Problem 36

Question

Describe geometrically the level surfaces for the functions. $f(x, y, z)=9 x^{2}-4 y^{2}-z^{2}$

Step-by-Step Solution

Verified

Answer

Level surfaces are cones for $ c=0 $, one-sheeted hyperboloids for $ c>0 $, and two-sheeted hyperboloids for $ c<0 $.

1Step 1: Understand the Function

The given function is $ f(x, y, z) = 9x^2 - 4y^2 - z^2 $. This is a quadratic function in the variables $ x $, $ y $, and $ z $. The form is similar to that of a quadratic in three-dimensional space commonly representing geometric shapes such as ellipsoids, hyperboloids, or paraboloids.

2Step 2: Set the Function to a Constant

To describe the level surfaces, we set $ f(x, y, z) $ equal to a constant $ c $. This yields the equation: \[ 9x^2 - 4y^2 - z^2 = c. \] Level surfaces are defined for different values of $ c $.

3Step 3: Analyze the Case $ c = 0 $

For $ c = 0 $, the equation becomes $ 9x^2 - 4y^2 - z^2 = 0 $. This surface is a cone because it can be rewritten as $ 9x^2 = 4y^2 + z^2 $, which represents a double cone opening along the x-axis in space.

4Step 4: Analyze the Case $ c > 0 $

For $ c > 0 $, the equation $ 9x^2 - 4y^2 - z^2 = c $ represents a one-sheeted hyperboloid. This is because there is more positive than negative quadratic term contribution, resulting in a hyperboloid with a connected middle section.

5Step 5: Analyze the Case $ c < 0 $

For $ c < 0 $, rearrange the equation to $ 9x^2 = 4y^2 + z^2 - c $. This equation represents a two-sheeted hyperboloid because the negative constant $ c $ shifts the balance to create two separate surfaces, one opening to positive and the other to negative x-values.

Key Concepts

Quadratic FunctionsHyperboloidsConesThree-Dimensional Geometry

Quadratic Functions

A quadratic function involves variables raised to the second power. In the realm of geometry, they play a significant role in determining the shapes of various surfaces. When we talk about a function like \[ f(x, y, z) = 9x^2 - 4y^2 - z^2 \]we are dealing with a quadratic equation in three dimensions. These functions often lead to the formation of symmetrical surfaces in geometry.

This particular function can represent shapes like ellipsoids, hyperboloids, or cones.
Quadratic functions are an essential tool for visualizing complex surfaces in 3D space.

They provide the foundational framework needed to explore the beautiful complexity of 3D geometry and the elegant forms it encompasses.

Hyperboloids

Hyperboloids come in two main types: one-sheeted and two-sheeted. In mathematical terms, a hyperboloid is generated by a quadratic equation. The equation\[ 9x^2 - 4y^2 - z^2 = c \]for various values of $ c $, illustrates how hyperboloids can be imagined as twisted surfaces that extend infinitely.

For $ c > 0 $, the resulting surface is a one-sheeted hyperboloid with a connected, saddle-like shape. It features a continuous top and bottom surface.
Conversely, when $ c < 0 $, the surface becomes a two-sheeted hyperboloid, consisting of two separate components facing away from each other.

These surfaces have unique properties, making them widely used in architecture and design for their curves and stability.

Cones

Cones in three-dimensional geometry are fascinating constructs that stem from quadratic equations. When a quadratic function like \[ 9x^2 - 4y^2 - z^2 = 0 \] is set to zero, it describes a conical surface.- This equation simplifies to \[ 9x^2 = 4y^2 + z^2 \], which depicts a double cone, with its vertex at the origin and extending symmetrically along the x-axis.- This means two pointed structures, or nappe, face opposite directions from the origin, demonstrating the distinct symmetrical properties of cones.This characteristic makes cones intriguing and visually compelling structures in both mathematical theory and practical applications ranging from optics to natural phenomena.

Three-Dimensional Geometry

Three-dimensional geometry involves the study of shapes and equations that exist in three dimensions: length, width, and height. It allows us to understand and visualize how objects fit into the space around us, using equations and coordinates.

The equation $ f(x, y, z) = 9x^2 - 4y^2 - z^2 $ influences the form and connections within this space.
By setting this function to different constants, we visualize various geometric surfaces like hyperboloids and cones in our three-dimensional world.

Understanding 3D geometry offers insights into how we can describe complex surfaces and predict their behavior when they interact with their environment. This field is crucial in many areas, including engineering, physics, and computer graphics, enhancing our capacity to design and comprehend the world.

Problem 36

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