This is an unsolved exercise given in my textbook, which I am having trouble with. The exercise seems simple, but for some reason I can't solve it. Help would be very nice!
Let $o(t)$ be a real continuous positive function in $[0,\infty)$, and $F(x,y,z)=o(\|(x,y,z)\|)(x,y,z)$ ($\|\cdot \|$ means the norm, e.g. $\sqrt{x^2+y^2+z^2}$). We need to prove that $F$ has a vector potential in $\mathbb{R}^3$.
Attempt: If $G$ is the potential, we know $G_x = o(\|(x,y,z)\|)x$, so I figured $G$ could be of the form $G(x,y,z)=\int_{0}^{x} o(\|(t,y,z)\|)t dt + h(y,z)$. So now, to find $h(y,z)$ I need to solve: $ G_y(x,y,z)=yo(\|(x,y,z)\|)=(\int_{0}^{x} o(\|(t,y,z)\|)t dt + h(y,z))_y $ and $ G_z(x,y,z)=zo(\|(x,y,z)\|)=(\int_{0}^{x} o(\|(t,y,z)\|)t dt + h(y,z))_z. $ But I don't really know how to differentiate $\int_{0}^{x} o(\|(t,y,z)\|)t$ by $y$ or $z$, which gets me stuck.
Thanks for reading. Please help me figure this out!