In a scalar field, the line integral of the gradient field is the difference between the value of the scalar field at the start and end points.
This generalises the fundamental theorem of calclulus.
We have a curve \(C\) on a plane.
Inside this is region \(D\).
We have two functions: \(L(x,y)\) and \(M(x,y)\) defined on the region and curve.
\(\oint_C (L dx + M dy)=\int \int_D (\dfrac{\delta M}{\delta x}-\dfrac{\delta L}{\delta y})dx dy\)
For type-I, we can integrate over y, then integrate over x.
For type-II, we can integrate over x, then integrate over y.