Question

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 - 5) j

f(x, y) = ___-______+ k

Answer 1

We need to have

`(\partial f)/(\partial x)=2x-4y`

`(\partial f)/(\partial y)=-4x+10y-5`

Integrate the first PDE with respect to
`x`:

`\displaystyle\int(\partial f)/(\partial x)\,\mathrm dx=\int(2x-4y)\,\mathrm dx\implies f(x,y)=x^2-4xy+g(y)`

Differentiate with respect to
`y` to get

`(\partial f)/(\partial y)=-4x+10y-5=-4x+(\mathrm dg)/(\mathrm dy)\implies(\mathrm dg)/(\mathrm dy)=10y-5`

`\implies g(y)=5y^2-5y+C`

So we have

`f(x,y)=x^2-4xy+5y^2-5y+C`

which means
`F` is indeed conservative.