= Morse lemma
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Let $f:M\to\mathbb R$ be <smooth>, and let $p$ be a <critical point> whose <Hessian> is nondegenerate. There are smooth local coordinates $y_1,\ldots,y_n$, centered at $p$, in which
$$
f=f(p)-y_1^2-\cdots-y_\lambda^2+y_{\lambda+1}^2+\cdots+y_n^2.
$$
The <integer> $\lambda$ is the <Morse index>. This is an exact local identity, rather than a quadratic approximation with a remainder.<Morse Theory by John Milnor>[Lemma 2.2]{ref}
To see the mechanism, apply the integral form of <Taylor's theorem> twice at the critical <point>, writing $f-f(p)=\sum_{i,j}x_ix_jh_{ij}(x)$ with smooth symmetric coefficients. Nondegeneracy allows successive smooth completion of squares; the resulting change of coordinates has <derivative> given by an <invertible linear map>, so the <inverse function theorem> makes it a valid coordinate system. <Sylvester's law of inertia> makes the number of negative squares independent of the coordinates.
For $f(x,y)=x^2-y^2$, the origin has index one and is a <saddle point>. For $f(x)=x^3$, the origin is critical but its <Hessian> is zero; no smooth coordinate change makes it a nondegenerate quadratic critical <point>. The normal form implies that every nondegenerate <critical point> is isolated, and supplies the local model for the <Morse handle-attachment theorem>.
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