= Solution
On the curved path, $\mathbf r(t)=(t,t,t^2)^T$ and $\mathbf r'=(1,1,2t)^T$. Direct substitution gives
$$
\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)=2t+(3a-6)t^2+4t^3,
$$
$$
\mathbf G(\mathbf r(t))\cdot\mathbf r'(t)=2t\cos(t^2)+3ct^2.
$$
Thus
$$
\boxed{\int_{C_2}\mathbf F\cdot d\mathbf r=1+(a-2)+1=a},
\qquad
\boxed{\int_{C_2}\mathbf G\cdot d\mathbf r=\sin1+c}.
$$
Both values agree with part (i), for every $a,b,c$. Agreement on these two selected paths alone does not establish a <conservative vector field>.
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