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 = (x 1 , x 2 , x 3) Parameter Space: ξ1 , ξ2 (directions along grid lines. Purely arbitrary. No con-nection to distances.) Describe surface: �x(ξ1, ξ2) = (x 1(ξ1, ξ2), x 2(ξ1, ξ2), x 3(ξ1, ξ2)) What is area, A? 1Lecture 7 8.251 Spring 2007 � A = infinitesimal rectangles ξ1,ξ2 Map to surfae: d�v1: infinitesimal vector corresponding to dξ1 on ξ1 To linear order, these 1, 2, 3, 4 points form a parallelogram d�v1: Mapping of bottom line of rectangle = ∂�x (ξ1, ξ2)dξ1 ∂ξ�1 d�v2: Mapping of left line of rectangle = ∂�x (ξ1, ξ2)dξ2 ∂ξ�2 dA = base height · = |d�v1d�v2sin θ � | · | | = |dv12dv22 − (d� d�v2)2v1 ·|�|� |�� � � �2 = dξ1dξ2 ∂�x ∂�x ∂�x ∂�x ∂�x ∂�x ∂ξ1 · ∂ξ2 ∂ξ2 · ∂ξ2 − ∂ξ1 · ∂ξ2 2� � �� � � �� � Lecture 7 8.251 Spring 2007 A = dA Important that this formula is reparameterization-invariant. Reparam. Invariance Choose another coordinate par. (ξ�1 , ξ�2). Can write as functions of our (ξ1, ξ2) coordinates. Must have: ��� �� � � �2 dA = dξ�1dξ�2 ∂�x ∂�x ∂�x ∂�x ∂�x ∂�x ∂ξ�1 · ∂ξ�2 ∂ξ�2 · ∂ξ�2 − ∂ξ�1 · ∂ξ�2 Metric d�x = ∂�x dξ1 + ∂�x dξ2 = ∂�x dξi ∂ξ1 ∂ξ2 ∂ξi implicit sum i=1,2 ∂�x ∂�x ds2 = |d�x|2 = d�x · d�x = ∂ξi ∂ξj dξidξj This is the metric. = gij (ξ1, ξ2)dξ1dξ2 ∂�x ∂�xWhere metric gij = ∂ξi ∂ξj 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 gij = ∂ξ1 ∂�x · ∂ξ1 ∂�x ∂ξ2 ∂�x · ∂ξ1 ∂�x ∂ξ1 · ∂ξ2 ∂ξ2 · ∂ξ2 3Lecture 7 8.251 Spring 2007 � A = dξ1dξ2√g where g = det(gij )
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