I'm teaching myself about connections and came across something I am not completely sure about. The text I am using defines a connection as taking in two vector fields and outputting a vector field. However, later when discussing covariant derivatives along a curve I see this equation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

D_t V(t) = \nabla_{\dot{\gamma}(t)} W,

[/tex]

where V is a vector field along the curve [tex]\gamma[/tex], W is an extension field of V, [tex]\nabla[/tex] is the connection and D_t takes in a vector field along [tex]\gamma[/tex] and gives a vector field along [tex]\gamma[/tex].

I understand that the left hand side is the value of the vector field D_t V at time t (a tangent vector at [tex]\gamma(t)[/tex]). However, the right hand side is confusing me since [tex]\dot{\gamma}(t)[/tex] is a tangent vector at [tex]\gamma(t)[/tex] and not a vector field. Since the value of a covariant derivative at a point p depends only on the value at p of the vector field we are differentiating along, is the right hand side the same as [tex](\nabla_X W) (\gamma(t))[/tex] where X is any vector field such that [tex]X(\gamma(t)) = \dot{\gamma}(t)[/tex]?

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Covariant Derivatives

**Physics Forums | Science Articles, Homework Help, Discussion**