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

Question

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

Quizplus · Accepted Answer

Stokes' Theorem relates the circulation of a vector field $\vec{F}$ around a closed curve C to the curl of $\vec{F}$ over the surface S bounded by C, requiring S to be smooth and oriented with a piecewise smooth, oriented boundary C, and $\vec{F}$ to be smooth on an open region containing S and C.

State Stokes' Theorem ∫CF⃗⋅dr⃗=∫Scurl⁡F⃗⋅dA⃗\int _ { C } \vec { F } \cdot \vec { d r } = \int _ { S } \operatorname { curl } \vec { F } \cdot \vec { d A }∫C​F⋅dr=∫S​curlF⋅dA

State Stokes' Theorem $\int _ { C } \vec { F } \cdot \vec { d r } = \int _ { S } \operatorname { curl } \vec { F } \cdot \vec { d A }$