Show that $\mathbf { F }$ 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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$.
$\mathbf { F } ( x , y , z ) = 21 x ^ { 2 } y \mathbf { i } + \left( 7 x ^ { 3 } + 15 y ^ { 2 } z ^ { 2 } \right) \mathbf { j } + 10 y ^ { 3 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0 , - 3 )$
A) 0
B) $- 154$
C) $- 107$
D) $- 89$

Question

Quizplus · Accepted Answer

The vector field $\mathbf{F}$ is conservative if its curl is zero. Calculating the curl of $\mathbf{F}$ shows it is zero, confirming $\mathbf{F}$ is conservative. The potential function $f$ can be found by integrating the components of $\mathbf{F}$. Evaluating the line integral of a conservative field over any path from $A$ to $B$ is simply $f(B) - f(A)$. Since $f(A) = f(B)$, the integral is 0.

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

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