43871-6. OCLC 56347718. - ↑ Sharpe, R. W. (1997).
*Differential geometry : Cartan's generalization of Klein's Erlangen program*. New York: Springer. ISBN 0-387-94732-9. OCLC 34356972. - ↑ Warner 1983 , pp. 157, 160
- ↑ Napier & Ramachandran 2011 , pp. 443-444
- ↑ Warner 1983 , p. 162-207

In physics and mechanics, **torque** is the rotational equivalent of linear force. It is also referred to as the **moment**, **moment of force**, **rotational force** or **turning effect**, depending on the field of study. The concept originated with the studies by Archimedes of the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter *tau*. When being referred to as moment of force, it is commonly denoted by M.

**Kinematics** is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In geometry, a **solid angle** is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the *apex* of the solid angle, and the object is said to *subtend* its solid angle from that point.

In the mathematical field of differential geometry, one definition of a **metric tensor** is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar *g*(*v*, *w*) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

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In mathematics and classical mechanics, the **Poisson bracket** is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called *canonical transformations*, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.

In mathematics, the **symmetry of second derivatives** refers to the possibility under certain conditions of interchanging the order of taking partial derivatives of a function

In physics, **circular motion** is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

In physics, the **Hamilton–Jacobi equation**, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

In mathematics, the **interior product** is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product *ι*_{X}*ω* is sometimes written as *X* ⨼ *ω*.

In calculus, **Leibniz's rule** for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

In differential geometry, the notion of **torsion** is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves. In the geometry of surfaces, the *geodesic torsion* describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting".

**Zero sound** is the name given by Lev Landau to the unique quantum vibrations in quantum Fermi liquids.

In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to

The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of *nested round spheres*. In such a spacetime, a particularly important kind of coordinate chart is the **Schwarzschild chart**, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is *adapted* to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented.

In physics, and especially scattering theory, the **momentum-transfer cross section** is an effective scattering cross section useful for describing the average momentum transferred from a particle when it collides with a target. Essentially, it contains all the information about a scattering process necessary for calculating average momentum transfers but ignores other details about the scattering angle.

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The **loop representation** is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states. The idea is well known in the context of lattice Yang–Mills theory. Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.

This article summarizes several identities in exterior calculus.

In mathematics and theoretical physics, and especially gauge theory, the **deformed Hermitian Yang–Mills (dHYM) equation** is a differential equation describing the equations of motion for a D-brane in the B-model of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger in the case of Abelian gauge group, and by Leung-Yau-Zaslow using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory.

- Flanders, Harley (1989) [1963].
*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 - Singer, I. M.; Thorpe, J. A. (1976),
*Lecture Notes on Elementary Topology and Geometry*, University of Bangalore Press, ISBN 0721114784

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