Divergence Theorem 68C08A

1. **Problem Statement:** State the Divergence theorem without proof. 2. **Divergence Theorem Formula:** The Divergence theorem relates the flux of a vector field through a closed surface to the divergence of the field inside the volume bounded by the surface. It is stated as: $$\iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{V} (\nabla \cdot \mathbf{F}) \, dV$$ where: - $S$ is a closed surface bounding the volume $V$. - $\mathbf{F}$ is a vector field. - $\mathbf{n}$ is the outward unit normal vector on $S$. - $\nabla \cdot \mathbf{F}$ is the divergence of $\mathbf{F}$. 3. **Explanation:** The theorem states that the total outward flux of the vector field $\mathbf{F}$ across the closed surface $S$ is equal to the integral of the divergence of $\mathbf{F}$ over the volume $V$ enclosed by $S$. This theorem is fundamental in vector calculus and is used to convert surface integrals into volume integrals and vice versa.