I'm working on the following Lemma of the proof of Milnor:
Suppose that the vector field $v$ on $U$ corresponds to $$ v' =df \circ v \circ f^{-1}$$ on $U'$ under a diffeomorphism $f: U \to U'$. The the index of $v$ at an isolated zero $z$ is equal to the index of $v'$ at $f(z)$. In his proof he used the notion of a "one-parameter family of embeddings". What is the definition of such a family? I never met one before. Many thanks for your help.
A "one parameter family of embeddings" is a function $$ t \mapsto f_t $$ where $t$ typically ranges over the reals (or something like the open interval $-1 < t < 1$), and each $f_t$ is an embedding of some fixed manifold $M$ into some other manifold $K$ (where $K$ is often $\Bbb R^n$).
As an example, considering $M = S^1$, and $K = S^1 \times S^1$, the function $$ f_t (\theta) = (t, \theta) $$ is, for each fixed $t$, an embedding of $S^1$ as one factor of $S^1 \times S^1$.
In short: it's a path in the space of embeddings. I suspect that many authors assume it's smooth as a function of $t$, but it should certainly be continuous as a function of $t$ at the very least.