$\text { Let } \mathbf { F } ( x , y , z ) = \arctan \frac { x } { y } \mathbf { i } + \ln \sqrt { x ^ { 2 } + y ^ { 2 } } \mathbf { j } + \mathbf { k } \text { and let } C \text { be the triangle }$ with vertices of $( 0,0,0 ) , ( 6,6,6 )$, and $( 0,0,12 )$, oriented counterclockwise. Use Stokes's Theorem to evaluate $\int _ { c } \mathbf { F } \cdot d \mathbf { r }$.
A) 0
B) 6
C) 9
D) 18
E) 12

Question

Quizplus · Accepted Answer

Stokes's Theorem relates a surface integral over a surface $S$ to a line integral over the boundary curve $C$ of $S$. The theorem states that $\int _ { c } \mathbf { F } \cdot d \mathbf { r } = \int _ { S } \nabla \times \mathbf { F } \cdot d \mathbf { S }$, where $\nabla \times \mathbf { F }$ is the curl of $\mathbf { F }$. For the given $\mathbf { F }$, the curl $\nabla \times \mathbf { F }$ is $\mathbf { 0 }$ because the partial derivatives of the components of $\mathbf { F }$ with respect to $x$, $y$, and $z$ do not vary in a way that would produce a non-zero curl. Since the curl of $\mathbf { F }$ is $\mathbf { 0 }$, the surface integral (and thus the line integral over $C$) is $0$.

With Vertices Of (0,0,0),(6,6,6)( 0,0,0 ) , ( 6,6,6 )(0,0,0),(6,6,6)

With Vertices Of $( 0,0,0 ) , ( 6,6,6 )$