External derivative

This article deals with the outer derivative of differential forms. For the "outer derivation" as a designation for the derivation of the outer function of a chain, see the chain rule

The outer derivative or Cartan derivative is a term used in differential geometry and analysis . It generalizes the derivation of functions to differential forms known from analysis . The name Cartan derivation is explained by the fact that Élie Cartan (1869–1952) is the founder of the theory of differential forms.

External derivative

definition

Let be a -dimensional smooth manifold and an open subset. With the space is here -forms on the manifold referred. So there is exactly one function for all of them , so that the following properties apply: ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {A}}^{k}(M)}$ ${\displaystyle k}$ ${\displaystyle M}$ ${\displaystyle k\in \mathbb {N} \cup \{0\}}$ ${\displaystyle \mathrm {d} \colon {\mathcal {A}}^{k}(U)\to {\mathcal {A}}^{k+1}(U)}$

1. ${\displaystyle \mathrm {d} }$is an antiderivation , which means for and applies . ${\displaystyle \alpha \in {\mathcal {A}}^{k}(U)}$ ${\displaystyle \beta \in {\mathcal {A}}^{l}(U)}$ ${\displaystyle \mathrm {d} (\alpha \wedge \beta )=\mathrm {d} \alpha \wedge \beta +(-1)^{k}\alpha \wedge \mathrm {d} \beta }$
2. Let , then is defined as the total differential . ${\displaystyle f\in C^{\infty }(U)}$ ${\displaystyle \,\mathrm {d} f}$
3. ${\displaystyle \mathrm {d} \circ \mathrm {d} =0}$
4. The operator behaves of course with regard to restrictions, that is: if are open sets and , then we have . ${\displaystyle U\subset V\subset M}$ ${\displaystyle \alpha \in {\mathcal {A}}^{k}(V)}$ ${\displaystyle \,\mathrm {d} (\alpha |U)=(\mathrm {d} \alpha )|U}$

Of course, it has to be proven that such an operator exists and is unique. This is called outer derivation or Cartan derivation and is usually referred to as. The index, which indicates the degree of the differential form to which the operator is applied, is omitted. ${\displaystyle \,\mathrm {d} }$

Formula for the outer derivative

One can also find the outer derivative using the formula

${\displaystyle {\begin{array}{rcl}\mathrm {d} \omega (X_{0},\ldots ,X_{k})&=&\sum _{i=0}^{k}(-1)^{i}X_{i}(\omega (X_{0},\ldots ,{\hat {X}}_{i},\ldots ,X_{k}))\\[0.5em]&+&\sum _{0\leq i<j\leq k}(-1)^{i+j}\omega ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k})\end{array}}}$

represent, the circumflex in means that the corresponding argument must be left out, denotes the Lie bracket . ${\displaystyle {\hat {}}}$ ${\displaystyle {\hat {X}}_{i}}$ ${\displaystyle [.,.]}$

Coordinate representation

Let be a point on a smooth manifold. The outer derivative of has the representation at this point ${\displaystyle p\in M}$ ${\displaystyle \omega \in {\mathcal {A}}^{k}(M)}$

${\displaystyle \mathrm {d} \omega |_{p}=\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}\sum _{i=1}^{n}\left.{\frac {\partial a_{i_{1},\ldots ,i_{k}}}{\partial x_{i}}}\right|_{p}\mathrm {d} x_{i}\wedge \mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}}$,

has the local representation ${\displaystyle \omega }$

${\displaystyle \omega =\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}\mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}.}$

Representation via antisymmetrization mapping

The outer derivative of -forms is simply given by the total derivative and is always covariant ( see also covariant derivative ) and antisymmetric. The external derivative of a -form can be viewed up to a multiple as an antisymmetrization of the formal tensor product of with the form: ${\displaystyle \mathrm {d} ^{0}}$ ${\displaystyle 0}$ ${\displaystyle k}$ ${\displaystyle \omega }$ ${\displaystyle \mathrm {d} ^{0}}$

${\displaystyle \mathrm {d} ^{k}\omega =(k+1)\operatorname {Alt} (\mathrm {d} ^{0}\otimes \omega )}$

In index notation :

${\displaystyle (\mathrm {d} ^{k}\omega )_{i_{1},\ldots ,i_{k+1}}=(k+1)\partial _{[i_{1}}\omega _{i_{2},\ldots ,i_{k+1}]}}$

Return transport

Let be two smooth manifolds and a once continuously differentiable function. Then the return transport is a homomorphism, so that ${\displaystyle M,\ N}$ ${\displaystyle f\colon M\to N}$ ${\displaystyle f^{*}\colon {\mathcal {A}}(N)\to {\mathcal {A}}(M)}$

1. ${\displaystyle f^{*}(\psi \wedge \omega )=f^{*}\psi \wedge f^{*}\omega }$ and
2. ${\displaystyle \,f^{*}(\mathrm {d} \omega )=\mathrm {d} (f^{*}\omega )}$

applies.

In words one also says: product formation or external differentiation are compatible with the "pullback" relation.

Adjoint outer derivative

In this section let be a pseudo-Riemannian manifold with index . The Hodge star operator is referred to below with . The operator ${\displaystyle (M,g)}$ ${\displaystyle i}$ ${\displaystyle \star }$

${\displaystyle \delta \colon {\mathcal {A}}^{k+1}(M)\to {\mathcal {A}}^{k}(M)}$

is defined by and for by ${\displaystyle \delta ({\mathcal {A}}^{0}(M))=0}$ ${\displaystyle \beta \in {\mathcal {A}}^{k+1}(M)}$

${\displaystyle \delta (\beta )=(-1)^{nk+1+i}\star \mathrm {d} \star \beta .}$

It is called the adjoint external derivative or co- derivative .

This operator is linear and it holds . Indeed, the operator to be adjoint is . If the manifold is also compact , then applies to the Riemannian metric and the relation ${\displaystyle \delta \circ \delta =0}$ ${\displaystyle \delta }$ ${\displaystyle \mathrm {d} }$ ${\displaystyle g}$ ${\displaystyle \omega ,\nu \in {\mathcal {A}}(M)}$

${\displaystyle g(\mathrm {d} \omega ,\nu )=g(\omega ,\delta \nu )}$.

For this reason one also notes as , since this is the adjoint operator . Similar duality relationships can also be defined for pseudo-Riemannian metrics , for example for the Minkowski metric of the special theory of relativity or the Lorentz metric of the general theory of relativity . ${\displaystyle \delta }$ ${\displaystyle \textstyle \mathrm {d} ^{*}}$

Generalization of further differential operators

The differential operators known from vector analysis can be extended to Riemannian manifolds with the help of the outer derivative and the Hodge-Star operator . In particular, a formula is obtained for the rotation which operates on n-dimensional spaces. In the following there is always a smooth Riemann manifold . ${\displaystyle \mathrm {d} }$ ${\displaystyle \star }$ ${\displaystyle M}$

Be and cross (flat and sharp) isomorphism

These two isomorphisms are induced by the Riemannian metric. They map tangential vectors to cotangent vectors and vice versa. To understand it, it is sufficient at this point to demonstrate the effect of the isomorphisms in three-dimensional space. Let the flat operator be a vector field in standard coordinates of ${\displaystyle F\in T_{p}\mathbb {R} ^{3}\cong \mathbb {R} ^{3}}$ ${\displaystyle F}$

${\displaystyle F^{\flat }=F^{1}\mathrm {d} x_{1}+F^{2}\mathrm {d} x_{2}+F^{3}\mathrm {d} x_{3}\in T_{p}^{*}\mathbb {R} ^{3}\cong {\mathcal {A}}^{1}(\mathbb {R} ^{3})}$.

The flat operator maps vector fields into their dual space. The Sharp operator is the inverse operation. If a co-vector field (or a 1-form) then applies (also standard coordinates) ${\displaystyle \nu \in T_{p}^{*}\mathbb {R} ^{3}\cong {\mathcal {A}}^{1}(\mathbb {R} ^{3})}$

${\displaystyle \nu ^{\sharp }=\nu _{1}{\frac {\partial }{\partial x_{1}}}+\nu _{2}{\frac {\partial }{\partial x_{2}}}+\nu _{3}{\frac {\partial }{\partial x_{3}}}\in T_{p}\mathbb {R} ^{3}}$.

Cross product

The cross product is not a differential operator and is also only defined for three-dimensional vector spaces in vector analysis. Nevertheless, especially for the definition of the rotation, it is very important: Let be a vector space and two elements of an outer power of , then the generalized cross product is defined by ${\displaystyle V}$ ${\displaystyle v,w\in \Lambda ^{k}V}$ ${\displaystyle V}$

${\displaystyle v\times w=\left(\star (v^{\flat }\wedge w^{\flat })\right)^{\sharp }}$.

For a reason for this definition see under external algebra .

gradient

Let it be a partially differentiable function and let the standard scalar product be given. The gradient of the function in the point is for any of that by the requirement ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle f}$ ${\displaystyle a\in \mathbb {R} ^{n}}$ ${\displaystyle h\in \mathbb {R} ^{n}}$

${\displaystyle \mathrm {d} f(a)h=\langle \nabla f(a),h\,\rangle }$

uniquely definite vector . With the help of differential forms calculus can the gradient on a Riemann manifold by ${\displaystyle \nabla f(a)}$ ${\displaystyle M}$

${\displaystyle \nabla f:=(\mathrm {d} f)^{\sharp }}$

define. Since the set of 0-forms is by definition equal to the set of functions that can be differentiated as often as desired, this definition generalizes the gradient of functions. This can be seen quickly with a short invoice. If it is a smooth function, then it holds ${\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} }$

${\displaystyle (\mathrm {d} f)^{\sharp }={\begin{pmatrix}{\frac {\partial f}{\partial x_{1}}}\mathrm {d} x^{1}+{\frac {\partial f}{\partial x_{2}}}\mathrm {d} x^{2}+{\frac {\partial f}{\partial x_{3}}}\mathrm {d} x^{3}\end{pmatrix}}^{\sharp }={\frac {\partial f}{\partial x_{1}}}{\frac {\partial }{\partial x_{1}}}+{\frac {\partial f}{\partial x_{2}}}{\frac {\partial }{\partial x_{2}}}+{\frac {\partial f}{\partial x_{3}}}{\frac {\partial }{\partial x_{3}}}.}$

In Euclidean vector spaces this is often noted as follows:

${\displaystyle (\mathrm {d} f)^{\sharp }={\begin{pmatrix}{\frac {\partial f}{\partial x_{1}}},{\frac {\partial f}{\partial x_{2}}},{\frac {\partial f}{\partial x_{3}}}\end{pmatrix}}^{\sharp }={\begin{pmatrix}{\frac {\partial f}{\partial x_{1}}}\\{\frac {\partial f}{\partial x_{2}}}\\{\frac {\partial f}{\partial x_{3}}}\end{pmatrix}}.}$

rotation

In vector analysis, the rotation is a mapping . The following applies to general vector fields ${\displaystyle \mathrm {rot} \colon T_{p}\mathbb {R} ^{3}\to T_{p}\mathbb {R} ^{3}}$

${\displaystyle \mathrm {rot} (f)=\nabla \times f=\left(\star \left(\mathrm {d} f^{\flat }\right)\right)^{\sharp }}$.

The following calculation shows that the known expression for the rotation is obtained for the dimension : ${\displaystyle n=3}$

${\displaystyle {\begin{array}{cl}&\mathrm {d} (f_{1}\cdot \mathrm {d} x_{1}+f_{2}\cdot \mathrm {d} x_{2}+f_{3}\cdot \mathrm {d} x_{3})\\=&\mathrm {d} f_{1}\wedge \mathrm {d} x_{1}+\mathrm {d} f_{2}\wedge \mathrm {d} x_{2}+\mathrm {d} f_{3}\wedge \mathrm {d} x_{3}\\[0.5em]=&{\frac {\partial f_{1}}{\partial x_{1}}}\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{1}+{\frac {\partial f_{1}}{\partial x_{2}}}\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{1}+{\frac {\partial f_{1}}{\partial x_{3}}}\cdot \mathrm {d} x_{3}\wedge \mathrm {d} x_{1}\\+&{\frac {\partial f_{2}}{\partial x_{1}}}\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{2}+{\frac {\partial f_{2}}{\partial x_{2}}}\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{2}+{\frac {\partial f_{2}}{\partial x_{3}}}\cdot \mathrm {d} x_{3}\wedge \mathrm {d} x_{2}\\+&{\frac {\partial f_{3}}{\partial x_{1}}}\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{3}+{\frac {\partial f_{3}}{\partial x_{2}}}\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{3}+{\frac {\partial f_{3}}{\partial x_{3}}}\cdot \mathrm {d} x_{3}\wedge \mathrm {d} x_{3}\\[0.5em]=&\left({\frac {\partial f_{3}}{\partial x_{2}}}-{\frac {\partial f_{2}}{\partial x_{3}}}\right)\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{3}+\left({\frac {\partial f_{1}}{\partial x_{3}}}-{\frac {\partial f_{3}}{\partial x_{1}}}\right)\cdot \mathrm {d} x_{3}\wedge \mathrm {d} x_{1}+\left({\frac {\partial f_{2}}{\partial x_{1}}}-{\frac {\partial f_{1}}{\partial x_{2}}}\right)\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\end{array}}}$

This formula is obtained immediately by inserting the definition of the gradient into that of the cross product.

divergence

There is also a generalization of divergence , which is

${\displaystyle \mathrm {div} (f)=\nabla \cdot f=\star \mathrm {d} (\star f\,^{\flat }).}$

Hodge-Laplace operator

The Hodge-Laplace operator is a special generalized Laplace operator . Such operators have an important meaning in differential geometry.

definition

Let be a smooth Riemannian manifold, then the Hodge-Laplace operator is defined by ${\displaystyle M}$

${\displaystyle \Delta =\mathrm {d} \delta +\delta \mathrm {d} \,.}$

A function is called harmonic if it satisfies Laplace's equation . The harmonic differential forms are defined analogously . A differential form is called harmonic if the Hodge-Laplace equation is fulfilled. The number of all harmonic forms is noted with. Because of the Hodge decomposition, this space is isomorphic to the corresponding De Rham cohomology group . ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle \Delta f=0}$ ${\displaystyle \omega \in {\mathcal {A}}(M)}$ ${\displaystyle \Delta \omega =0}$ ${\displaystyle {\mathcal {H}}^{k}(M)}$ ${\displaystyle M}$

properties

The Hodge-Laplace operator has the following properties:

1. ${\displaystyle \,\star \Delta =\Delta \star }$, so if is harmonious, then is also harmonious. ${\displaystyle \omega }$ ${\displaystyle \star \omega }$
2. The operator is self-adjoint with respect to a Riemannian metric g, that is, applies to all ; . ${\displaystyle \,\Delta }$ ${\displaystyle \omega ,\nu \in {\mathcal {A}}(M)}$ ${\displaystyle \,g(\Delta \omega ,\nu )=g(\omega ,\Delta \nu )}$
3. It is necessary and sufficient for the equation that and hold. ${\displaystyle \,\Delta \omega =0}$ ${\displaystyle \,\mathrm {d} \omega =0}$ ${\displaystyle \,\delta \omega =0}$

Dolbeault operator

→ Main article : Complex differential form

Two other differential operators related to the Cartan derivative are the Dolbeault and the Dolbeault transverse operator on manifolds. In this way one can introduce the spaces of the differential forms of the degree , which are noted by, and naturally get the mappings ${\displaystyle (p,q)}$ ${\displaystyle {\mathcal {A}}^{p,q}}$

${\displaystyle \partial \colon {\mathcal {A}}^{p,q}\to {\mathcal {A}}^{p+1,q}}$

and

${\displaystyle {\overline {\partial }}\colon {\mathcal {A}}^{p,q}\to {\mathcal {A}}^{p,q+1}}$

with . These differential operators have the representations in local coordinates ${\displaystyle \mathrm {d} =\partial +{\overline {\partial }}}$

${\displaystyle \partial \left(\sum _{I,J}f_{I,J}\mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}\right)=\sum _{I,J,K}{\frac {\partial f_{I,J}}{\partial z_{K}}}\mathrm {d} z_{K}\wedge \mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}}$

and

${\displaystyle {\overline {\partial }}\left(\sum _{I,J}f_{I,J}\mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}\right)=\sum _{I,J,K}{\frac {\partial f_{I,J}}{\partial {\overline {z}}_{K}}}\mathrm {d} {\overline {z}}_{K}\wedge \mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}.}$

literature

• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications . Springer , Berlin 2003, ISBN 3-540-96790-7 .
• S. Morita: Geometry of Differential Forms . AMS , ISBN 0-8218-1045-6 .

Footnotes

1. ↑ Ivan Avramidi, Notes on Differential Forms (PDF; 112 kB) , 2003
2. ↑ This is related to a terminology used in physics according to which one differentiates between polar and axial vectors; for example, the cross product of two polar vectors gives an axial vector. The quantities of theoretical mechanics (“ angular momentum ” or “ torques ”) designated as or are e.g. B. axial vectors. ${\displaystyle \mathbf {L} }$ ${\displaystyle \mathbf {D} }$