Question

Use Stoke's Theorem to evaluate / 2xy²zdx + 2x²yzdy + (x²y² – 2z)dz where C is the curve given by X = cost, y = sint, z = sint, 0 < t < 27 traversed in the direction of increasing t.

Answer 1

The question involves using Stoke's Theorem to convert a line integral around curve C into a surface integral over the surface bounded by C. Calculation of the curl of the given vector field is necessary, leading to evaluating the corresponding surface integral.

Step-by-step explanation:

The student is asking to evaluate a line integral using Stoke's Theorem. This involves converting the line integral around a closed curve C into a surface integral over the surface S bounded by C. The given curve is parametrized by X = cos(t), y = sin(t), z = sin(t) for 0 < t < 2π. Stoke's Theorem states that the line integral of a vector field F around a closed curve C is equal to the surface integral of the curl of F over the surface S bounded by C when traversed in the specified direction.

To apply Stoke's Theorem, we first need to calculate the curl of the vector field F given by its components F = (2xy²z, 2x²yz, x²y² – 2z). After calculating the curl, we have to evaluate the surface integral over the surface bounded by the curve C. Because C is given by a parametrization rather than a specific surface, we would need to use the parametrization to describe the surface S for this purpose.