曲面的微分几何.pdf

1

1.1 x. graph {( ) } x x x x v

( ) v , x . x ( ) x , v , v

d x , d e d v (1.1) . d x

d r

1.2 / / d d d d d (1.2) d d 1

∧ d , ∧ d ( ) det d (1.3)

1 ii / ∧ ( ) ( ) , . , x

1.3 d d (1.4) 1 d , ( ) d d d d d d d d d (1.5) , ( ) d d d d d (1.6) ( ) d d d d d d d (1.7) d d d d d

1 iii dd . dd d ( d d ) d ( d d ) (1.8) d d d

d d d d , , d d d d

dd . ( ) d d ̸ (1.9) d 0, d

1.4 Stokes Ω Ω d (1.10) d , d d d (1.11) Ω Ω Stokes Stokes 0 d d

2 iv

2 , ̸ , rank ∇ ̸ , ̸ immersion ( ) ( ) ( ) ( ) , .

2.1 ( )( ) ( ) (2.1) I d d d d d d d 1. I , I , 2. I , d d d , I. d d, cos d d ⟨d d⟩ I (d d) (2.2) d d d d , d d ∥ ∥d d √det d d (2.3) .

3 v

2.2 , , Schmidt , , I ∥ ∥ d ∥ ∥ d

3 , , , , d pushforward, d d , d ( ) ( ) (3.1) d

3 vi Jacobian ( ) ∇ ( ), ∇ , ( ) d (3.2)

( ) . d d d d

d d . ,