If $\mathbf { F } ( x , y ) = ( 6 x - 5 y ) \mathbf { i } - \left( 5 x - 6 y ^ { 2 } \right) \mathbf { j }$ is a conservative field, then its potential function $f ( x , y )$ is

A) $3 x ^ { 2 } - 5 x y + 2 y ^ { 3 } + C$
B) $3 x ^ { 2 } + 5 x y + 2 y ^ { 3 } + C$
C) $3 x ^ { 2 } - 5 x y - 2 y ^ { 3 } + C$
D) $3 x ^ { 2 } + 5 x y - 2 y ^ { 3 } + C$
E) $- 3 x ^ { 2 } - 5 x y + 2 y ^ { 3 } + C$

Question

If $\mathbf { F } ( x , y ) = ( 6 x - 5 y ) \mathbf { i } - \left( 5 x - 6 y ^ { 2 } \right) \mathbf { j }$ is a conservative field, then its potential function $f ( x , y )$ is

A) $3 x ^ { 2 } - 5 x y + 2 y ^ { 3 } + C$
B) $3 x ^ { 2 } + 5 x y + 2 y ^ { 3 } + C$
C) $3 x ^ { 2 } - 5 x y - 2 y ^ { 3 } + C$
D) $3 x ^ { 2 } + 5 x y - 2 y ^ { 3 } + C$
E) $- 3 x ^ { 2 } - 5 x y + 2 y ^ { 3 } + C$

Quizplus · Accepted Answer

For a vector field $\mathbf{F}(x, y) = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j}$ to be conservative, it must be the gradient of some potential function $f(x, y)$, meaning $\mathbf{F} = \nabla f$. This implies $P = \frac{\partial f}{\partial x}$ and $Q = \frac{\partial f}{\partial y}$. Integrating $P = 6x - 5y$ with respect to $x$ gives $3x^2 - 5xy + g(y)$, and integrating $Q = -(5x - 6y^2)$ with respect to $y$ gives $-5xy + 2y^3 + h(x)$. Since both must equal $f(x, y)$ except for a constant, and comparing the integrated results leads to $f(x, y) = 3x^2 - 5xy + 2y^3 + C$, which matches choice A.

If F(x,y)=(6x−5y)i−(5x−6y2)j\mathbf { F } ( x , y ) = ( 6 x - 5 y ) \mathbf { i } - \left( 5 x - 6 y ^ { 2 } \right) \mathbf { j }F(x,y)=(6x−5y)i−(5x−6y2)j

If $\mathbf { F } ( x , y ) = ( 6 x - 5 y ) \mathbf { i } - \left( 5 x - 6 y ^ { 2 } \right) \mathbf { j }$