chap8 - normal direction to a plane is constant; more...

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Chapter 8: Hints and (partial) Solutions Exercise 8.1 (a) For this problem 0 (prime) denotes a covariant derivative and · (dot) or d dt a usual derivative. Use covariant derivative definition (pg 45): ( X + Y ) 0 = ± d dt [ X + Y ] ² ( t ) - ³± d dt [ X + Y ] ² ( t ) · N ( α ( t )) ´ N ( α ( t )) = ˙ X ( t ) + ˙ Y ( t ) - ˙ X ( t ) + ˙ Y ( t ) · N ( α ( t )) i N ( α ( t )) = ˙ X ( t ) + ˙ Y ( t ) - h ˙ X ( t ) · N ( α ( t )) + ˙ Y ( t )] · N ( α ( t )) i N ( α ( t )) = ˙ X ( t ) - h ˙ X ( t ) · N ( α ( t )) i N ( α ( t )) + ˙ Y ( t ) - h ˙ Y ( t )] · N ( α ( t )) i N ( α ( t )) = X 0 ( t ) + Y 0 ( t ) Exercise 8.2 Hint : When is X 0 = ˙ X ? and why would it help in this problem? It’s important to note that the
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Unformatted text preview: normal direction to a plane is constant; more precisely, you should be able to argue that N ( p ) = ( p,a/ k a k ) is an orientation of the surface S (a plane) where a = ( a 1 ,a 2 ,...,a n +1 )....
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