Find the value of the line integral $ \int_{c} \mathbf{F} \cdot d \mathbf{r} $ on the closed path consisting of line segments from $( 0,2 )$ to $( 0,0 )$, from $( 0,0 )$ to $( 2,0 )$, and then from $( 2,0 )$ to $( 0,2 )$, where $\mathbf { F } ( x , y ) = 3 y e ^ { 3 x y } \mathbf { i } + 3 x e ^ { 3 x y } \mathbf { j }$.
(Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)
A) 99
B) 12
C) 36
D) 30
E) 0

Question

Quizplus · Accepted Answer

The vector field $\mathbf{F} = 3ye^{3xy}\mathbf{i} + 3xe^{3xy}\mathbf{j}$ is conservative because it can be derived from the potential function $\phi(x, y) = e^{3xy}$, since $\nabla \phi = \mathbf{F}$. For conservative fields, the line integral over any closed path is zero.

Find the Value of the Line Integral ∫cF⋅dr \int_{c} \mathbf{F} \cdot d \mathbf{r} ∫c​F⋅dr

Find the Value of the Line Integral $\int_{c} \mathbf{F} \cdot d \mathbf{r}$