Differential forms

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Wedge product

The wedge product is the product in an exterior algebra. If and are differential k-forms of degrees and , respectively, then(1)It is not (in general) commutative, but it is associative,(2)and bilinear(3)(4)(Spivak 1999, p. 203), where and are constants. The exterior algebra is generated by elements of degree one, and so the wedge product can be defined using a basis for :(5)when the indices are distinct, and the product is zero otherwise.While the formula holds when has degree one, it does not hold in general. For example, consider :(6)(7)(8)If have degree one, then they are linearly independent iff .The wedge product is the "correct" type of product to use in computinga volume element(9)The wedge product can therefore be used to calculate determinants and volumes of parallelepipeds. For example, write where are the columns of . Then(10)and is the volume of the parallelepiped spanned by ...

Exterior power

The th exterior power of an element in an exterior algebra is given by the wedge product of with itself times. Note that if has odd degree, then any higher power of must be zero. The situation for even degree forms is different. For example, if(1)then(2)(3)(4)

Exterior derivative

The exterior derivative of a function is the one-form(1)written in a coordinate chart . Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k-forms using the formula(2)when is a -form and where is the wedge product.The exterior derivative of a -form is a -form. For example, for a differential k-form(3)the exterior derivative is(4)Similarly, consider(5)Then(6)(7)Denote the exterior derivative by(8)Then for a 0-form ,(9)for a 1-form ,(10)and for a 2-form ,(11)where is the permutation tensor.It is always the case that . When , then is called a closed form. A top-dimensional form is always a closed form. When then is called an exact form, so any exact form is also closed. An example of a closed form which is not exact is on the circle. Since is a function defined up to a constant multiple of , is a well-defined one-form, but there is no function for which it is the exterior derivative.The exterior derivative..

Exact form

A 1-form is said to be exact in a region if there is a function that is defined and of class (i.e., is once continuously differentiable in ) and such that .

Multilinear form

A multilinear form on a vector space over a field is a map(1)such that(2)and(3)for every and any indexes .For example, the determinant of a square matrix of degree is an -linear form for the columns or rows of a matrix.

Exact differential

A differential of the form(1)is exact (also called a total differential) if is path-independent. This will be true if(2)so and must be of the form(3)But(4)(5)so(6)There is a special notation encountered especially often in statistical thermodynamics. Consider an exact differential(7)Then the notation , sometimes referred to as constrained variable notation, means "the partial derivative of with respect to with held constant." Extending this notation a bit leads to the identity(8)where it is understood that on the left-hand side is treated as a variable that can itself be held constant.


Given a differential operator on the space of differential forms, an eigenform is a form such that(1)for some constant . For example, on the torus, the Dirac operator acts on the form(2)giving(3)i.e., .

Lagrange bracket

Let be any functions of two variables . Then the expression(1)is called a Lagrange bracket (Lagrange 1808; Whittaker 1944, p. 298).The Lagrange brackets are anticommutative,(2)(Plummer 1960, p. 136).If are any functions of variables , then(3)where the summation on the right-hand side is taken over all pairs of variables in the set .But if the transformation from to is a contact transformation, then(4)giving(5)(6)(7)(8)Furthermore, these may be regarded as partial differential equations which must be satisfied by , considered as function of in order that the transformation from one set of variables to the other may be a contact transformation.Let be independent functions of the variables . Then the Poisson bracket is connected with the Lagrange bracket by(9)where is the Kronecker delta. But this is precisely the condition that the determinants formed from them are reciprocal (Whittaker 1944, p. 300; Plummer 1960,..

Dirac notation

A notation invented by Dirac which is very useful in quantum mechanics. The notation defines the "ket" vector, denoted , and its conjugate transpose, called the "bra" vector and denoted . The "bracket" is then defined by .Dirac notation satisfies the identitieswhere is the complex conjugate.

Sylvester's signature

Diagonalize a form over the rationals towhere all the entries are integers and , , ... are relatively prime to . Then Sylvester's signature is the sum of the -parts of the entries.

Stokes' theorem

For a differential (k-1)-form with compact support on an oriented -dimensional manifold with boundary ,(1)where is the exterior derivative of the differential form . When is a compact manifold without boundary, then the formula holds with the right hand side zero.Stokes' theorem connects to the "standard" gradient, curl, and divergence theorems by the following relations. If is a function on ,(2)where (the dual space) is the duality isomorphism between a vector space and its dual, given by the Euclidean inner product on . If is a vector field on a ,(3)where is the Hodge star operator. If is a vector field on ,(4)With these three identities in mind, the above Stokes' theorem in the three instances is transformed into the gradient, curl, and divergence theorems respectively as follows. If is a function on and is a curve in , then(5)which is the gradient theorem. If is a vector field and an embedded compact 3-manifold with boundary in , then(6)which..

Inexact differential

An infinitesimal which is not the differential of an actual function and which cannot be expressed asthe way an exact differential can. Inexact differentials are denoted with a bar through the . The most common example of an inexact differential is the change in heat encountered in thermodynamics.


A bra is a vector living in a dual vector space to that containing kets . Bras and kets are commonly encountered in quantum mechanics. Bras and kets can be considered as 1-vectors and 1-forms (or vice versa), although this is almost always done only in a finite-dimensional vector space.Considered as an inner product, the bra and ket form an angle bracket (bra+ket = bracket) .

Rectifiable current

The space of currents arising from rectifiable sets by integrating a differential form is called the space of two-dimensional rectifiable currents. For a closed bounded rectifiable curve of a number of components in , bounds a rectifiable current of least area. The theory of rectifiable currents generalizes to -D surfaces in .

Hodge star

On an oriented -dimensional Riemannian manifold, the Hodge star is a linear function which converts alternating differential k-forms to alternating -forms. If is an alternating k-form, its Hodge star is given bywhen , ..., is an oriented orthonormal basis.

Bilinear form

A bilinear form on a real vector space is a functionthat satisfies the following axioms for any scalar and any choice of vectors and . 1. 2. 3. . For example, the function is a bilinear form on .On a complex vector space, a bilinear form takes values in the complex numbers. In fact, a bilinear form can take values in any vector space, since the axioms make sense as long as vector addition and scalar multiplication are defined.

Angle bracket

An angle bracket is the combination of a bra and ket (bra+ket = bracket) which represents the inner product of two functions or vectors (or 1-forms),in a function space, orin a vector space. Here, represents the adjoint.The expression is also commonly used to denote the expectation value of a variable .

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