Flow chart for Surface Integrals int int_S vector F dot n hat d S. The flow chart starts with the question: Is S the boundary of a solid? Two arrows extend from this question, one for yes and one for no. The yes arrow points to: Divergence Theorem. The no arrow points to the question: Does vector F = del cross vector G? Two arrows extend from this question, one for yes and one for no. The yes arrow points to: Stokes Theorem. The no arrow points to: Parameterize S and do a double integral.