Stokes' Theorem and Gauss' divergence theorem

Question 1

Stokes’ theorem (conditions)

Answer 1

Suppose D C |R^2 is a region in the plane that is bounded by a simple piecewise smooth curve Γ C D.

Suppose also that r : D -> |R^3 is a (piecewise) smooth parametrisation of an orientable surface S and C = r(Γ)

Suppose also that C is oriented so that it inherits the counterclockwise orientation of Γ through r.

Finally let F: S-> |R^3 be a smooth vector field.

Question 2

Stokes’ theorem

Answer 2

∮(over C) F . dr = ∫∫ (over S) curl F . n dS.

Question 3

Gauss’s (divergence) theorem

Answer 3

Let V be a solid in |R^3 that is bounded by a (piecewise) smooth orientable closed surface S and n the normal vector to S pointing outward from V. If F: V-> |R^3 is a smooth vector field, then
∫∫(over S) F . n dS = ∫∫∫ (over V) div F dx dy dz.