Problem 51 - Calculus Volume 3

Question

Find the level surface for the functions of three variables and describe it. $$w(x, y, z)=x^{2}+y^{2}-z^{2}, c=4$$

Step-by-Step Solution

Answer

The level surface is a hyperboloid of one sheet centered at the origin, given by \( \frac{x^2}{4} + \frac{y^2}{4} - \frac{z^2}{4} = 1 \).

Step 1: Understand the Given Function

The function given is \( w(x, y, z) = x^2 + y^2 - z^2 \). We need to find the level surface for this function when \( w = c \), where \( c = 4 \). This means the equation that describes the level surface is \( x^2 + y^2 - z^2 = 4 \).

Step 2: Relate to Standard Forms

The equation \( x^2 + y^2 - z^2 = 4 \) is similar to the standard form of a hyperboloid. In particular, this resembles the equation of a hyperboloid of one sheet, which is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \). Here, \( a = b = 2 \) and the completed equation becomes \( \frac{x^2}{4} + \frac{y^2}{4} - \frac{z^2}{4} = 1 \).

Step 3: Analyze the Level Surface

The equation \( \frac{x^2}{4} + \frac{y^2}{4} - \frac{z^2}{4} = 1 \) describes a hyperboloid of one sheet centered at the origin. The constants \( 4 \) in the denominators of \( x^2 \) and \( y^2 \) represent half the lengths of the axes in the \( x \) and \( y \) directions. The negative sign in front of \( z^2 \) indicates that the hyperboloid is oriented along the \( z \)-axis.

Key Concepts

Level Surfaces

Level surfaces are a fascinating concept in multivariable calculus. These surfaces represent the set of points in space where a function of three variables is constant. For example, with the function \( w(x, y, z) = x^2 + y^2 - z^2 \), a level surface is where \( w = c \). This creates an equation \( x^2 + y^2 - z^2 = c \), providing a snapshot of the function's behavior at a specific value of \( c \).
When thinking about level surfaces, it's like slicing through a 3D shape to reveal a cross-section that is constant at that specific level. These are crucial for understanding contours and shapes in a more advanced way. Level surfaces help visualize and analyze the topology of the function, giving insight into its growth or decay across three axes.

Hyperboloids

Hyperboloids are 3D surfaces that have a unique saddle-like shape. They are a type of quadric surface that can appear in various forms, such as hyperboloids of one sheet or two sheets.
For the equation \( x^2 + y^2 - z^2 = 4 \), we have a hyperboloid of one sheet. This happens because the terms \( x^2 \) and \( y^2 \) both have positive coefficients, while \( z^2 \) has a negative coefficient. By standardizing the equation as \( \frac{x^2}{4} + \frac{y^2}{4} - \frac{z^2}{4} = 1 \), we confirm this surface type.
In mathematics and engineering, hyperboloids serve various practical purposes. They can be seen in cooling towers, structural designs, and antenna dishes, thanks to their ability to handle stress and disperse signals effectively.

Equations of Surfaces

Understanding the equations of surfaces allows us to describe and analyze complex shapes in three dimensions. Such equations set constraints on points \( (x, y, z) \) in space to form elaborate and often symmetric shapes. The standard form of hyperboloid equations is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \), which imposes specific conditions on the geometric properties of the surface.
In our case, \( x^2 + y^2 - z^2 = 4 \) is an example of converting a simple equation into its standard form to understand its characteristics better. The equation tells us about the orientation, symmetry, and scaling of the surface. Such mathematical constructs are vital tools in physics, computer graphics, and architecture, where precise modeling of surfaces is required.