Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ z^{2}=3 y $$

Step-by-Step Solution

Answer

The graph is a parabolic cylinder, \( z^2 = 3y \), extending along the x-axis.

Step 1: Identify the Type of Surface

The given equation is \( z^2 = 3y \). Recognize this as a parabolic cylinder in three-space. The equation represents a parabola that opens along the y-axis due to the square term \( z^2 \), while the lack of an x-term indicates that it is constant for every x-value, hence forming a cylinder.

Step 2: Analyze the Shape in 2D

To understand the surface, first consider the equation \( z^2 = 3y \) in the yz-plane (where x=0). This is a parabola opening in the positive y-direction with vertex at the origin (0,0).

Step 3: Extend to Three Dimensions

In three dimensions, the two-dimensional parabola from the yz-plane extends infinitely along the x-axis, forming a parabolic cylinder. Every cross-section parallel to the yz-plane remains a parabola.

Step 4: Sketch the Graph

Start by sketching the parabola \( z^2 = 3y \) on the yz-plane. Then, extend the parabola along the x-axis to create the cylindrical shape. Each plane parallel to yz and at a different value of x will also show a parabola of the same shape.

Key Concepts

Parabolic Cylinder

A parabolic cylinder is a fascinating geometrical structure that emerges from quadratic equations. It's called a cylinder not because it looks like a round pipe but because it retains its shape along one of the axes, specifically the x-axis in this case. The defining characteristic of a parabolic cylinder comes from its equation, such as \( z^2 = 3y \). Here’s what makes it unique:

• The squared term \( z^2 \) signifies that its cross-section is a parabola.
• Since the equation lacks an x-term, the parabola is stretched infinitely along the x-axis.
• Each slice parallel to the yz-plane is identical, showing the same parabolic shape.

Due to these properties, a parabolic cylinder is often visualized as a surface that appears to "roll out" a parabola along one direction.

Three-space

Three-space, or 3D space, allows for greater exploration of complex shapes than two-dimensional space. It's better understood with coordinates (x, y, z), representing width, depth, and height.
In the context of three-space, the equation \( z^2 = 3y \) helps us visualize a shape that extends along the x-axis. Think about how:

• Each point in space is represented by its three coordinates, making shapes more dynamic.
• We can understand the complex unity of shapes, like cylinders, as they stretch through different planes.
• Three-space enables precise sketching and better realization of mathematical surfaces.

This space is crucial in fields such as physics and engineering, where understanding of spatial dimensions is key to solving real-world problems.

Graph Sketching

Graph sketching is a visual method to bring equations to life. For mathematicians and students alike, it is an essential skill to interpret abstract formulas into visual representations.
Consider our parabolic cylinder example, \( z^2 = 3y \):

• Begin by sketching its 2D form in the yz-plane \( z^2 = 3y \), visualizing it as a basic parabola.
• Recognize the absence of an x-term which implies stretching the shape along the x-axis to form the complete three-dimensional structure.
• Imagine slicing through the surface at any x-value, still seeing a parabolic 2D slice.

This process requires logical steps to check how equations transition from abstract ideas to tangible forms, providing clarity, especially in learning complex mathematical concepts.

Mathematical Surfaces

Mathematical surfaces encompass a wide range of shapes residing in three-space. They include everything from simple planes to more complicated forms like our parabolic cylinder.

• Surfaces are represented by equations, which define their shape and extent.
• Understanding these surfaces requires recognizing their inherent properties, like symmetry, curvature, and how they interact with axes.
• Our parabolic cylinder case reveals a continuous surface stretched infinitely across one axis, showcasing the interaction of algebra with geometry.

These surfaces are crucial not just in theoretical examinations, but in practical applications like architecture and computer graphics, where the math defines how objects will be visualized and manipulated.