= Solution
On an <affine chart> $\operatorname{Spec}A\subseteq V$, the <Module of Kähler differentials> $\Omega_{A/k}$ is generated by symbols $da$ subject to $k$-linearity and $d(ab)=a\,db+b\,da$. It represents $k$-<derivations>. These <modules> commute with <localization>, so their associated <quasi-coherent sheaves> glue to the <Kähler differential sheaf> $\Omega^1_{V/k}$. It is coherent: if $A=k[x_1,\dots,x_N]/I$ and $I=(f_1,\dots,f_s)$, then it has the <finite presentation of a module>
$$
A^s\xrightarrow{(\partial f_j/\partial x_i)}A^N\longrightarrow\Omega_{A/k}\longrightarrow0.
$$
For a <closed point> $P$, put $R=\mathcal O_{V,P}$ with <maximal ideal> $\mathfrak m$ and residue field $k$. The <Zariski tangent space> is $\operatorname{Hom}_k(\mathfrak m/\mathfrak m^2,k)$, equivalently $\operatorname{Der}_k(R,k)$. A derivation kills constants and $\mathfrak m^2$, so it factors through $a\mapsto a-a(P)$; conversely every <linear functional> on $\mathfrak m/\mathfrak m^2$ defines such a derivation by the product rule. The <universal property of Kähler differentials> therefore identifies this space with
$$
\boxed{T_{V,P}=\operatorname{Hom}_k(\Omega^1_{V,P}/\mathfrak m_P\Omega^1_{V,P},k).}
$$
In particular, $\Omega^1_{V,P}/\mathfrak m_P\Omega^1_{V,P}\cong\mathfrak m_P/\mathfrak m_P^2$, the <algebraic cotangent space>.
A point is a <smooth point of a variety> when its <local ring> is a <regular local ring>; over this algebraically closed field this says $\dim_kT_{V,P}=n=\dim V$. Tensor the <finite presentation of a module> above with $k(P)$. The tangent space is the <kernel> of the <Jacobian matrix> $J(P)$, hence has dimension $N-\operatorname{rank}J(P)$. This proves the <Jacobian criterion>
$$
\boxed{P\text{ smooth}\iff\operatorname{rank}J(P)=N-n.}
$$
For a smooth <irreducible variety> $V$, choose at each point an invertible $(N-n)$-minor and shrink the <affine chart> so that it remains invertible. Its relations eliminate $N-n$ of the generators of $\Omega_{A/k}$, yielding a surjection $A^n\to\Omega_{A/k}$. At the <generic point>, its target has dimension $n$: over a perfect ground field, a separating transcendence basis of the <function field> has differentials forming a basis. Equivalently this follows from the assumed <density of the smooth locus>. The <kernel> therefore becomes zero over the <fraction field> of the <integral domain> $A$. As a submodule of $A^n$, it is a <torsion-free module>, so it is already zero. Thus these maps give local <isomorphisms> with $\mathcal O_V^n$, proving <local freeness of differentials on a smooth variety> with rank $n$.
For an <affine chart> $\operatorname{Spec}A\subseteq V$, write $W\cap\operatorname{Spec}A=\operatorname{Spec}(A/I)$ and $\mathcal M=\widetilde M$. The <restriction of a module sheaf to a closed subvariety> is $\widetilde{M\otimes_AA/I}=\widetilde{M/IM}$. The <Conormal exact sequence for Kähler differentials> is
$$
I/I^2\xrightarrow{\bar f\mapsto df\otimes1}\Omega_{A/k}\otimes_AA/I\longrightarrow\Omega_{(A/I)/k}\longrightarrow0.
$$
It follows from the generators and relations: passing to $A/I$ imposes precisely the additional relations $df=0$ for $f\in I$. The first map is well defined because $d(I^2)$ lies in $I\Omega_{A/k}$. Glue these exact <module> sequences, using <exactness of localization>, to obtain
$$
\mathcal I_W/\mathcal I_W^2\longrightarrow\Omega_V^1|_W\longrightarrow\Omega_W^1\longrightarrow0.
$$
In the final assertion, interpret <locally principal subvariety> as a proper local hypersurface. Its ideal on each chart is $(f)$ with $f\ne0$. Since $V$ is an <irreducible variety> and reduced, $f$ is a <non-zero-divisor>, and $A/(f)\to(f)/(f^2)$, $\bar a\mapsto af$, is an <isomorphism>. These local rank-one descriptions make the <conormal sheaf> $\mathcal I_W/\mathcal I_W^2$ invertible.
Because $W$ is not contained in the singular locus of $V$, there is a dense open subset of $W$ where both $V$ and $W$ are <smooth varieties>. At a <closed point> there, $\dim W=\dim V-1$ by the <Krull principal ideal theorem>. The tangent description $T_{W,P}=\ker(df:T_{V,P}\to k)$ then forces $df\ne0$. Hence the conormal map is injective at the <generic point> of $W$. Its <kernel> is a subsheaf of a <line bundle> on the integral <variety> $W$, so it is a <torsion-free sheaf>; a <torsion-free sheaf> with zero generic fibre is zero. This proves <conormal injectivity for a generically smooth Cartier divisor>. If zero equations were allowed in the phrase locally principal, $W=V$ would be a counterexample to invertibility; the proper-hypersurface convention is essential.
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