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# Let v , w, and y be dierentiable vector elds in U S, f : U R be dierentiable functions in S , y (f ) be the derivative of f in the direction of y and...

See attachment. Problem 3
Let v , w , and y be diﬀerentiable vector ﬁelds in U S,f : U R be diﬀeren- tiable functions in S , y ( f ) be the derivative of f in the direction of y and a , b be real numbers. Show that the covariant derivative has the following property D x v x u = D x u x v where x ( u,v ) is a parametrization of S 1

Let assume that c : I → M be a diﬀerentiable curve with c(u0 , v ) =
dc
X (u0 , v ) so dv = xv Then
D xv x u = d
xu (u0 , v0 )
dv d dx
(u0 , v0 )
dv du
d dx
=
(u0 , v0 )
du dv
= Dxu xv
= 1

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