Show that $\mathrm { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$, and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$, where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$.
$\mathbf { F } ( x , y , z ) = 24 x ^ { 3 } y \mathbf { i } + \left( 6 x ^ { 4 } + 4 y z ^ { 2 } \right) \mathbf { j } + 4 y ^ { 2 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( 0,0 , - 2 )$
Select the correct answer.
A) $- 89$
B) 0
C) $- 154$
D) $- 107$

Question

Quizplus · Accepted Answer

The vector field $\mathbf{F}$ is conservative if its curl is zero. Calculating the curl of $\mathbf{F}$ shows that it is zero, confirming that $\mathbf{F}$ is conservative. Therefore, there exists a potential function $f$ such that $\mathbf{F} = \nabla f$. By integrating the components of $\mathbf{F}$, we find $f(x, y, z) = 6x^4y + 2y^2z^2$. The line integral of a conservative vector field over a curve from $A$ to $B$ is simply $f(B) - f(A)$. Evaluating $f$ at $A(0,0,0)$ and $B(0,0,-2)$, both give 0, so the integral is 0.

Show That F\mathrm { F }F Is Conservative, and Find a Function

Show That $\mathrm { F }$ Is Conservative, and Find a Function