Here is another construction which goes back to Chern's proof of the Gauss-Bonnet theorem.

Suppose that $\pi: E\to M$ is an oriented rank $2k$ real vector bundle over the manifold $M$. Assume additionally that $E$ is equipped with a metric $g$, and a connection $\nabla$ compatible with the metric.

In your case $2k=\dim M =2$, $E= TM$, $g$ is a Riemann metric and the connection $\nabla$ is the Levi-Civita connection.

Denote by $S(E)$ the unit sphere bundle of $E$. Using the connection $\nabla$ one can **explicitly** construct a form

$$\Psi(\nabla) \in \Omega^{2k-1}(S(E)) $$

with the property that the integral over each fiber of $S(E)$ is equal to $-1$. The form $\Psi(\nabla)$ is known as the *global angular form* determined by $g$ and $\nabla$.

The connection $\nabla$ also **explicitly** defines via the Chern-Weil construction a closed $2k$-form $\newcommand{\be}{\boldsymbol{e}}$

$$\be(\nabla)\in \Omega^{2k}(M). $$

This the the so-called *Euler form* which in your case is $\frac{1}{2\pi} K dA$.

$\newcommand{\bR}{\mathbb{R}}$ Now choose a smooth function $\rho :[0,\infty)\to [0,\infty)$ which ir equal to $-1$ near $0$ and equal to $0$ on $[1, \infty)$. Let $r: E\to \bR$ denote the radial distance along the fibers of the vector bundle. Define

$$\omega(\nabla)=-\rho'(r) dr\wedge \Psi(\nabla) + \rho(r)\pi^*\be(\nabla). $$

One can then show the following.

**1.** $\omega(\nabla)$ is a closed form representing the Thom class of $E$.

**2.** If $\zeta_0: M\to E$ denotes the zero section, then

$$\zeta_0^* \omega(\nabla)=\be(\nabla). $$

**3.** Suppose that $\zeta:M\to E$ is a section with transversal zero set. For $t>0$ we set $\zeta_t:=t\zeta$. Note that $\zeta$ and $\zeta_t$ have the same zero sets. Let $Z=\zeta^{-1}(0)$ so that $Z$ is a codimension $2k$-submanifold of $M$ whose normal bundle carries a natural orientation. Then the following hold.

**a.** For any neighborhood $\newcommand{\eN}{\mathscr{N}}$ $\eN$ of $Z$ in $M$ there exists $T=T(\eN)$ such that the support of $\zeta_t^* \omega(\nabla)$ is contained in $\eN$ for any $t>T$.

**b.** The form $\zeta_t^*\omega(\nabla)$ is Poincare dual to the cohomology class determined by $Z$.

**c.** As $t\to \infty$ the forms $\zeta_t^*\omega(\nabla)$ converge in the sense of currents to the current of integration determined by $SZ$.

In your case, the zero set consists of finitely many points, the $2$-form $\zeta_t^*\omega(\nabla)$ has the description

$$\zeta_t^*\omega(\nabla)=\rho_t dA $$

where the function $\rho_t$ is concentrated near the zeros of the section, and forms high (positive or negative) peaks around these points.

The statement **3.c.** above contains as a special case the Poincare-Hopf theorem.

For more details see Section 8.3.2 of these notes.