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

Let the surface be defined by the scalar function f(x, y, z) = x³ + y³ + z³ - 3. The normal vector to the surface at any point is given by the gradient vector ∇f.

At the point (1, 2, -1), the gradient vector is:

The magnitude of this vector is:

The unit normal vector n is given by ∇f / |∇f|:

According to Stoke's Theorem, the line integral ∫_C F · dr = ∫∫_S (∇ × F) · n dS, where F = e^xi + 2yj - k.

First, calculate the curl of F:

Since the curl of the vector field F is the zero vector, the surface integral of the curl over any surface S bounded by the curve C is zero.

Therefore, the value of the line integral ∫_C (e^xdx + 2ydy - dz) is 0.