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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$.
$\mathbf { F } ( x , y , z ) = 27 x ^ { 2 } y \mathbf { i } + \left( 9 x ^ { 3 } + 24 y ^ { 3 } z ^ { 2 } \right) \mathbf { j } + 12 y ^ { 4 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0,1 )$
A) $- 89$
B) 0
C) $- 154$
D) $- 107$

Question

Quizplus · Accepted Answer

A vector field $\mathbf{F}$ is conservative if it can be written as the gradient of a scalar potential function $f$, that is, $\mathbf{F} = \nabla f$. For a conservative field, the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ from point $A$ to point $B$ depends only on the values of $f$ at $A$ and $B$, and not on the path $C$. Since both $A$ and $B$ have a component where $y=0$, and all terms in $\mathbf{F}$ involve $y$ to at least the first power, the potential function $f$ evaluated at $A$ and $B$ will be the same, leading to a line integral of $0$.

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

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