State Stokes' Theorem.
A)If S is a smooth oriented surface with smooth, oriented boundary C, then

\int _ { C } \vec { F } \cdot \vec { d r } = \int _ { S } \operatorname { curl } \vec { F } \cdot \vec { d A }

B)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, then

\int _ { S } \vec { F } \cdot \vec { d r } = \int _ { C } \operatorname { curl } \vec { F } \cdot \vec { d A }

C)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, then

\int _ { C } \vec { F } \cdot \vec { d r } = \int _ { S } \operatorname { cur } \vec { F } d V

D)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, and if

\vec { F }

is a smooth vector field on an open region containing S and C, then

\int _ { C } \vec { F } \cdot \vec{ d r } = \int _ { S } \operatorname { curl } \vec { F } \cdot \vec { d A }

E)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, then

\int _ { C } { \vec { F } } \cdot \vec { d r } = \int _ { S } d i v { \vec { F } } d V

Let S Be the Triangle with Vertices at (12,0,0), (0,6,0)and x+2y+2z=12x + 2 y + 2 z = 12x+2y+2z=12

Essay