Suppose That Curl $\vec { F } ( 1,2,1 ) = 3 \vec { i } - 2 \vec { j } + - 5 \vec { k }$

Essay

Suppose that curl $\vec { F } ( 1,2,1 ) = 3 \vec { i } - 2 \vec { j } + - 5 \vec { k }$ curl $\vec { F } ( 0,2,1 ) = 6 \vec { i } + 2 \vec { j } + 5 \vec { k }$ and curl ${ \vec { F } } ( 1,3 , - 1 ) = - 3 \vec { i } + 2 \vec { j } + 10 \vec { k }$ Estimate the following line integrals.
(a) $\int _ { C _ { 1 } } \vec { F } \cdot d \vec { r }$ where C₁ is given by $\vec { r } ( t ) = \vec { i } + ( 3 + 0.1 \cos t ) \vec { j } + ( 0.1 \sin t - 1 ) \vec { k } , \quad 0 \leq t \leq 2 \pi$ (b) $\int _ { c _ { 2 } } \vec { F } \cdot d \vec { r }$ where C₂ is given by $\vec { r } ( t ) = 0.1 \sin t \vec { i } + 2 \vec { j } + ( 1 + 0.1 \cos t ) \vec { k } , 0 \leq t \leq 2 \pi$ (c) $\int _ { C _ { 3 } } \vec { F } \cdot d \vec { r }$ where C₃ is given by $\vec { r } ( t ) = ( 1 + 0.1 \cos t ) \vec { i } + ( 2 + 0.1 \sin t ) \vec { j } + \vec { k } , \quad 0 \leq t \leq 2 \pi$