Answer
$\Phi = 0$
Work Step by Step
The flux of the vector field \(\mathbf{F}(x, y, z) = z^3 \mathbf{i} - x^3 \mathbf{j} + y^3 \mathbf{k}\) across the sphere \(x^2 + y^2 + z^2 = a^2\) is given by \[ \Phi = \iint \mathbf{F} \cdot \mathbf{n} \, dS = \iiint \nabla \cdot \mathbf{F} \, dV \] where \(\mathbf{G}\) is the region bounded by the sphere. Since \[ \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(z^3) + \frac{\partial}{\partial y}(-x^3) + \frac{\partial}{\partial z}(y^3) = 0 \] Using the Divergence Theorem, we have \[ \Phi = \iiint \nabla \cdot \mathbf{F} \, dV = \iiint (0) \, dV = 0 \] Finally, \[ \text{flux} = \Phi = 0 \]
