Let $C$ be a smooth curve of degree $d$ in $\mathbb{P}^2$ over $\mathbb{C}$. Say $C$ is defined by $p(x,y,z)=0$, with $p$ a homogeneous degree $d$ polynomial.

In vector calculus one learns that the gradient of $p$ is normal to $C$ at every point of the curve.

In algebraic geometry, the invertible sheaf associated to the normal bundle $N_{C|\mathbb{P}^2}$ to $C$ in $\mathbb{P}^2$, is given by $\mathcal{O}_{\mathbb{P}^2}(d) _{|C}$.

Is there any relationship between the gradient and the bundle or the sheaf?