By Sjamaar R.
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Additional info for Manifolds and differential forms
N , has n-dimensional volume 1. ) Its image A 0, 1 n under the map A is a parallelepiped with edges Ae1 , Ae2 , . . , Aen , the columns of A. Hence A 0, 1 n ✷ ▲❉ ❊ ▼ 36 3. PULLING BACK FORMS ◆✌❖ ❙◗ ❘❯❚ ❚❱❘❁❚ vol A ❲ X ❳ ❘❨❚ det A ❚ vol X. ❚ ❖ has n-dimensional volume vol A 0, 1 n det A det A vol 0, 1 n . This rule n generalizes as follows: if X is a measurable subset of R , then e2 X Ae2 A AX e1 ❚ Ae1 ❚ So det A can be interpreted as a volume change factor. (A set is measurable if it has a well-defined, finite or infinite, n-dimensional volume.
1) dci ✘ t ✛ dt. 1. E XAMPLE . Let U be the punctured plane R2 ✙☞✩ 0 ✪ . Let c : ✔ 0, 2π ✖✗✒ U be the usual parametrization of the circle, c ✘ t ✛✫✓✬✘ cos t, sin t ✛ , and let α be the angle form, ✙ y dx ✚ x dy α✓ . 8), so 2π ✭ c α ✓✮✭ 0 dt ✓ 2π . A curve c : ✔ a, b ✖✯✒ U can be reparametrized by substituting a new variable, t ✓ p ✘ s ✛ , where s ranges over another interval ✔ a¯ , b¯ ✖ . We shall assume p to be a ¯ Such one-to-one mapping from ✔ a¯ , b¯ ✖ onto ✔ a, b✖ satisfying p ✰✱✘ s ✛✜✓ ✲ 0 for a¯ ✳ s ✳ b.
Ai , . . , a j , . . , an ö ÷ùø det ó a1 , . . , a j , . . , ai , . . , an ö for any i ÷ ú j; (iii) normalization: det ó e1 , e2 , . . , en ö dard basis vectors of Rn . ÷ 1, where e1 , e2 , . . , en are the stan- We also write det A instead of det ó a1 , a2 , . . , an ö , where A is the matrix whose columns are a1 , a2 , . . , an . Axiom (iii) lays down the value of det I. Axioms (i) and (ii) govern the behaviour of oriented volumes under the elementary column operations on matrices.
Manifolds and differential forms by Sjamaar R.