1. We are given the vector field $F(x,y,z) = yi + zj + xk$ and the parameterizations $x = t$, $y = t^2$, and $z = t^3$. 2. Substitute the parameterizations into the vector field: $$F(t) = t^2 i + t^3 j + t k$$ 3. The user provides the answer 412/15 for this case, which likely refers to evaluating some integral or magnitude related to this vector field over a parameter interval (not explicitly stated). 4. The second vector field is $F(x,y,z) = e^x i + e^y j + e^z k$. 5. Substituting $x = t$, $y = t^2$, $z = t^3$ gives: $$F(t) = e^t i + e^{t^2} j + e^{t^3} k$$ 6. The user states the answer as $e^{2} + e^{4} + e^{8} - 3$, which suggests an evaluation or integral result at specific values of $t$ (again not explicitly given). 7. Summary: - For the first field with polynomial parameterization, the answer given is $\frac{412}{15}$. - For the exponential field with the same parameterization, the answer given is $e^{2} + e^{4} + e^{8} - 3$. Since no explicit problem such as line integral or evaluation limits was stated, this explanation focuses on substitution and the expressions of $F(t)$ for both cases.