On Stokes's Theorem

Remember this form of Green's Theorem: where C is a simple closed positively-oriented curve that encloses a closed region, R, in the xy-plane. It measures circulation along the boundary curve, C. Stokes's Theorem generalizes this theorem to more interesting surfaces. Stokes's Theorem For F(x,y,z) = M(x,y,z)i+N(x,y,z)j+P(x,y,z)k, M, N, P have continuous first-order partial derivatives. S is a 2-sided surface with continuously varying unit normal, n, C is a piece-wise smooth, simple closed curve, positively-oriented that is the boundary of S, T is the unit tangent vector to C, then ∲F·dr = ∫∫ ∇⨯F·k dA ⇀ ⇀ ⇀ ⇀ ˆ F(x,y) = M(x,y)i + N(x,y)j, ˆ ˆ ˆ ˆ ∲F·T ds = ∫∫ (∇⨯F)·n dS ⇀ ⇀ ˆ C C S ˆ R ˆ ˆ ˆ ⇀