A Morse gradient zero contributes $(-1{)}^{\lambda }$ to the index

Given: A closed smooth manifold $M$, a Morse function $f$, a Riemannian metric $g$ and a critical point $p$ of $f$ of Morse index $\lambda$.

The gradient ${\mathrm{grad}}_{g}f$ is a smooth vector field on $M$ and it vanishes exactly at the critical points of $f$ (The Riemannian gradient is the metric dual of the differential, The Riemannian gradient vanishes exactly at the critical points).

In coordinates, differentiating the gradient formula at a critical point gives $D({\mathrm{grad}}_{g}f{)}_p=g(p{)}^{-1}{H}_p$, because $d{f}_p=0$ kills the derivatives of the inverse metric. Thus the linearization (vertical derivative) of ${\mathrm{grad}}_{g}f$ is the Hessian bilinear form ${\mathrm{Hess}}_{p}f$, read through $g$; the critical Hessian of a Morse function is nondegenerate with $\lambda$ negative and $n-\lambda$ positive entries in its inertia normal form, so the linearization is invertible and $p$ is a nondegenerate zero of ${\mathrm{grad}}_{g}f$ (The intrinsic Hessian of a smooth function at a critical point, At a critical point, the intrinsic Hessian agrees with the Levi-Civita Hessian, Nondegenerate critical points, nullity, index, and coindex, Nondegenerate zero of a vector field).

A nondegenerate zero has index equal to the sign of the determinant of its linearization (The index of a nondegenerate vector-field zero), the sign of the determinant of a nondegenerate symmetric form with $\lambda$ negative and $n-\lambda$ positive entries in its inertia normal form is $(-1{)}^{\lambda }$ (Nondegenerate critical points, nullity, index, and coindex), and negating the field multiplies the index by $(-1{)}^n$ (Negation scales the local index by $(-1{)}^n$); the index of the negated function swaps, ${\mathrm{ind}}_p(-f)=n-\lambda$ (Index and coindex swap under negation).