If work is given by the $1$-form $3dx+4dy-dz$, find all points that can be reached from the origin without work

Question

I have no idea how to approach this question. There's no mention of how "work" functions in my notes. As an attempt I listed the points $$(\pm3, \pm4,\pm1)$$ but it's a guess.

Answer 1

HINT: You want points $P$ so that $\int_0^P 3\,dx+4\,dy-dz = 0$. Note first that the integral is path-independent. Why? Indeed, you can find a function $f$ so that $df = 3\,dx+4\,dy-dz$, and then the integral is equal to $f(P)-f(0)$.