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Chapter 9: Problem 12

What kind of surfaces are the level surfaces \(f(x, y, z)=\) const? $$f=4 y^{2}-z$$

Short Answer

Expert verified
Level surfaces are parabolic cylinders parallel to the x-axis.

Step by step solution

01

Understanding the Given Function

The function is given as \( f(x, y, z) = 4y^2 - z \). This function is defined in three-dimensional space, involving variables \( x \), \( y \), and \( z \). However, notice that \( f \) does not explicitly contain the variable \( x \), indicating that the level surfaces are parallel to the \( x \)-axis.
02

Setting Up the Level Surface Equation

A level surface for a function of three variables is defined by setting \( f(x, y, z) = c \), where \( c \) is a constant. In this case, the level surface equation becomes \( 4y^2 - z = c \).
03

Rearranging the Equation

To understand the type of surface, rearrange the equation to solve for \( z \): \[ z = 4y^2 - c \]. This rearranged equation reveals the shape of the level surface.
04

Analyzing the Surface Equation

The equation \( z = 4y^2 - c \) represents a paraboloid. Specifically, it is a parabolic cylinder opening along the \( z \)-axis since the equation contains only \( y \) and \( z \). The surface is translated along the \( z \)-axis by the constant \( c \).
05

Conclusion on Level Surfaces

Since the given function does not depend on \( x \), every level surface is a parabolic cylinder parallel to the \( x \)-axis. Each value of \( c \) alters the translation along the \( z \)-axis, but they maintain the same parabolic shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabolic Cylinder
A parabolic cylinder is a type of three-dimensional surface that can be visualized by considering the equation of a parabola that exists in a two-dimensional plane. When this parabola extends infinitely along a third dimension, it forms a cylindrical structure, hence the name "parabolic cylinder". In our case, the equation is given as \( z = 4y^2 - c \). Let's break down this equation:
  • The term \( 4y^2 \) represents a parabola opening along the \( z \)-axis. This parabola is only affected by changes in the \( y \) and \( z \) coordinates.
  • The \( x \) variable is absent, indicating the cylinder extends parallel to the \( x \)-axis.
  • Altering the constant \( c \) translates the parabola up or down the \( z \)-axis, but keeps the same parabolic shape.
With this insight, we understand that a parabolic cylinder maintains a constant cross-sectional shape and extends infinitely along one direction. Its distinguishing feature is the parabolic cross-section.
Three-Dimensional Space
Three-dimensional space is the framework used to describe locations or dimensions involving three parameters: commonly known as \( x \), \( y \), and \( z \). Understanding three-dimensional space is fundamental in this exercise because it defines where the level surfaces of a function like \( f(x, y, z) = 4y^2 - z \) reside.
  • Each point in this space can be defined by three coordinates: \( x \), \( y \), and \( z \).
  • Level surfaces in three-dimensional space are objects where function values remain constant, which is essential for comprehending how these surfaces manifest.
  • In this scenario, even though the function involves three variables, it varies with only \( y \) and \( z \), indicating independence from \( x \).
Visualizing these concepts allows us to understand how level surfaces, like parabolic cylinders, can exist in such spaces and how they interact with each other along different axes.
Level Surface Equation
A level surface equation establishes a relationship that allows for identifying surfaces within three-dimensional space where the function values remain constant. Specifically, it involves setting the function equal to a specific constant value, \( f(x, y, z) = c \). For our example, this means setting \( 4y^2 - z = c \).
  • By solving this equation, we determine the specific shapes and orientations of level surfaces.
  • In the problem's context, the equation \( z = 4y^2 - c \) reveals parabolic cylinders, which are the level surfaces defined by the constant \( c \).
  • The constant \( c \) shifts the surface up or down along the \( z \)-axis, without affecting its parabolic form.
The level surface equation is a powerful tool for grasping how changes in the equation parameters translate into tangible geometric surfaces within three-dimensional environments. It provides a direct pathway to visualizing and understanding complex geometrical relationships and shapes.

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Most popular questions from this chapter

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