Lecture 7 8 251 Spring 2007 Lecture 7 Topics Area formula for spacial surfaces Area formula for spatial surfaces spatial as opposed to space time Consider 2D surface in 3D space 3D Space x x1 x2 x3 Parameter Space 1 2 directions along grid lines Purely arbitrary No con nection to distances Describe surface x 1 2 x1 1 2 x2 1 2 x3 1 2 What is area A 1 Lecture 7 8 251 Spring 2007 A in nitesimal rectangles 1 2 Map to surfae d v1 in nitesimal vector corresponding to d 1 on 1 To linear order these 1 2 3 4 points form a parallelogram x 1 2 d 1 1 x 1 2 d 2 2 d v1 Mapping of bottom line of rectangle d v2 Mapping of left line of rectangle dA base height d v1 d v2 sin dv1 2 dv2 2 d v1 d v 2 2 2 x x x x x x 1 2 d d 1 2 2 2 1 2 2 Lecture 7 8 251 Spring 2007 A dA Important that this formula is reparameterization invariant Reparam Invariance Choose another coordinate par 1 2 Can write as functions of our 1 2 coordinates Must have 2 x x x x x x 1 2 dA d d 1 2 2 2 1 2 Metric d x x 1 x d 2 d 2 1 x i d i implicit sum i 1 2 ds2 d x 2 d x d x x x i j d i d j This is the metric 1 2 1 gij d d Where metric gij 2 x x i j Called the induced metric induced because metric not made up but rather determined inherited from the metric in the space the surface was embedded in x x x x 1 1 2 1 gij x x x x 1 2 2 2 3 Lecture 7 8 251 Spring 2007 A d 1 d 2 g where g det gij 4
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