JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II MATHEMATICS - II Previous Question Paper (November/December - 2020 - R13)

The work done by a force field F along a path C is given by the line integral W = ∫ F · dr. Since the force field F = 3x²i + j + zk is conservative, we can express it as the gradient of a scalar potential function φ(x, y, z) such that F = ∇φ.

By inspection, φ(x, y, z) = x³ + y + z²/2. The work done is independent of the path and depends only on the endpoints (x₁, y₁, z₁) and (x₂, y₂, z₂):

To evaluate the surface integral ∫ F · n ds over the surface S (part of the sphere x² + y² + z² = 1 in the first octant), we apply the Divergence Theorem: ∫ F · n ds = ∬ ∇ · F dV.

First, calculate the divergence of F = yzi + zxj + xyk:

Since the divergence of the vector field is zero, the integral over the closed volume bounded by the sphere and the coordinate planes is zero. The surface integral over the curved surface S plus the integrals over the three flat faces (in the xy, yz, and zx planes) must sum to zero. On the flat faces, the normal vectors are -k, -i, and -j respectively, and the dot products with F are also zero. Therefore, the integral over the curved surface S is 0.