Determine whether or not the given vector field is conservative.

$($a$)F(x,y)=(6x-5y)i+(2x-3)j$

$($b$)F(x,y)=(7+8xy)i+(4x^2-3y^2)j$

Solution

$($a$)$ Let $P(x,y)=6x-5y$ and $Q(x,y)=2x-3.$ Then $\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{y}}=$ and $\frac{\text{d}\text{e}\text{l}\text{Q}}{\text{d}\text{e}\text{l}\text{x}}=$ Since $\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{y}},\frac{\text{d}\text{e}\text{l}\text{Q}}{\text{d}\text{e}\text{l}\text{x}},F$ is $-$ Select$-,v.$

$($b$)$ Let $P(x,y)=7+8xy$ and $Q(x,y)=4x^2-3y^2.$ Then $\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{y}}=$ and $\frac{\text{d}\text{e}\text{l}\text{Q}}{\text{d}\text{e}\text{l}\text{x}}=.$

Also, the domain of $F$ is the entire plane