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Unformatted text preview: 3.1. EXTERIOR ALGEBRA 63 Notation. To deal with forms it is convenient to introduce multi-indices. We will denote a p-tuple of integers from 1 to n by a capital letter I = ( i 1 , . . . , i p ) , where 1 i 1 , . . . , i p n . For a p-tuple of the same integers ordered in an increasing order we define I = ( i 1 , . . . , i p ) . where 1 i 1 < i 2 < < i p n . We call I an increasing p-tuple associated with I . Therefore, a collection of p-forms p !Alt i 1 i p , where 1 i 1 < i 2 < < i p n , forms a basis in the space p . Thus, every p-form p has the form = X 1 i 1 < < i p n i 1 i p p !Alt i 1 i p . Therefore, the dimension of the space p is equal to the number of distinct increasing p-tuples of integers from 1 to n . Theorem 3.1.5 The dimension of the space p of p-forms is dim p = n p !...
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