JNTU B.Tech II MATHEMATICS-II Previous Question Paper (September - 2021 - R13)

Given the scalar field φ = x³ + y³ + z³ - 3xyz, the vector field F is defined as ∇φ. We need to find the divergence of F, which is div F = ∇ · (∇φ) = ∇²φ, the Laplacian of φ.

First, calculate the partial derivatives of φ:

Now, calculate the second-order partial derivatives:

The divergence is the sum of these second partial derivatives:

To evaluate the surface integral ∬_S F · n dS where F = xzi + yzj + z²k and S is the upper hemisphere x² + y² + z² = 1 (z ≥ 0), we use the Divergence Theorem. Since S is not a closed surface, we close it with the base disk S₁ in the xy-plane (z = 0, x² + y² ≤ 1).

By the Divergence Theorem, ∬_S+S1 F · n dS = ∭_V (∇ · F) dV.

Calculate ∇ · F:

Using spherical coordinates (x = ρ sinθ cosφ, y = ρ sinθ sinφ, z = ρ cosθ), the volume integral is:

Now calculate the flux through the base S₁. On S₁, z = 0 and the outward normal n = -k. Thus, F · n = (xzi + yzj + z²k) · (-k) = -z² = 0. Therefore, ∬_S1 F · n dS = 0.

Since ∬_S F · n dS + ∬_S1 F · n dS = π, we have ∬_S F · n dS = π - 0 = π.