I am trying to justify a sentence in Milnor's H-cobordism book. I feel like my explanation is too simple. It occurs on page 64.
In the book, it asserts that given a smooth homotopy $h_t: \mathbb{R}^n \to \mathbb{R}^n$ (between a map $h$ and the identity on $\mathbb{R}^n$), we can express it in the form $$h_t(x) = x_1h^1(t,x) + \ldots + x_nh^n(t,x), \hspace{2cm} x = (x_1,\ldots, x_n)$$ where $h^i(t,x)$ is a smooth function of $t$ and $x$, for $i = 1,\ldots, n$ and $h^i(t,0) = \frac{\partial h_t}{\partial x_i} (0)$.
I feel like my justification is too naive/elementary and as a result, the functions I've defined aren't smooth:
Note that \begin{align*} h_t(x) =&\, \int_0^1 \frac{d h_t(sx_1, \ldots, sx_n)}{ds} ds\\ =& \, \sum_{i=1}^n x_i\cdot\underbrace{\left[\int_0^1 \frac{\partial h_t}{\partial x_i} (sx_1, \ldots, sx_n)\right]}_{:= h^i(t,x)} \end{align*}
I would gladly appreciate any feedback/nudges in the right direction.