Parametric Representation of Surface Above Cone

Find the parametric representation of the surface which is the part of the sphere $x^2 + y^2 + z^2 = 4$ that lies above the cone with equation $z = \sqrt{x^2 + y^2}$.

When tasked with finding the parametric representation of a surface, especially one defined as the intersection or section of more fundamental geometric shapes, it is crucial to understand both the parametric and implicit forms of those shapes. In this instance, the broader challenge involves a sphere and a cone, two classical surfaces in multivariable calculus. Recognizing that both the sphere and cone are symmetric shapes will aid in leveraging their respective equations effectively. The sphere, given by all points equidistant from a center, and the cone, defined similarly by its equation, interact in three-dimensional space at specific features that may lead to a simpler form in polar or spherical coordinates.

The parametric representation transforms these standard forms into a function of one or two parameters—often angles when dealing with spheres and cones. The problem is simplified dramatically by shifting from Cartesian coordinates to spherical coordinates, especially given the spherical symmetry. The conditions given by the boundaries, like the part of the sphere above the cone, suggest specific limits on the parameters that outline the surface fully.

The fundamental advantage of a parametric form is how it allows one to describe complicated geometric shapes succinctly. Solving such problems typically involves setting up the range of parameters that satisfy both the sphere and cone conditions. In practical terms, this means deriving the radial and angular limits or expressions that translate the intersection of the surfaces. The goal is to express every point on the desired surface in terms of these parameters, delivering a concise description in a multidimensional space.