Show that 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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 6 x y \mathbf { i } + \left( 3 x ^ { 2 } + 4 y z ^ { 2 } \right) \mathbf { j } + 4 y ^ { 2 } z \mathbf { k }$ ; $A ( 0,0,0 )$ and $B ( 2 , - 2,0 )$
A) -113
B) -24
C) -131
D) -178

Question

Quizplus · Accepted Answer

The vector field F is conservative if its curl is zero. Calculating the curl of F, we find it is zero, indicating F is conservative. To find f, integrate the components of F: $f(x, y, z) = 3x^2y + y^2z^2$. The integral from A to B is $f(B) - f(A) = f(2, -2, 0) - f(0, 0, 0) = -24$.

Show That F Is Conservative, and Find a Function F F=∇f\mathbf { F } = \nabla fF=∇f

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