In order to show that two definitions of smoothness in algebraic geometry coincide I would like to see a direct proof of the following fact:
Let $R$ be a commutative ring and $f_1,\dots,f_k\in R[x_1,\dots,x_n]$ polynomials such that the Jacobian $J=(\partial f_{j}/\partial x_i)_{i,j}$ has total rank in every residue field $k(P)$ for $P\in\text{Spec}(R)$.
Then the following $R$-algebra is flat $$A=R[x_1,\dots,x_n]/(f_1,\dots,f_k).$$
The idea is that $\text{Spec }A\rightarrow \text{Spec }R$ should be a smooth morphism under this hypothesis and smooth morphisms are flat by definition.
Edit: The two definitions that I am trying to compare are the following
(Gortz and Wedhorne Algebraic Geometry I)
A morphism $f:X\rightarrow Y$ of schemes is smooth of relative dimension $d$ if $\forall x\in X$ there exists an open neighborhoods $U$ of $x$ such that $f(U)\subseteq V$, $V=\text{Spec}(R)$ and an open immersion $$j:U\rightarrow R[T_1,\dots,T_n]/(f_1,\dots, f_{n-d})$$ of $R$-schemes for suitable $n$ and $f_i$ such that the Jacobian matrix $$J_{f_1,\dots,f_{n-d}(x)}=\left ( \frac{\partial f_i}{\partial T_j}(x)\right )\in M_{(n-d)\times n}(\kappa(x))$$ has rank $n-d$.
(Qing Liu, Hartshorne, Wikipedia)
A morphism $f:X\rightarrow Y$ is called smooth if it is locally of finite presentation, flat and for each $y\in Y$ the fiber $X_y$ is a smooth $\kappa(y)$-variety.
Even though this definitions are stated in full generality I think it can be wise to restrict them to the locally noetherian case (otherwise we can't speak about regular points for example). Maybe later the locally noetherian case implies the finite presentation case by some base change trick.
The implication 2.$\implies$ 1. is a bit hard but is based in the fact that if $f:X\rightarrow Y$ is a smooth morphism then it is locally a complete intersection, that is, locally in $x$ we can decompose $f$ as a regular immersion $X\rightarrow \mathbb{A}^{n+1}_Y$ followed by a projection $\mathbb{A}^{n+1}_Y\rightarrow Y$ (the sketch is in Liu Remark 3.19 and the references therein).
Now for 1.$\implies$ 2., it is clear that 1. implies locally of finite presentation and smooth fibers. The problem was the flatness part and that's the reason of the question.