The divergence of a vector field is a measure of how much the vector field spreads out or converges at a point. It is defined as follows:
$$\text{div}(\overrightarrow {\text{V}}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$$
where $\overrightarrow {\text{V}} = (P, Q, R)$.
In this case, we have:
$$\overrightarrow {\text{V}} = 5{\text{xy}}\overrightarrow {\text{i}} + 2{{\text{y}}^2}\overrightarrow {\text{j}} + 3{\text{y}}{{\text{z}}^2}\overrightarrow {\text{k}}$$
Therefore, the divergence of $\overrightarrow {\text{V}}$ is:
$$\text{div}(\overrightarrow {\text{V}}) = 5y + 4y^2 + 6yz$$
To evaluate the divergence at the point $(1, 1, 1)$, we substitute $x = 1$, $y = 1$, and $z = 1$ into the expression for $\text{div}(\overrightarrow {\text{V}})$. This gives:
$$\text{div}(\overrightarrow {\text{V}}) = 5(1) + 4(1)^2 + 6(1)(1) = 9$$
Therefore, the divergence of the velocity vector at $(1, 1, 1)$ is $\boxed{9}$.
Here is a brief explanation of each option:
- Option A: $9$. This is the correct answer.
- Option B: $10$. This is incorrect because the divergence of $\overrightarrow {\text{V}}$ at $(1, 1, 1)$ is $9$, not $10$.
- Option C: $14$. This is incorrect because the divergence of $\overrightarrow {\text{V}}$ at $(1, 1, 1)$ is $9$, not $14$.
- Option D: $15$. This is incorrect because the divergence of $\overrightarrow {\text{V}}$ at $(1, 1, 1)$ is $9$, not $15$.