Understanding the meaning of immersion for Manifolds

Question

Let $f: X \rightarrow Y$ be a smooth map of manifolds. $f$ is an immersion at $x \in X$ if $df_x: T_x(X) \rightarrow T_y(Y)$ is an injective map where $y = f(x).$ $T_x(X)$ is the tangent plane of $X$ at $x.$ To my understanding, the map $f$ takes a vector of $T_x(X)$ and we compute the directional derivative of $f$ in that given direction. That is, $df_x(v) = \lim_{t \rightarrow 0} \frac{f(x + tv) - f(x)}{t}.$ However, how can this map ever be injective. Two vectors pointing in the same direction with different magnitude are different vectors but they obviously yield the same directional derivative. Or do we equate vectors with the same direction, regardless of magnitude?

Answer 1

Try computing $df_{x}(\lambda v)$ in terms of $df_{x}(v)$ for scalar $\lambda \not = 0$ using your definition. I think you'll find that it is not simply equal to $df_{x}(v)$.