91Ó°ÊÓ

Parametric equations are a powerful tool in mathematics, allowing us to describe complex curves and surfaces in a more intuitive way. These equations define the coordinates of points on a curve or surface as functions of one or more parameters. For example, in the given problem, the parametric equations are:

• \( x = \sqrt{u} \cos v \)
• \( y = \sqrt{u} \sin v \)
• \( z = u \)

Here, \( u \) and \( v \) are parameters. Parametric equations are especially useful when modeling shapes in three dimensions, such as surfaces and volumes. They allow us to explore how the shape changes with different values of the parameters. This flexibility makes parametric equations a crucial tool in engineering, computer graphics, and physics.

The Pythagorean identity is a fundamental concept in trigonometry and plays a crucial role in simplifying parametric equations. Specifically, it states that:\[\cos^2 v + \sin^2 v = 1\]This identity is derived from the Pythagorean Theorem and works for any angle \( v \). In the context of solving the given problem, the identity helps eliminate the parameter \( v \) from the parametric equations for \( x \) and \( y \):

• By squaring \( x = \sqrt{u} \cos v \), we get \( x^2 = u \cos^2 v \).
• Squaring \( y = \sqrt{u} \sin v \) gives \( y^2 = u \sin^2 v \).
• Adding both equations, \( x^2 + y^2 = u(\cos^2 v + \sin^2 v) = u \).

This simplification using the Pythagorean identity is key to converting parametric equations into rectangular coordinates. It enables us to find the relationship between \( x \), \( y \), and \( z \) without the parameters, ultimately leading to the equation \( z = x^2 + y^2 \).

A paraboloid is a quadratic surface that can either open upward or downward, depending on its specific equation. In the exercise, we derive the equation of a paraboloid by transforming the parametric equations into rectangular coordinates, leading to:\[z = x^2 + y^2\]This indicates an upward-opening paraboloid with its vertex at the origin (0, 0, 0). Paraboloids have the unique geometric property where all cross-sections parallel to the xy-plane are circles, and sections parallel to the xz or yz planes are parabolas. For the derived paraboloid:

• The base at \( z = 0 \) is a circle with radius \( \sqrt{u} \). However, because \( u \leq 4 \), the paraboloid is confined to \( 0 \leq z \leq 4 \).
• At different heights \( z \), the radius of the cross-sectional circles increases as \( \sqrt{z} \).

Understanding paraboloids is essential in various fields, such as optics, where parabolic mirrors focus light, and in engineering structures influenced by quadratic shapes.