lecturenotes2.20100204.4b6af9a4560990.70088729

lecturenotes2.20100204.4b6af9a4560990.70088729 - 2 Tangent...

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2T a n g e n t V e c t o r s De f nition 11 A curve is a subset C ( s ) of allowable con f gurations parametrized by a single parameter s . De f nition 12 Two curves C 1 ( s ) and C 2 ( s ) that pass through C 0 at s =0 are equivalent if d ds ξ ( C 1 ( s )) ¯ ¯ ¯ ¯ s =0 = d ds ξ ( C 2 ( s )) ¯ ¯ ¯ ¯ s =0 . A tangent vector at C 0 is the collection of all curves, which pass through C 0 at s and are equivalent to any one curve in this collection. The coordinates of a tangent vector relative to a chart are given by the components of d ds ξ ( C 1 ( s )) ¯ ¯ ¯ ¯ s =0 . Proposition 13 Consider two coordinate charts ( C 1 , ξ 1 ) and ( C 2 , ξ 2 ) on the collection of allowable con f gu- rations, such that C 1 C 2 6 = and ξ 1 and ξ 2 are related by the coordinate transformation ξ 2 = f ( ξ 1 ) . Let C ( s ) be a curve that passes through C 0 C 1 C 2 at s . It follows that d ds ξ 2 ( C ( s )) ¯ ¯ ¯ ¯ s =0 = ξ 1 f ( ξ 1 ( C 0 )) · d ds ξ 1 ( C ( s )) ¯ ¯ ¯ ¯ s =0 . The equivalence of curves is thus independent of chart. 4
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Proposition 14 Consider a coordinate chart ( C , ξ ) ,where ξ =( ξ 1 , ξ 2 ) T and suppose that ξ 1 = g ( ξ 2 ) on a restricted collection ˜ C C of allowable con f gurations, such that ³ ˜ C , ξ 2 ´ is a chart. Let ˜ C ( s ) be a curve in ˜ C that runs through C 0 at s =0 .I tfo l
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This note was uploaded on 11/17/2011 for the course TAM 412 taught by Professor Weaver during the Spring '08 term at University of Illinois, Urbana Champaign.

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lecturenotes2.20100204.4b6af9a4560990.70088729 - 2 Tangent...

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