Problem 11 - Calculus: Early Transcendental Functions

Question

Sketch a graph of the parametric surface. \(x=u \cos v, y=u \sin v, z=u^{2}\)

Step-by-Step Solution

Answer

The given parametric equations describe an upward opening paraboloid.

Step 1: Understand the parametric equations

First of all, notice that the given parametric equations describe a surface in 3D. Here, \(u\) and \(v\) are parameters: \n- \(x = u cos v\) , \n- \(y = u sin v\) , \n- \(z = u^{2}\). \nThese parametric equations have the form of a cylindrical coordinate transformation where \(u\) is the radial distance, \(\theta\) is the angle with the x-axis, and \(z\) is the height.

Step 2: Description of the Parametric Surface

With the parametric equations, we can describe the surface. For \(u > 0\) and \(0 \leq v < 2\pi \), the formulas give us the whole of the surface. And for \(u = 0\) along with any value for v, z also becomes 0. Therefore, the parametric surface described is a paraboloid opening upward.

Step 3: Sketch the Parametric Surface

Because the parametric equations describe an upward opening paraboloid, it can be sketched as follows: Start plotting the points by letting u and v vary. It's similar to a parabola but instead of having a vertex and an axis, it has a vertex and an axis of revolution (a line through the vertex). It's bell-shaped and symmetric about the z-axis.

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height component. In this system, any point in 3D space can be described using a radius (distance from the origin), an angle, and a height. The angle is typically measured from the positive x-axis in the xy-plane. This makes cylindrical coordinates especially useful for problems involving symmetry around an axis, such as cylindrical or conical surfaces.

The transformation from cylindrical coordinates \(r, \theta, z\) to Cartesian coordinates is given by:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(z = z\)

In the given parametric surface, we can identify that the variable \(u\), often acting like a radius, and \(v\), behaving as an angle, translate directly from cylindrical coordinates. This allows us to systematically understand the spatial arrangement of the points on the surface.

Paraboloid

A paraboloid is a surface generated by a parabola revolving around an axis. It can either open upwards or downwards depending on the equation's sign. This surface is symmetrical about its axis of revolution.

In the case we explore, the paraboloid is described by the parametric equations \(x = u \cos v\), \(y = u \sin v\), and \(z = u^2\). The equation for \(z\) reveals the surface as part of an upward-opening paraboloid since \(z\) is always positive while \(u\) is non-negative. This reflects the characteristic bowl shape. The surface extends infinitely upwards as \(u\) increases.

The geometric shape is determined by letting \(u\) vary while \(v\) forms a complete circle from 0 to \(2\pi\), thereby creating a symmetric paraboloid about the z-axis. The surface is smooth, continuous, and opens infinitely as \(z\) increases.

Parametric Equations

Parametric equations describe a set of points as functions of one or more parameters. They are especially powerful in representing curves and surfaces in three dimensions.

For the equation given as a parametric surface, we use parameters \(u\) and \(v\) to map a surface in 3D space. This mapping allows us to precisely define the x, y, and z coordinates of any point on the surface.

Properties

• They smoothly and continuously trace out the surface.
• The parameters can control aspects like the radius and angular position.
• They help easily derive characteristics and behavior of the surfaces like curves, bounds, and intersections.

The specific equations \(x = u \cos v\), \(y = u \sin v\), and \(z = u^2\) show how the parametric form can create complex shapes that are otherwise difficult to express with standard Cartesian equations. Parametric equations are thus a powerful tool for visualizing and computing various three-dimensional forms.