### Problem:

Evaluate the line integral of the vector field \(\mathbf{F}(x, y, z) = y\mathbf{i} + xz\mathbf{j} + z^2\mathbf{k}\) along the curve \(C\), where \(C\) is given by the parametric equations:
\[
\mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + t^3\mathbf{k}, \quad t \in [0, 1].
\]

### Solution Steps:

1. **Find \(\mathbf{r}'(t)\):** Differentiate \(\mathbf{r}(t)\) with respect to \(t\).
2. **Evaluate \(\mathbf{F}(\mathbf{r}(t))\):** Substitute the parametric equations into \(\mathbf{F}(x, y, z)\).
3. **Compute the dot product \(\mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\).**
4. **Integrate the result with respect to \(t\) over \([0, 1]\).**

Would you like me to work out the solution or provide hints for each step?

Explore how to evaluate the line integral of the vector field \( \mathbf{F}(x, y, z) = y\mathbf{i} + xz\mathbf{j} + z^2\mathbf{k} \) along a parametric curve. Includes detailed steps for finding \( \mathbf{r}'(t) \), computing \( \mathbf{F}(\mathbf{r}(t)) \), and integrating over the interval \([0, 1]\). Suitable for advanced calculus students.

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