Improvements on analytic modelling of stellar spots

In this work, we present the solution of the stellar spot problem using the Kelvin-Stokes theorem. Our result is applicable for any given location and dimension of the spots on the stellar surface. We present explicitly the result up to the second degree in the limb-darkening law. This technique can be used to calculate very efficiently mutual photometric effects produced by eclipsing bodies occulting stellar spots and to construct complex spot shapes.