Let $\nabla$ be the Euclidean connection on a manifold $M = \mathbb{R}^n$. The definition I'm following is if $X, Y$ is are smooth vector fields on $M$ with $Y$ given by: $$ Y = \sum_{i}^{n} Y^i \frac{\partial}{\partial x_i} $$
Then the Euclidean connection is defined as:
$$ \nabla_Y X = \sum_{i}^{n} X(Y^i)\frac{\partial}{\partial x_i} $$
I want to show that given a curve $\gamma: I \rightarrow M $, a vector field whose euclidean connection with $\gamma'$ is $0$ has to be constant.
What I don't understand is how we can act on $\gamma^i$ with $X$ when $\gamma ^i$ is a map from $I \subset \mathbb{R}$ into $M$, unlike $Y^i$ which is $C^\infty(M)$
The definition of the Euclidean connection should be
$$\nabla_XY=\sum_{i}^{n} X(Y^i)\frac{\partial}{\partial x_i}.$$
(This makes $\nabla$ satisfy the standard definition of a connection.) Using this, it should be easy to define $\nabla_{\gamma'}Y$ for a curve $\gamma$.