Answer
The flux across the sphere is zero.
Work Step by Step
Divergence Theorem: It states that the flux across the boundary of $R$ is equal to the net contraction or expansion of the vector field $F$ within the region $R$.This implies that
$\int\int F \cdot n d C=\int\int\int \nabla \cdot F dR$
Here, $R$ defines the connected and region in space which is oriented by boundary $C$ and n denotes the outward flux unit normal vector on boundary $C$.
In order to compute the net outward flux of the given vector field, we will firstly compute $\nabla \cdot F$.
Now, we have:
$\nabla \cdot F=\lt \dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z} \gt \cdot \lt 2x-y, x, -2x\gt\\=\dfrac{\partial}{\partial x}(2z-y) +\dfrac{\partial}{\partial y}(x)+\dfrac{\partial}{\partial z}(-2x)\\=0$
Thus, $\int\int F \cdot n d C=\int\int\int \nabla \cdot F dR=0$
This implies that the flux across the sphere is zero by using the divergence theorem.
