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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 1
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 y ( fv ) = y ( f ) v + fD y ( v ) 1

Let assume that c be a diﬀerentiable curve with c(t0 ) = p and with
= y (p) Then
d
Dy (f v ) = f (c(t)).v (c(t))
dt
df (c(t))
dv (c(t))
=
.v (c(t)) + f (c(t)).
dt
dt
= y (f ).v + f.Dy (v ) dc
|...

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