Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r}$, where $C$ is given by the vector function $\mathbf{r}(t)$.
\[
\begin{array}{l}
\mathbf{F}(x, y)=x y \mathbf{i}+9 y^{2} \mathbf{j} \\
\mathbf{r}(t)=14 t^{4} \mathbf{i}+t^{6} \mathbf{j}, \quad 0 \leq t \leq 1
\end{array}
\]

Step 1 ：First, we need to find the derivative of the vector function \(\mathbf{r}(t)\) with respect to \(t\), which gives us \(\mathbf{r}'(t) = \begin{bmatrix} 56t^3 \\ 6t^5 \end{bmatrix}\).

Step 2 ：Next, we substitute \(\mathbf{r}(t)\) into \(\mathbf{F}(x, y)\) to get \(\mathbf{F}(\mathbf{r}(t)) = \begin{bmatrix} 14t^{10} \\ 9t^{12} \end{bmatrix}\).

Step 3 ：We then compute the dot product \(\mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\), which results in \(54t^{17} + 784t^{13}\).

Step 4 ：Finally, we integrate this from \(t=0\) to \(t=1\), which gives us the value of the line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\).

Step 5 ：The final answer is \(\boxed{59}\).