Final answer:
The student's question involves evaluating a surface integral of the curl of a vector field over a surface defined by an ellipsoid. Stokes' Theorem or direct calculation with suitable coordinates should be employed to find the solution after calculating the curl of the given vector field.
Step-by-step explanation:
The student has asked to evaluate a surface integral of the curl of a vector field ℒ over a given surface S. To compute this integral, one typically uses Stokes' Theorem, which relates the surface integral of the curl of a vector field over an open surface to the line integral of the vector field around the boundary curve of the surface.
However, in cases where the vector field is complex or the surface has a simple symmetry, a direct computation might be preferred as it can simplify the process.
The given vector field is ℒ(x, y, z) = yi - xj + zx³y²k. The surface S is the portion of the ellipsoid x² + y² + 3z² = 1 that lies below the xy-plane (z ≤ 0). The integral can be calculated directly or by converting the surface integral into a line integral using Stokes' theorem.
In practical scenarios, the calculation of such integrals is often facilitated by taking advantage of symmetries and employing the appropriate polar, cylindrical, or spherical coordinates to simplify the antiderivatives. In this case, since the curl has not been provided, a direct answer to the integral cannot be given. Instead, the student would need to first compute the curl of ℒ before applying Stokes' theorem or directly integrating over S.