Answer: f = x^2-7yx+4y^2-6y
Step-by-step explanation:
A vector field is conservative if the work done by that vector field around any loop is 0 or the end points only matter when calculating the work done, not the path in between (think of gravity). The curl(F) must equal 0.
curl(F) = ∇ x F =0
= det( [tex]\left[\begin{array}{ccc}\frac{d}{dx} &\frac{d}{dy} \\(2x-2y)&(-7x+8y-6)\end{array}\right][/tex] ) (***partial derv)
-7 - (-7) = 0
Yes, it is conservative, which means we can find the potentional energy function by turning the vector field into the gradient of scalar function.
[tex]f_x=2x-7y\\f_1=x^2-7yx+C(y)\\f_1y=-7x +C'(y)\\\\C'(y)=8y-6\\C(y)=4y^2-6y\\\\f=x^2-7yx+4y^2-6y\\[/tex]
