Problem 66 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ y^{2}-x^{2}-z^{2}=1 $$

Step-by-Step Solution

Answer

This is a hyperboloid of two sheets, symmetric about the x-z plane.

Step 1: Identify the Surface Type

The given equation is a quadratic equation in three variables: \( y^2 - x^2 - z^2 = 1 \). By comparing the equation with the standard form of a hyperboloid of two sheets \( \frac{y^2}{a^2} - \frac{x^2}{b^2} - \frac{z^2}{c^2} = 1 \), we can identify this as a hyperboloid of two sheets.

Step 2: Determine the Characteristics

In this hyperboloid of two sheets, \( a^2 = 1 \), \( b^2 = 1 \), and \( c^2 = 1 \). It opens along the y-axis because the \( y^2 \) term is positive, whereas \( x^2 \) and \( z^2 \) terms are negative.

Step 3: Sketch the Surface

To sketch the hyperboloid of two sheets, note that for any constant \( y = k \), the cross-sections are hyperbolae in the \((x, z)\)-plane. The surfaces appear as two separate sheets symmetric with respect to the x-z plane, starting from \( y = -rac{1}{2} \) and \( y = rac{1}{2} \). As \(|y|\) increases, the sheets are farther apart.

Key Concepts

Quadratic Surfaces

Quadratic surfaces are fascinating in the realm of mathematics, particularly in multivariable calculus and 3D geometry. These surfaces are defined by second-degree polynomial equations in three variables, often taking forms such as ellipsoids, paraboloids, hyperboloids, and others.

Their significance lies in their symmetrical properties and the way they can represent complex 3D objects concisely. Understanding quadratic surfaces allows one to explore various sections and projections, crucial for fields like architecture and engineering. By looking into the equation \( y^2 - x^2 - z^2 = 1 \), we understand it is specifically a hyperboloid of two sheets. Recognizing this is key to drafting precise sketches and models in practical applications.

3D Coordinate Systems

The 3D coordinate system is essential for graphing and analyzing equations involving three dimensions. It extends the familiar 2D Cartesian system with an additional z-axis. Each point in 3D space is represented as a tuple \((x, y, z)\), describing its position relative to the origin.

• The x-axis generally runs horizontally.
• The y-axis runs vertically.
• The z-axis comes out from or into the page or screen.

By placing our surfaces within this system, such as the hyperboloid of two sheets, we can analyze how changes in each variable affect the overall shape. Recognizing positive and negative correlations between variables can significantly aid in visualizing spatial changes and comprehending how equations like \( y^2 - x^2 - z^2 = 1 \) transform in 3D space.

Sketching 3D Surfaces

When sketching a 3D surface such as a hyperboloid of two sheets, it's helpful to identify the main features of its equation. For \( y^2 - x^2 - z^2 = 1 \):

• The equation suggests the surface opens along the y-axis, with the positive \(y^2\) term.
• The negative \(x^2\) and \(z^2\) terms indicate the surface extends symmetrically along these axes.
• The hyperboloid presents two separate 'sheets' or pieces.

To construct a graph, start by plotting key cross-sections for fixed values of \(y\). These cross-sections display hyperbolae in the \((x, z)\)-plane. Adjusting \(y\) informs how the sheets' positions fluctuate. Understanding these cross-sections fosters precision in sketching and aids in recognizing patterns in surface geometry.

Conic Sections

Conic sections are the curves obtained when a plane intersects a cone. They include ellipses, parabolas, and hyperbolas. Intriguingly, these curves have direct relationships to quadratic surfaces in 3D.

• For hyperboloids, like in our equation \( y^2 - x^2 - z^2 = 1 \), the intersection in vertical planes parallel to the axis of symmetry are hyperbolas.
• These hyperbolas help reveal the shape and structure of the 3D surface.
• The nature of intersections—whether they rise steeply or flatten—varies significantly based on the coefficients and signs in the surface equations.

Exploring conic sections shows us how a simple equation can unveil diverse and complex geometrical shapes, enhancing our understanding of both theoretical mathematics and practical applications in modeling and design.