Problem 5 - Thomas Calculus

Question

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.). The surfaces are labeled ( a \(-(1)\) . $$x=y^{2}-z^{2}$$

Step-by-Step Solution

Answer

The equation defines a hyperbolic paraboloid surface.

Step 1: Identify the Form

The given equation is \( x = y^2 - z^2 \). We need to recognize the form and pattern this equation fits into. This equation has the form \( x = Ax^2 - Ay^2 \), which resembles a hyperbolic paraboloid.

Step 2: Compare with Standard Form

A standard hyperbolic paraboloid can be written as \( z = x^2/a^2 - y^2/b^2 \). In our equation, \( x = y^2 - z^2 \), the roles of \( x \), \( y \), and \( z \) are rearranged compared to the standard form. However, it maintains the characteristic one-way subtraction between squares, confirming it's a hyperbolic paraboloid.

Step 3: Identify the Surface Type

Based on the comparison with the standard form, the surface defined by the equation \( x = y^2 - z^2 \) is a hyperbolic paraboloid. Hyperbolic paraboloids are saddle-shaped surfaces.

Key Concepts

Quadratic Surfaces

Quadratic surfaces are geometric objects in three-dimensional space that are defined by quadratic equations. A quadratic equation in three dimensions typically involves variables \(x, y,\) and \(z\) raised to the second power, combined together. Common examples include an equation like \(Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0\). Quadratic surfaces can take various forms, and each form corresponds to a different type of surface. This could include:

• Ellipsoids
• Hyperboloids (one-sheet or two-sheet)
• Paraboloids (elliptic or hyperbolic)
• Cylinders and cones

Quadratic surfaces are defined by their specific mathematical properties and are used to model various physical phenomena. In engineering and physics, they describe things like lenses, thermal distribution patterns, and stress surfaces. Understanding these surfaces requires recognizing these quadratic forms and their characteristics.

Surface Identification

Identifying a surface from its equation involves recognizing the equation's structure and comparing it with standard forms of known surfaces. The standard form provides a template that indicates the unique shape the equation represents. For instance, comparing the equation \(x = y^2 - z^2\) from our example, with the standard form \(z = x^2/a^2 - y^2/b^2\), reveals it as a hyperbolic paraboloid. This identification hinges on:

• Recognizing the one-way subtraction of squares, a unique trait of hyperbolic paraboloids.
• Understanding how shifting the roles of variables still fits a standard form.

A hyperbolic paraboloid is often referred to as a saddle surface due to its distinct upward and downward curvatures – much like a horse's saddle.
Accurate surface identification is crucial in fields such as architecture and graphic design, where specific shapes need to be recreated or analyzed for stability and aesthetic appeal.

Equations of Surfaces

The equations of surfaces can tell us a lot about the shape and properties of a 3D object. By carefully analyzing a surface's equation, you can determine whether it's curved, flat, symmetrical, or asymmetrical. Consider the general format of a surface equation like \(x = y^2 - z^2\):

• It includes terms where the variables are squared, such as \(y^2\) and \(z^2\), indicating curvature.
• The presence of subtraction between these terms (\