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Chapter 12: Problem 57

Is the statement true or false? Give reasons for your answer. Any surface which is the level surface of a threevariable function \(g(x, y, z)\) can also be represented as the graph of a two-variable function \(f(x, y)\)

Short Answer

Expert verified
False: Not every level surface can be expressed as a graph of a two-variable function.

Step by step solution

01

Understanding Level Surfaces

A level surface of a three-variable function \( g(x, y, z) \) is a set of points \((x, y, z)\) such that \( g(x, y, z) = c \), where \( c \) is a constant. This surface is a collection of all points in 3D space that satisfy this equation.
02

Concept of Graph of a Two-Variable Function

A graph of a two-variable function \( f(x, y) \) represents points \((x, y, f(x, y))\) in 3D space. This means for every \((x, y)\), there is a unique \( z \, (= f(x, y))\) value.
03

Analyzing Relationship

To represent a level surface \( g(x, y, z) = c \) as \( f(x, y) = z \), every combination \( (x, y) \) must yield a single \( z \) value. However, level surfaces can have multiple \( z \) values or different complex geometries (e.g., spheres, ellipsoids) for the same \( (x, y) \), which do not satisfy the function graph criterion.
04

Conclusion of Verification

Since a level surface often does not provide a unique \( z \) for each \( (x, y) \), and can represent more complex shapes, it cannot always be expressed as the function graph \( z = f(x, y) \) of two variables. Therefore, the statement is false.

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Key Concepts

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

Level Surfaces
Level surfaces are a crucial concept in multivariable calculus. They are essentially the collection of all points \((x, y, z)\) in three-dimensional space that satisfy \(g(x, y, z) = c\), where \(c\) is a constant. To visualize level surfaces, imagine cutting through a three-dimensional object at different heights. Each cross-section represents a level surface, akin to contour lines on a topographic map that signify elevation. Level surfaces can be diverse in shape, including planes, spheres, or more complex forms. They help us understand the structure of three-dimensional functions by showing how values of \(g\) are distributed across space.
Three-Variable Function
A three-variable function \(g(x, y, z)\) takes three input values and produces a single output. Such functions describe phenomena that depend on three independent variables, such as temperature variations in a room, which can change with position and height. Generally, these functions are represented in three-dimensional space, and their level surfaces provide insight into regions of equal value, much like isotherms on a weather map. Understanding these functions is vital for describing and working with three-dimensional models in physics, engineering, and more.
Two-Variable Function
Two-variable functions \(f(x, y)\) are calculated based on two inputs to give a single output. These functions are often depicted in three-dimensional space, where each \((x, y)\) point on a plane gets lifted to form the third dimension, \(z = f(x, y)\). This setup creates a surface known as the graph of the function.In simpler terms, think of this as building a landscape where the base is the \(xy\)-plane, and the heights at any given point are determined by the value of the function \(f(x, y)\). Understanding two-variable functions is essential as they build the foundation for exploring more complex scenarios like those with three variables.
3D Space
Three-dimensional space, frequently abbreviated as 3D space, constitutes the framework where we can visualize three-variable functions. In 3D space, every point is defined by three coordinates: \(x\), \(y\), and \(z\). By understanding 3D space, we stretch beyond the confines of flat, two-dimensional behavior and explore the complex interactions that occur when three dimensions are involved.
  • **Coordinate System**: Typically represented by Cartesian coordinates \((x, y, z)\).
  • **Visualizing Shapes**: 3D space allows us to graph surfaces and understand their interactions.
  • **Applications**: Used in fields like physics, engineering, and computer graphics to model real-world phenomena.
3D space naturally fuses with three-variable functions, providing a canvas to represent and solve multifaceted problems.
Function Graph
A function graph in multivariable contexts typically represents the relationship between input and output variables. For a function \(f(x, y)\), the graph is a surface in 3D space where each point \((x, y)\) is mapped to its height \(z = f(x, y)\).This graphical representation provides:
  • **Intuitive Understanding**: Easily visualize how variables affect the output.
  • **Connections**: Highlights relationships among variables and their collective behavior.
  • **Analysis Tool**: Facilitates examination of functions, derivatives, and integrals in multivariable calculus.
In conclusion, while the graph of a two-variable function is a structured surface, it cannot capture the full complexity of level surfaces from three-variable functions. These differences underscore the intricate nature of multivariable calculus.

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Most popular questions from this chapter

Are the statements true or false? Give reasons for your answer. If \(H=f(t, d)\) is the function giving the water temperature \(H^{\circ} \mathrm{C}\) of a lake at time \(t\) hours after midnight and depth \(d\) meters, then \(t\) is a function of \(d\) and \(H\)

Match the functions (a)-(d) with the shapes of their typical level curves (I)-(IV). A. $$\quad f(x, y)=\frac{y}{x^{2}+1}$$ B. $$\quad f(x, y)=\frac{1}{x^{2}+2 y^{2}}$$ C. $$f(x, y)=\frac{x^{2}+1}{y^{2}+1}$$ D. $$f(x, y)=\frac{x}{x^{2}+y^{2}+1}$$ I. Circles II. Parabolas III. Hyperbolas IV. Ellipses

Describe in words the level surfaces of the function \(g(x, y, z)=x+y+z\)

Give an example of: A function \(f(x, y, z)\) whose level sets are paraboloids.

Explain what is wrong with the statement. In 3 -space, \(y=1\) is a line parallel to the \(x\) -axis.
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