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LectureNotes13U

# LectureNotes13U - Math 497C Curves and Surfaces Fall 2004...

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Math 497C Nov 11, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 13 2.9 The Covariant Derivative, Lie Bracket, and Rie- mann Curvature Tensor of R n Let A R n , p A , and W be a tangent vector of A at p , i.e., suppose there exists a curve γ : ( , ) A with γ (0) = p and γ (0) = W . Then if f : A R is a function we define the (directional) derivative of f with respect to W at p as W p f := ( f γ ) (0) = df p ( W ) . Similarly, if V is a vectorfield along A , i.e., a mapping V : A R n , p V p , we define the covariant derivative of V with respect to W at p as W p V := ( V γ ) (0) = dV p ( W ) . Note that if f and V are C 1 , then by definition they may be extended to an open neighborhood of A . So df p and dV p , and consequently W p f and W p V are well defined. In particular, they do not depend on the choice of the curve γ or the extensions of f and V . Exercise 1. Let E i be the standard basis of R n , i.e., E 1 := (1 , 0 , . . . , 0), E 2 := (0 , 1 , 0 , . . . , 0) , . . . , E n := (0 , . . . , 0 , 1). Show that for any functions f : R n R and vectorfield V : R n R n ( E i ) p f = D i f ( p ) and ( E i ) p V = D i V ( p ) ( Hint: Let u i : ( , ) R n be given by u i ( t ) := p + tE i , and observe that ( E i ) p f = ( f u i ) (0), ( E i ) p V = ( V u i ) (0)). 1 Last revised: November 29, 2004 1

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The operation is also known as the standard Levi-Civita connection of R n . If W is a tangent vectorfield of A , i.e., a mapping W : A R n such that W p is a tangent vector of A for all p A , then we set Wf ( p ) := W p f and ( W V ) p := W p V.
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