In Milnor‘s book 'Morse Theory' p12 and p13, the proof of Theorem 3.1, which is
Let $f$ be a smooth real valued function on a manifold $M$. Let $a < b$ and suppose that the set $f^{-1}[a,b]$, consisting of all $p \in M$ with $a < f(p) < b$, is compact, and contains no critical points of $f$. Then $M^a$ is diffeomorphic to $M^b$. Here $M^a=\{x\in M,f(x)\leq a \}$
he claim
Choose a Riemannian metric on $M$, $<.,.>$,define $grad f$ by $<Y, grad f> = Y(f)$. Define $\rho:M \rightarrow\mathbb{R}=1/<grad f, grad f>$, and $X_q = \rho(q) (grad f)_q$, then $X$ generates a 1-parameter group of diffeomorphisms $\phi_t:M \rightarrow M$. By caculation, $t\rightarrow f(\phi_t(q))$is linear with derivative $+1$ as long as $f(\phi_t(q))$ lies between $a$ and $b$. And then he claim $\phi_{b-a}:M\rightarrow M$ is clearly a diffeomorphism from $M^a$ onto $M^b$, I wonder why it is such a diffeomorphism, I can not find it clear.
Note that \begin{align} \frac{d}{dt} (f \circ \phi_q)(t) &= (df)_{\phi_q(t)} (X_{\phi_q(t)}) \\ &= (df)_{\phi_q(t)} \Big( \rho(\phi_q(t)) \cdot \text{grad}f)\Big)\\ &= \rho(\phi_q(t)) \cdot \text{grad} f(f) \\ &= \rho(\phi_q(t)) \cdot \langle\text{grad} f, \text{grad} f \rangle_{\phi_q(t)} \\ &= 1 \end{align} This means that $(f \circ \phi_q)(t) = t+f(q)$, and so the formula help to give $\phi_{b-a}(M^a) = M^b$.