Answer
See proof
Work Step by Step
Let $\vec{F} = (y + z)\hat{i} - xz^3\hat{j} + (x^2\sin y)\hat{k}$ be a vector field, and let $\vec{P}$ be a point in the domain. If $\vec{F}$ has a source (sink) at $\vec{P}$, then: \[ \nabla \cdot \vec{F}(\vec{P}) > 0 \quad \text{or} \quad \nabla \cdot \vec{F}(\vec{P}) < 0. \] In our case: \[ \nabla \cdot \vec{F} = \frac{\partial}{\partial x}(y + z) - \frac{\partial}{\partial y}(xz^3) + \frac{\partial}{\partial z}(x^2\sin y) = 0. \] Hence, $\vec{F}$ doesn't have sources or sinks in the entire domain.
