91Ó°ÊÓ

#tag_title#Step 2: Calculate the Divergence of \(\mathbf{F}\)# #tag_content#Now let's calculate the divergence of the vector field \(\mathbf{F}\). The divergence of a vector field \(\mathbf{G}\) with components \(G_x, G_y, G_z\) is given by \(\nabla \cdot \mathbf{G} = \frac{\partial G_x}{\partial x} + \frac{\partial G_y}{\partial y} + \frac{\partial G_z}{\partial z}\). Applying this to our vector field \(\mathbf{F}\), we get: $$ \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(a_yz - a_z y) + \frac{\partial}{\partial y}(a_zx - a_x z) + \frac{\partial}{\partial z}(a_x y - a_y x) $$ Differentiating with respect to each variable, we have: $$ \nabla \cdot \mathbf{F} = 0 - a_z +a_x - a_y + 0 + 0 = - (a_z + a_y) + a_x $$ #tag_title#Step 3: Apply the Divergence Theorem# #tag_content#Now that we have the divergence of \(\mathbf{F}\), we can use the divergence theorem to find the net outward flux. Recall that the divergence theorem states: $$ \iint_{S} \mathbf{F} \cdot \mathbf{\hat{n}} dS = \iiint_{V} \nabla \cdot \mathbf{F} dV $$ Since the divergence is constant (it does not depend on \(x, y, z\)), we can write: $$ \iint_{S} \mathbf{F} \cdot \mathbf{\hat{n}} dS = \iiint_{V} (-a_y - a_z + a_x) dV $$ Since we know the volume of the enclosed region \(V\), we can write this as: $$ \iint_{S} \mathbf{F} \cdot \mathbf{\hat{n}} dS = (-a_y - a_z + a_x) \cdot \mathrm{Volume}(V) $$ To summarize, the net outward flux of the vector field \(\mathbf{F}\) across any smooth closed surface in \(\mathbb{R}^{3}\) is given by the expression \((-a_y - a_z + a_x) \cdot \mathrm{Volume}(V)\).