Understanding Covariant Derivatives Along a Curve

• eok20

eok20

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:

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

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

I understand that the left hand side is the value of the vector field D_t V at time t (a tangent vector at $$\gamma(t)$$). However, the right hand side is confusing me since $$\dot{\gamma}(t)$$ is a tangent vector at $$\gamma(t)$$ 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 $$(\nabla_X W) (\gamma(t))$$ where X is any vector field such that $$X(\gamma(t)) = \dot{\gamma}(t)$$?

Thanks.