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