please help with this! Use the Divergence Theorem to calculate the flux of the vector fields F₁ = −4zi + 4xk and F2 = −4zi + (1 − 4y)j + 4xkthrough the surface S given by the sphere of radius a centered at the origin with outwards orientation. Be sure that you are able toexplain your answers geometrically.With W giving the interior of the sphere,Ss¹[§ F₁ · dÃ = √wdV ==and√s F₂ · dÃ = √w dv==

Given That : Vector Fields are F1→=-4zi^+4xk^ and F2→=-4zi^+1-4yj^+4xk^ To Find : Using the divergence theorem calculate the flux of the given vector field through a surface S of radius centered at the origin.Considered the given vector fields F1→=-4zi^+4xk^ . a) ∫∫SF→1dA→=∫∫∫W∇·F1→dV=∫∫∫W∂∂xi^+∂∂yj^+∂∂zk^·-4zi^+4xk^dV=∫∫∫W∂∂x-4z+∂∂y0+∂∂z4xdV=∫∫∫W0+0+0dV=∫∫∫W0dV=0 ⇒∫∫SF→1dA→=∫W0dV=0 Here W is the interior of the sphere that is it denotes the volume of the sphere of radius a. The flux of the vector field F→1 through the sphere of radius a is zero and the flux across the surface S is the volume of the fluid crossing per unit of time. Therefore, the rate of change of the volume of the fluid crossing S per unit of time is zero. b) Considered the given vector field F2→=-4zi^+1-4yj^+4xk^ . ∫∫SF→2dA→=∫∫∫W∇·F2→dV=∫∫∫W∂∂xi^+∂∂yj^+∂∂zk^·-4zi^+1-4yj^+4xk^dV=∫∫∫W∂∂x-4z+∂∂y1-4y+∂∂z4xdV=∫∫∫W-4dV=-4∫∫∫WdV=-4V Since V is the volume of the sphere of radius a which is equal to V=43πa3. ∫∫SF→2dA→=-4V=-443πa3=-163πa3 ⇒∫∫SF→2dA→=-4∫WdV=-163πa3 Here, the flux is negative. The flux of the vector field F→2 through the sphere of radius a is in the downward direction is -163πa3. This means the negative amount of fluid crosses the surface S in the downward direction.