Question

Determine whether or not f is a conservative vector field. if it is, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y) = (2x − 6y) i + (−6x + 10y − 9) j\

Answer 1

$\mathbf f$ is conservative if we can find a scalar function $f$ such that $\nabla f=\mathbf f$. This is equivalent to solving the system of PDEs,

$\dfrac{\partial f}{\partial x}=2x-6y$

$\dfrac{\partial f}{\partial y}=-6x+10y-9$

Integrate both sides of the first PDF with respect to $x$:

$f(x,y)=x^2-6xy+g(y)$

where $g$ is some function independent of $x$. Then differentiatng with respect to $y$, we have

$\dfrac{\partial f}{\partial y}=-6x+\dfrac{\mathrm dg}{\mathrm dy}=-6x+10y-9$

$\implies\dfrac{\mathrm dg}{\mathrm dy}=10y-9\implies g(y)=5y^2-9y+C$

and so $\mathbf f$ is indeed conservative, with

$f(x,y)=x^2-6xy+5y^2-9y+C$