Let $F(x,y,z)=(f_x +xye^{xz^2},f_y + x^2e^{xz^2},f_z + 2x^3ye^{xz^2})$ where $f:\mathbb{R}^3 \to \mathbb{R}$ is a $C^1$ function such that $f(x,y,z)=f(-x,-y,-z)$. Compute the work of $F$ through the curve parametrized by $\sigma:[0,\pi]\to \mathbb{R}^3, \sigma(t)=(\sin t (1+e^{t^2}),\cos t, t(\pi-t)^{4/3})$.
I'm incapable of solving this exercise. I'd be grateful to anyone who can provide any hint whatsoever.