Solution
= Solution
For $\mathbf r(t)=(t,t^2,t^2)$, the velocity is $(1,2t,2t)$. Because $y=z$ throughout this path, both fields again agree pointwise, now as $(3t^8,2t^7,t^7)$. Thus
$$
\boxed{\int\mathbf F\cdot d\mathbf r=\int\mathbf G\cdot d\mathbf r=\int_0^1(3t^8+4t^8+2t^8)dt=1.}
$$
These equal values alone do not establish <path independence> for either field on all of space.