Find the conservative vector field for the potential function $h ( x , y , z ) = 16 x y \ln ( x + y )$ by finding its gradient.
A) $\mathbf { G } ( x , y , z ) = \left( \frac { 33 x y } { x + y } + 33 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 33 x y } { x + y } + 33 x \ln ( x + y ) \right) \mathbf { j }$
B)$\mathbf { G } ( x , y , z ) = \left( \frac { 770 x y } { x + y } + 770 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 770 x y } { x + y } + 770 x \ln ( x + y ) \right) \mathbf { j }$
C)$\mathbf { G } ( x , y , z ) = \left( \frac { 49 x y } { x + y } + 49 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 49 x y } { x + y } + 49 x \ln ( x + y ) \right) \mathbf { j }$
D)$\mathbf { G } ( x , y , z ) = \left( \frac { 16 x y } { x + y } + 16 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 16 x y } { x + y } + 16 x \ln ( x + y ) \right) \mathbf { j }$
E)$\mathbf { G } ( x , y , z ) = \left( \frac { 257 x y } { x + y } + 257 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 257 x y } { x + y } + 257 x \ln ( x + y ) \right) \mathbf { j }$

Question

Find the conservative vector field for the potential function $h ( x , y , z ) = 16 x y \ln ( x + y )$ by finding its gradient. 
A) $\mathbf { G } ( x , y , z ) = \left( \frac { 33 x y } { x + y } + 33 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 33 x y } { x + y } + 33 x \ln ( x + y ) \right) \mathbf { j }$
B)$\mathbf { G } ( x , y , z ) = \left( \frac { 770 x y } { x + y } + 770 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 770 x y } { x + y } + 770 x \ln ( x + y ) \right) \mathbf { j }$
C)$\mathbf { G } ( x , y , z ) = \left( \frac { 49 x y } { x + y } + 49 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 49 x y } { x + y } + 49 x \ln ( x + y ) \right) \mathbf { j }$
D)$\mathbf { G } ( x , y , z ) = \left( \frac { 16 x y } { x + y } + 16 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 16 x y } { x + y } + 16 x \ln ( x + y ) \right) \mathbf { j }$
E)$\mathbf { G } ( x , y , z ) = \left( \frac { 257 x y } { x + y } + 257 y \ln ( x + y ) \right) \mathbf { i } + \left( \frac { 257 x y } { x + y } + 257 x \ln ( x + y ) \right) \mathbf { j }$

Quizplus · Accepted Answer

The gradient of the potential function is $
abla h = (16y\ln(x+y) +16xy/(x+y), 16x\ln(x+y)+16xy/(x+y))$. By comparing this to the general form of a conservative vector field $\mathbf{F} = (M,N)$, we see that $M=16y\ln(x+y) +16xy/(x+y)$ and $N=16x\ln(x+y)+16xy/(x+y)$. Therefore, the conservative vector field for $h$ is $\mathbf{G} = (16y\ln(x+y) +16xy/(x+y))\mathbf{i} +(16x\ln(x+y)+16xy/(x+y))\mathbf{j}$, which is option D.

Find the Conservative Vector Field for the Potential Function h(x,y,z)=16xyln⁡(x+y)h ( x , y , z ) = 16 x y \ln ( x + y )h(x,y,z)=16xyln(x+y)

Find the Conservative Vector Field for the Potential Function $h ( x , y , z ) = 16 x y \ln ( x + y )$