Answer
a) $P_1$ is a source and and $P_2$ is a sink.
b) $P_1$ is a source and and $P_2$ is a sink.
Work Step by Step
The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, $S$ is a closed surface and $E$ is the region inside that surface.
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
a) We can see at the point $P_1$ that the vectors that end near that point are shorter than the vectors that start near that point and thus the net flow is outwards, so $P_1$ is a source. On the other hand, we can see at the point $P_2$ that the vectors that end near that point are longer than the vectors that start near that point and thus the net flow is inwards, so $P_2$ is a sink.
Hence, $P_1$ is a source and $P_2$ is a sink.
b) $div F=\dfrac{\partial x}{\partial x}+\dfrac{\partial y^2}{\partial y}=1+2y$
We can see that the y-value of $P_1$ is positive thus, we have div F $\gt 0$ and so $P_1$ is a source. We can also see that the y-value of $P_2$ is less than $-1$, thus, we have div F $\lt 0$ and so $P_2$ is a sink.
Hence, $P_1$ is a source and $P_2$ is a sink.
