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Chapter 12: Problem 28

Describe geometrically the domain of each of the indicated functions of three variables. \(f(x, y, z)=\sqrt{x^{2}+y^{2}-z^{2}-9}\)

Short Answer

Expert verified
The domain is the set of points outside or on the hyperboloid of two sheets given by \(x^2 + y^2 - z^2 = 9\).

Step by step solution

01

Understand the function

The function given is \(f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9}\). We need to determine for which values of \(x\), \(y\), and \(z\) this function is defined. A square root function is defined when the expression inside the square root is non-negative.
02

Set up the inequality

The function \(f(x, y, z)\) will be defined when the inside of the square root is non-negative, i.e., \(x^2 + y^2 - z^2 - 9 \geq 0\). This simplifies to \(x^2 + y^2 - z^2 \geq 9\).
03

Interpret the inequality

The inequality \(x^2 + y^2 - z^2 \geq 9\) describes a region in three-dimensional space. Geometrically, it represents a set of points \((x, y, z)\) such that the subtraction of the square of the \(z\)-coordinate from the sum of the squares of the \(x\) and \(y\) coordinates is at least 9.
04

Identify the surface

The surface described by \(x^2 + y^2 - z^2 = 9\) is a hyperboloid of two sheets. The inequality \(x^2 + y^2 - z^2 \geq 9\) includes all points outside or on this hyperboloid.
05

Geometrical domain

Therefore, the domain of the function is the set of all points outside or on the hyperboloid of two sheets described by \(x^2 + y^2 - z^2 = 9\) in three-dimensional space.

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Key Concepts

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

Hyperboloid
A hyperboloid is a fascinating geometrical shape in three-dimensional space. Imagine it as a surface that curves outward and adds a complex twist to simple cylinders or spheres. There are two main types of hyperboloids鈥攐ne-sheeted and two-sheeted. The equation given in the exercise is for a two-sheeted hyperboloid. Here's why:
  • Its equation is of the form: \(x^2 + y^2 - z^2 = 9\).
  • The plus signs corresponding to \(x^2\) and \(y^2\) mean that in those directions, the surface opens outwards.
  • The minus sign with \(z^2\) shows that the sheets open away from each other along the z-axis.
This configuration crafts what's known as a hyperboloid of two sheets. A helpful way to think about it is like two "bowls" facing away from one another, forming a top and bottom open structure.
Three-dimensional space
Three-dimensional space is where length, width, and height are all available for an object to move and exist. It is where most objects in our world exist, unlike a two-dimensional space that limits them to just a flat plane. In mathematics,
  • the points in space are identified using three numbers, known as coordinates: \((x, y, z)\).
  • 'x', 'y', 'z' axes all intersect at the origin point, designated by \((0, 0, 0)\).
  • This coordinate system helps to locate any point or shape within this 3D environment.
In this problem, the hyperboloid exists in three-dimensional space. It uses the x, y, and z coordinates to describe its shape and position. Understanding this setup is crucial for visualizing the object in space and determining how it might interact with other shapes or concepts.
Domain of functions
In calculus, the domain of a function is the set of all possible inputs for which the function is defined. For functions involving square roots, like the one in the exercise, the expression under the square root must be non-negative so the domain consists only of points where this condition is true. Thus,
  • The inequality \(x^2 + y^2 - z^2 \geq 9\) outlines the domain.
  • This means every point \((x, y, z)\) either on or outside the hyperboloid satisfies this inequality.
  • These points provide valid input values that ensure the function has real-number outputs.
Understanding domains helps to predict and chart where certain functions can be calculated, ensuring that computational attempts won't run into undefined territory. Knowing the domain creates a safer mathematical environment for exploration.
Functions of three variables
Functions of three variables are an extension of the familiar functions of one or two variables, providing a means to explore more complex relationships. A three-variable function is often written in the form \(f(x, y, z)\), showcasing how it takes three distinct inputs:
  • Each input corresponds to one of the three dimensions: x, y, or z, affecting the resulting outcome of the function.
  • The interplay between these variables is what gives rise to intricate surfaces or shapes, like the hyperboloid in this exercise.
  • Such functions allow the mapping of diverse conditions and help describe real-world phenomena that depend on three independent variables.
Visualizing these functions requires thinking in three dimensions, often using graphs or simulations that show how changes in x, y, or z influence the function's results. This insight is handy in fields ranging from physics to engineering, where variables often weave together in complex ways.

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Most popular questions from this chapter

Find the minimum distance from the origin to the line of intersection of the two planes $$ x+y+z=8 \text { and } 2 x-y+3 z=28 $$

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