Suppose a vector field is given by F = (4x, −2y2, z2) and S is the surface of a cylinder definedby x2 + y2 = 4 and 0 ≤ z ≤ 3.Use a suitable volume integral given by the Divergence Theorem to show that the surfaceintegral across the total surface of the cylinder is given by 84π. You may use an appropriateJacobian without proof if stated correctly.[12 marks]2

The Divergence Theorem states that the surface integral of a vector field across a closed surface is equal to the volume integral of the divergence of the field throughout the volume enclosed by the surface.

Step 1: Compute the divergence of F. The divergence of F is given by ∇ • F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z = 4 - 4y + 2z.

Step 2: Set up the volume integral. The volume V of the cylinder is given by the region x² + y² ≤ 4, 0 ≤ z ≤ 3. We can use cylindrical coordinates (r, θ, z) where x = r cos θ, y = r sin θ, and z = z. The volume element in cylindrical coordinates is dV = r dr dθ dz. The limits of integration for r, θ, and z are 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, and 0 ≤ z ≤ 3, respectively.

Step 3: Evaluate the volume integral. ∫∫∫_V ∇ • F dV = ∫ (from 0 to 2) ∫ (from 0 to 2π) ∫ (from 0 to 3) (4 - 4r sin θ + 2z) r dr dθ dz.

Step 4: Compute the integral. After computing the integral, you should find that the surface integral across the total surface of the cylinder is 84π.