External derivative

In Mathematical, the external derivative , operator of the differential Topology, extends the concept of the Différentielle of a function to the differential forms moreover high degree. It is important in the theory of integration of the variety S, and it is the Différentielle employed to define the cohomology of De Rham and Alexander-Spanier. Its current form was invented by Élie Cartan.

Definition

The derivative external of a differential form of degree K is a differential form of degree K + 1 .

For ω = f_I dx_I of form K on R ^N , the definition is the following one:

$d\; \{\backslash \; Omega\}\; =\; \backslash \; sum\_\; \{i=1\}\; ^n\; \backslash \; frac\; \{\backslash \; partial\; f\_I\}\; \{\backslash \; partial\; x\_i\}\; dx\_i\; \backslash \; wedge\; dx\_I.$

For the forms K general: Σ _I F _I dx _I (where the Multi-index I exceeds all the ordered subsets of {1,…, N } of cardinality K ), we do nothing but linearly extend . Note that if $i\; =\; I$ above then $dx\_i\; \backslash \; wedge\; dx\_I\; =\; 0$ (see Produit external).

Properties

External differentiation satisfied three important properties:

• the Linearity

• the rule of the Produces external (see Antidérivation)

• and D ² = 0, a formula codifying the equality of the mixed derivative partial, such as

$d\; (D\; \backslash \; Omega)\; =0\; \backslash ,\; \backslash !$

in any time.

It can be shown that this external derivative is only determined by these properties and its agreement with the differential on forms 0 (functions).

The core of D contains the closed forms , and the image of the exact forms (cf exact Différentielle ).

Invariant formula

Being given ω of form K and vector fields arbitrary smooth V₀, V₁,…, V_k we have

$d\; \backslash \; Omega\; (V\_0,\; V\_1,\dots \; V\_k)\; =\; \backslash \; sum\_i\; (-\; 1)\; ^i\; V\_i\; \backslash \; Omega\; (V\_0,\dots ,\; \backslash \; hat\; V\_i,\dots ,\; V\_k)$

where $\backslash ,\; \backslash !$ indicates the Crochet of Dregs and $\backslash \; Omega\; (V\_0,\dots ,\; \backslash \; hat\; V\_i,\dots ,\; V\_k)\; =\; \backslash \; Omega\; (V\_0,\dots ,\; V\_\; \{i-1\},\; V\_\; \{i+1\}\dots ,\; V\_k).$

In particular, for forms 1 we have:

$d\; \backslash \; Omega\; (X,\; Y)\; =X\; (\backslash \; Omega\; (Y))there\; (\backslash \; Omega\; (X))-\; \backslash \; Omega\; ().$

Bond with the vector calculus

The following correspondence reveals approximately a dozen formulas of the vector Calculus as only of the special cases of the three rules of external differentiation above.

Gradient

For a form 0, which is a function smoothes F : R ^N → R , we have

$df\; =\; \backslash \; sum\_\; \{i=1\}\; ^n\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; x\_i\}\; \backslash ,\; dx\_i.$

Then

$df\; (V)\; =\; \backslash \; langle\; \backslash \; mbox\; \{grad\}\; F,\; V\; \backslash \; rangle,$

where grad F indicates the Gradient F and $\backslash \; langle\; \backslash \; cdot,\; \backslash \; cdot\; \backslash \; rangle$ is the scalar Produit.

Rotational

For $\backslash \; omega=\; \backslash \; sum\_\; \{I\}\; f\_i\; \backslash ,\; dx\_i$ of form 1 on R ³ ,

$d\; \backslash \; omega=\; \backslash \; sum\_\; \{I,\; J\}\; \backslash \; frac\; \{\backslash \; partial\; f\_i\}\; \{\backslash \; partial\; x\_j\}\; dx\_j\; \backslash \; wedge\; dx\_i,$

who restricted to the three Dimension S $\backslash \; omega=\; U\; \backslash ,\; dx+v\; \backslash ,\; dy+w\; \backslash ,\; dz$ is

$d\; \backslash \; Omega\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; v\}\; \{\backslash \; partial\; X\}\; -\; \backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; there\}\; \backslash \; right)\; dx\; \backslash \; wedge\; Dy$

+ \ left (\ frac {\ partial W} {\ partial there} - \ frac {\ partial v} {\ partial Z} \ right) Dy \ wedge dz + \ left (\ frac {\ partial U} {\ partial Z} - \ frac {\ partial W} {\ partial X} \ right) dz \ wedge dx.

Then, for the vector field V= '' we have $d\; \backslash \; Omega\; (U,\; W)\; =\; \backslash \; langle\; \backslash \; mbox\; \{belch\}\; \backslash ,\; V\; \backslash \; times\; U,\; W\; \backslash \; rangle$ where belch V indicates rotational V , × is the vector Product, and <•, •> is the scalar Produit.

Divergence

For $\backslash \; Omega\; =\; \backslash \; sum\_\; \{I,\; J\}\; h\_\; \{I,\; J\}\; \backslash ,\; dx\_i\; \backslash \; wedge\; \backslash ,\; dx\_j,$ of form 2

$d\; \backslash \; Omega\; =\; \backslash \; sum\_\; \{I,\; J,\; K\}\; \backslash \; frac\; \{\backslash \; partial\; h\_\; \{I,\; J\}\}\; \{\backslash \; partial\; x\_k\}\; dx\_k\; \backslash \; wedge\; dx\_i\; \backslash \; wedge\; dx\_j.$

For three dimensions, with $\backslash \; Omega\; =\; p\; \backslash ,\; Dy\; \backslash \; wedge\; dz+q\; \backslash ,\; dz\; \backslash \; wedge\; dx+r\; \backslash ,\; dx\; \backslash \; wedge\; dy$ one obtains

$d\; \backslash \; Omega\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; p\}\; \{\backslash \; partial\; X\}\; +\; \backslash \; frac\; \{\backslash \; partial\; Q\}\; \{\backslash \; partial\; there\}\; +\; \backslash \; frac\; \{\backslash \; partial\; R\}\; \{\backslash \; partial\; Z\}\; \backslash \; right)\; dx\; \backslash \; wedge\; Dy\; \backslash \; wedge\; dz\; =\; \backslash \; mbox\; \{div\}\; V\; dx\; \backslash \; wedge\; Dy\; \backslash \; wedge\; dz,$

where V is a vector Field defined by $V\; =.$

Examples

For $\backslash \; sigma\; =\; U\; \backslash ,\; dx\; +\; v\; \backslash ,\; dy$ of form 1 on R ² we have

$d\; \backslash \; sigma\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; \{v\}\}\; \{\backslash \; partial\; \{X\}\}\; -\; \backslash \; frac\; \{\backslash \; partial\; \{U\}\}\; \{\backslash \; partial\; \{there\}\}\; \backslash \; right)\; dx\; \backslash \; wedge\; dy$

what is exactly the shape 2 being made integrate in the Théorème of Green.

See too

• Derived covariante external
• Theorem of Green
• Theorem from Stokes

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