Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. If it is not, enter NONE. F (x, y) = (2x - 4y) i + (-4x + 10y - 7) j f (x, y) = Incorrect: Your answer is incorrect. + K

Step-by-step explanation:

Given that,

F (x, y) = (2x - 4y) i + (-4x + 10y - 7) j

Let P=2x-4y

And Q=-4x+10y+7

For the function to the conservative

Then,

∂P/∂y = ∂Q/∂x

∂P/∂y = - 4

∂Q/∂x = - 4

Since,

∂P/∂y = ∂Q/∂x

Then, F is conservative

b. Now,

We are looking for a function f so that ∇f = {fx, fy}={2x-4y, - 4x+10y-7}

∇f means

∇f = ∂f/∂x •i + ∂f/∂y •j

Then, equating them

∂f/∂x = 2x-4y

∂f=2x-4y ∂x

Integrate both sides

∫ ∂f = ∫2x-4y ∂x

f=2x²/2 - 4xy + f (y)

f (x, y) = x²-4xy + f (y)

∂f/∂y = - 4x+f' (y)

Comparing this to

∂f/∂y = - 4x + 10y - 7.

Then, f' (y) = 10y-7

Integrated both side

f' (y) = 10y-7

∂f/∂y=10y-7

∂f=10y-7∂y

∫ ∂f = ∫ 10y-7 ∂y

f (y) = 10y²/2-7y + C

Here, C=0

f (y) = 5y²-7y

Substituting this into

f (x, y) = x²-4xy + f (y)

f (x, y) = x²-4xy + 5y²-7y

Then this is the required function