Let $\mathbf { F } ( x , y , z ) = 2 x y ^ { 3 } z ^ { 4 } \mathbf { i } + 3 x ^ { 2 } y ^ { 2 } z ^ { 4 } \mathbf { j } + 4 x ^ { 2 } y ^ { 3 } z ^ { 3 } \mathbf { k }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the elliptical path $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k } , \quad 0 \leq t \leq 2 \pi$ .
A) $\pi$

B) $\pi \sqrt { 2 }$
C)$\pi \sqrt { 3 }$
D)$2 \pi$
E)$\frac { \sqrt { 2 } } { 2 } \pi$
F)$\frac { \sqrt { 3 } } { 2 } \pi$

G)$\frac { \pi } { 2 }$

H)$0$

Question

Let $\mathbf { F } ( x , y , z ) = 2 x y ^ { 3 } z ^ { 4 } \mathbf { i } + 3 x ^ { 2 } y ^ { 2 } z ^ { 4 } \mathbf { j } + 4 x ^ { 2 } y ^ { 3 } z ^ { 3 } \mathbf { k }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the elliptical path $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k } , \quad 0 \leq t \leq 2 \pi$ .
A) $\pi$

B) $\pi \sqrt { 2 }$
C)$\pi \sqrt { 3 }$ 
D)$2 \pi$
E)$\frac { \sqrt { 2 } } { 2 } \pi$ 
F)$\frac { \sqrt { 3 } } { 2 } \pi$

G)$\frac { \pi } { 2 }$

H)$0$

Quizplus · Accepted Answer

The vector field $\mathbf{F}(x, y, z) = 2xy^3z^4\mathbf{i} + 3x^2y^2z^4\mathbf{j} + 4x^2y^3z^3\mathbf{k}$ is conservative because it is possible to find a scalar potential function $\phi(x, y, z)$ such that $\nabla \phi = \mathbf{F}$. The line integral of a conservative vector field over a closed path (which the elliptical path is, since it starts and ends at the same point as $t$ goes from $0$ to $2\pi$) is always zero. This is a fundamental property of conservative fields, reflecting that the work done along a closed path is zero.

Let Evaluate the Line Integral ∫CF⋅dr\int _ { C } \mathbf { F } \cdot d \mathbf { r }∫C​F⋅dr

Let Evaluate the Line Integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$