Math 2.0

Linear and multilinear algebra; matrix theory Definition

Wedge product of forms

Builds on Differential form

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open. For indices j1,…,jm∈{1,…,n}j_1, \dots, j_m \in \{1, \dots, n\}, let dxj1∧⋯∧dxjmdx_{j_1} \wedge \dots \wedge dx_{j_m} be 00 if two indices are equal. Otherwise it is sgn⁡(σ) dxI\operatorname{sgn}(\sigma) \, dx_I, where II is the increasing rearrangement of the indices and σ\sigma is the permutation that sorts them. For differential forms α=∑IfI dxI\alpha = \sum_I f_I \, dx_I of degree kk and β=∑JgJ dxJ\beta = \sum_J g_J \, dx_J of degree ll on UU, the wedge product is the (k+l)(k + l)-form α∧β=∑I,JfI gJ  dxi1∧⋯∧dxik∧dxj1∧⋯∧dxjl.\alpha \wedge \beta = \sum_{I, J} f_I \, g_J \; dx_{i_1} \wedge \dots \wedge dx_{i_k} \wedge dx_{j_1} \wedge \dots \wedge dx_{j_l}.

Remarks

The product is bilinear and associative. It is graded commutative: β∧α=(−1)kl α∧β\beta \wedge \alpha = (-1)^{k l} \, \alpha \wedge \beta. In particular, dxi∧dxj=−dxj∧dxidx_i \wedge dx_j = -dx_j \wedge dx_i and dxi∧dxi=0dx_i \wedge dx_i = 0. For a 00-form ff, the product f∧βf \wedge \beta is the form fβf \beta. In the plane, (a dx+b dy)∧(c dx+d dy)=(ad−bc) dx∧dy(a \, dx + b \, dy) \wedge (c \, dx + d \, dy) = (a d - b c) \, dx \wedge dy. The coefficient is the determinant, which is the signed area of the parallelogram spanned by (a,b)(a, b) and (c,d)(c, d). For 1-forms α\alpha and β\beta and vectors v,wv, w, (α∧β)(v,w)=α(v)β(w)−α(w)β(v)(\alpha \wedge \beta)(v, w) = \alpha(v) \beta(w) - \alpha(w) \beta(v).

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementDifferential form, definition, no evidence levelDifferential formdefinitionWedge product of forms, definition, no evidence levelWedge product of formsdefinition
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Limit of a function of several variables concept
  6. Differentiable function of several variables definition
  7. Partial derivative definition
  8. Differentiable on an open set definition
  9. Higher partial derivatives definition
  10. Iterated derivative definition
  11. C^k function characterised by continuous derivatives statement
  12. Differential form definition
  13. Wedge product of forms definition

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11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Differential_form, Exterior product
  2. Web https://en.wikipedia.org/wiki/Exterior_algebra, Areas in the plane
  3. Web https://en.wikipedia.org/wiki/Exterior_algebra, Alternating product
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