I'm learning about Green's Theorem ,it simply says that work done to move say a boat along a closed loop is like a giant circulation and it must equal the sum of all circulation (curl) in all inifinitesimally small area ,but what I can't understand is that how the work done by the field along the boundary(macroscopic circulation-work done to rotate the boat once along the curve) same as the work done to rotate in each small area?
Does the curl vector tell us the work done by a field to rotate an object?
To develop an intuition on the curl, it is better to think in terms of the velocity field of a fluid (say, two dimensional) instead of in terms of the work of a force.
If you consider an infinitesimal disk of radius $r$ around your point, its motion is approximately a translation and a rigid rotation (I believe this is intuitively clear).
The key fact is that the circulation of the velocity along its circumference is an infinitesimal of order $r^2$ as $r\to 0$. The reason for that is that your disk is moving as a whole and rotating. The translational component does not contribute to the curl since opposite points on the circumference have opposite contributions. Therefore you can assume that the velocity at the center is zero.
The velocity of a rigid disk increases linearly with the radius, namely $v=\omega r$ where $\omega$ is the angular velocity. Summing up, the circulation is $\omega r\cdot 2\pi r=2\pi\omega r^2$ (as always, in first approximation). This is proportional to the area of the disk $\pi r^2$ and the coefficient of proportionality is $2\omega$. This is the curl. When you "vectorize" this computation your curl is a vector, perpendicular to the plane of rotation and with magnitude $2\omega$. For finite loops, you just break them into infinitesimal pieces (say squares for simplicity), and the circulation along boundaries shared by neighboring squares cancels.