Question

Determine whether the vector field is conservative and if so find the potential function, f(x,y)=(6x 5y)i

Answer 1

I'm assuming you mean

`\vec f(x,y) = (6x + 5y)\,vec\imath`

It's a bit odd that there is no
`\vec\jmath`-component, but at any rate we want to find a scalar function
`f(x,y)` for which

`\\abla f(x,y) = \vec f(x,y)`

This would mean

`(\partial f)/(\partial x) = 6x + 5y`

and

`(\partial f)/(\partial y) = 0`

The second equation tells us
`f(x,y)` is a function that only depends on
`x`. But the first equation tells us
`f(x,y)` is a function of both
`x` and
`y`. No such function exists, so the given field is not conservative.