By Sjamaar R.

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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.

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