Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$. We admit that if functions $\varphi_1, \varphi_2, \ldots, \varphi_n$ are continuous on $\mathbb{R}$ and take values in $\mathbb{R}^n$, then the function $\Phi$ defined on $\mathbb{R}$ by: $$\Phi(t) = \operatorname{det}(\varphi_1(t), \varphi_2(t), \ldots, \varphi_n(t))$$ is continuous on $\mathbb{R}$. Using question I.B.2 and the multilinearity of the determinant, show that in a neighbourhood of 0 $$\operatorname{det}\left(f(t_1), f(t_2), \ldots, f(t_n)\right) = t^n \mathrm{jac}_f(0) + \mathrm{o}\left(t^n\right)$$
Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
We admit that if functions $\varphi_1, \varphi_2, \ldots, \varphi_n$ are continuous on $\mathbb{R}$ and take values in $\mathbb{R}^n$, then the function $\Phi$ defined on $\mathbb{R}$ by:
$$\Phi(t) = \operatorname{det}(\varphi_1(t), \varphi_2(t), \ldots, \varphi_n(t))$$
is continuous on $\mathbb{R}$.
Using question I.B.2 and the multilinearity of the determinant, show that in a neighbourhood of 0
$$\operatorname{det}\left(f(t_1), f(t_2), \ldots, f(t_n)\right) = t^n \mathrm{jac}_f(0) + \mathrm{o}\left(t^n\right)$$