Answer:
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
