Nonlinear Equations - UFRJ

Chapter 4
Differential forms
Through this section, vectors are represented boldface
such as x and coordinates are represented as x j . Whenever we are
speaking about a collection of vectors x 1 , . . . , x n , x ij is the j-th coordinate
of the i-th vector.
4.1 Multilinear algebra over R
Let A k be the space of alternating k-forms in R n , that is the space
of all k-linear forms α : (R n ) k → R such that, for all permutation
σ ∈ S k (the permutation group of k elements), we have:
α(u σ1 , . . . , u σk ) = (−1) |σ| α(u 1 , . . . , u k ).
Above, |σ| is minimal so that σ is the composition of |σ| elementary
permutations (permutations fixing all elements but two).
The canonical basis of A k is given by the forms dx i1 ∧ · · · ∧ dx ik ,
Gregorio Malajovich, Nonlinear equations.
28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011.
Copyright c○ Gregorio Malajovich, 2011.
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