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In the definition of affine connection, there is $\nabla_X (fY) = \mathrm df(X)Y + f\nabla_XY$ where $X,Y$ are vector fields (or their generalizations) on a smooth manifold $M$ and $f$ is smooth function from a smooth manifold $M$ to $\mathbb{R}$.

My question is, how would $\mathrm df(X)Y$ be valued like. Would it be valued as the following form: $\mathrm df(X)$ is treated like differential form and $\mathrm df(X)$ at the point $x$ of $M$ is $\mathrm df_p(X_p)$ where $X_p$ is a vector at $p$ of the vector field $X$. Is this correct?

Zeus
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Often people write what you write as $df(X) Y$ as $(X f) Y$. To define $X f$, we take any curve $x(t)$ passing through $p$ such that $x(0) = p$ and $x^{(1)}(0) = X_p$ and evaluate $$ X(f) = \left. \frac{d}{dt} f(x(t)) \right|_{t=0}$$

muzzlator
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