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

$\dfrac{\partial f}{\partial x}=2x-4y$

$\dfrac{\partial f}{\partial y}=-4x+10y-5$

Integrate the first PDE with respect to $x$:

$\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int(2x-4y)\,\mathrm dx\implies f(x,y)=x^2-4xy+g(y)$

Differentiate with respect to $y$ to get

$\dfrac{\partial f}{\partial y}=-4x+10y-5=-4x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=10y-5$

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

So we have

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

which means $F$ is indeed conservative.