Determine Rectangular Equation from Parametrization

Given the parametrization x = u, y = v, and $z=\sqrt{u^2 + v^2}$, determine the rectangular equation.

In this problem, you are given a parametrization of a surface in three-dimensional space and asked to find the corresponding rectangular equation. This requires an understanding of the relationship between parametric and rectangular forms of equations, where parametric equations are defined using parameters, often to represent curves or surfaces, and rectangular equations are expressed in terms of traditional Cartesian coordinates.

A critical step in converting from a parametric form to a rectangular form is eliminating the parameters by expressing the parametric equations in terms of just the Cartesian coordinates x, y, and z. This often involves algebraic manipulation to solve for one variable in terms of the others. Here, the parametrization also includes a square root, reminding us of the potential appearance of common quadratic expressions or distances, such as using the Pythagorean theorem in three dimensions.

This problem helps reinforce the understanding of how surfaces can be represented in different ways, and building the skill to transition between these forms is essential, especially in fields such as physics and engineering, where different forms may provide substantial insights into the nature of the surfaces in question. The challenge lies in recognizing the geometric form that the parametric equation describes and translating that into the appropriate Cartesian form, often looking out for familiar structures like spheres, cylinders, or planes within the equations.

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