Parametrization of Paraboloid over Unit Disk

Parametrize the paraboloid defined by the equation $z = x^2 + y^2$ over the closed unit disk in the xy-plane.

In this problem, we're tasked with parameterizing the surface of a paraboloid given by the equation $z = x^2 + y^2$, confined to the closed unit disk in the xy-plane. This is a common problem type in multivariable calculus to explore surface parameterizations over specific regions. The challenge lies in expressing a surface in terms of two parameters, which in this case are commonly the polar coordinates due to the nature of the region — a unit disk.

To tackle this problem, one should start by recalling that the unit disk in the xy-plane can be described using polar coordinates. This is because polar coordinates are well suited for circular regions. Therefore, by setting x equal to r times cos theta and y equal to r times sin theta, where r is between 0 and 1 and theta varies from 0 to 2 pi, we can conveniently describe the region of interest. These substitutions simplify the parameterization of both the xy-plane and the given surface.

Once the planar region is parameterized, the next step is to express the paraboloid itself in terms of r and theta. Substitute the expressions for x and y into the surface equation $z = x^2 + y^2$. Utilizing the properties of trigonometric identities and the fact that $r^2 = x^2 + y^2$ links all three parameters — r, theta, and z — together. Explore these relationships to formulate a concise parametric representation of the paraboloid. This approach not only highlights the use of polar coordinates in surface parameterization but also emphasizes how parametric equations provide a flexible means to transform and analyze geometric figures within multivariable calculus.