Here as another way to define curl:

1) Curl is a vector describing "instantaneous rotation". The line integral over a gradient vector field is zero on any closed curve, so whatever instantaneous rotation is, it should have the property $curl(\nabla f) = 0$.

2) Every symmetric $3\times 3$ matrix is $D^2f$ for some homogeneous polynomial $f$ in three variables. Thus a vector field $F:\mathbb{R}^3 \to \mathbb{R}^3$ is locally well-approximated by an irrotational gradient vector field when $DF$ is symmetric.

3) The function $g(\vec v) = \vec v \times \vec p$, where $\vec p$ is fixed but arbitrary, is both the general rotational velocity field about an axis $\vec p$ through the origin, with angular velocity $||\vec p ||$, as well as the linear function described by an arbitrary antisymmetric matrix. When $DF$ is antisymmetric, $F$ is locally well-approximated by a rotational velocity field.

4) $DF = \frac{(DF + DF^T) + (DF - DF^T)}{2}$. The first summand is irrotational, the second, a rotational velocity field. Define the instantaneous rotation field for F to be the linear function given by $(DF - DF^T)$ (ignoring the factor of $2$). The vector $\vec p$ which gives us the corresponding function $g$ is easily determined from the coefficients of $(DF - DF^T)$, and is precisely $curl(F)$.