Solution
= Solution
For $\mathbf r(t)=(t,t,2t^3)$, the conservative field is $(6t^7,4t^7,t^5)$ and the tangent is $(1,1,6t^2)$. Therefore
$$
\boxed{\int_A^B\mathbf F_1\cdot d\mathbf r=\int_0^1 16t^7\,dt=2}.
$$
The two different paths give the same value because their <line integrals> depend only on the potential difference $\phi_1(B)-\phi_1(A)=2$. There is no <scalar potential> for $\mathbf F_2$ to which the conditional request could be applied.