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Chapter 12: Problem 29

Sketch the surfaces in Exercises \(13-44.\) HYPERBOLOIDS $$z^{2}-x^{2}-y^{2}=1$$

Short Answer

Expert verified
The surface is a hyperboloid of one sheet centered on the origin, opening along the \(z\)-axis.

Step by step solution

01

Identify the Surface Type

The given equation is \(z^2 - x^2 - y^2 = 1\). This is the standard form of a hyperboloid of one sheet. It is characterized by one variable being positive and the others being negative, with the right side equal to a positive constant.
02

Analyze the Traces in the Coordinate Planes

Examine the traces of the surface in the coordinate planes:- In the \(xy\)-plane (\(z = 0\)), the trace is given by \(-x^2 - y^2 = 1\), which has no real solution.- In the \(yz\)-plane (\(x = 0\)), the trace is \(z^2 - y^2 = 1\), a hyperbola.- In the \(xz\)-plane (\(y = 0\)), the trace is \(z^2 - x^2 = 1\), also a hyperbola.
03

Analyze the Horizontal Cross-Sections

Analyze horizontal cross-sections (parallel to the \(xy\)-plane) by setting \(z = c\):- The equation becomes \(c^2 - x^2 - y^2 = 1\), which suggests: - If \(c^2 > 1\), then \(x^2 + y^2 = c^2 - 1\), a circle. - If \(c^2 = 1\), it degenerates to a point circle \(x^2 + y^2 = 0\). - If \(c^2 < 1\), there is no real solution as you can't have negative radius squared.
04

Describe the Overall Shape

Based on the traces and cross-sections, the surface has a hyperboloid of one sheet structure. It opens along the \(z\)-axis and forms circles in horizontal cross-sections as \(z\) moves away from zero, indicating the shape narrows at the center and widens further along the \(z\)-axis.

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Key Concepts

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

Hyperboloids
A hyperboloid is a type of quadratic surface that can appear in two variations: one sheet or two sheets. In the equation \( z^2 - x^2 - y^2 = 1 \), we see an example of a hyperboloid of one sheet, which is known for having a saddle-like shape.
This surface is three-dimensional and exhibits symmetry, often looking like a stretched or compressed hourglass.
Characteristics of a hyperboloid include:
  • It contains hyperbolas in vertical cross-sections.
  • In horizontal cross-sections, it forms circles when the constant is positive and sufficiently large.
  • This form has one continuous surface unlike the two-sheet hyperboloid, which consists of two disconnected parts.
Understanding hyperboloids is critical in multivariable calculus as they lay the foundation for more complex geometric and physical applications.
Coordinate Planes
Coordinate planes play a crucial role in visualizing and analyzing multivariable calculus problems. They represent two-dimensional sections that can help simplify complex surfaces into more manageable shapes.
In the context of the hyperboloid \( z^2 - x^2 - y^2 = 1 \), examining the surface's traces in each coordinate plane provides essential insights:
  • In the xy-plane \((z = 0)\): We encounter the equation \(-x^2 - y^2 = 1\), which doesn't form a real surface, indicating the absence of a visible trace.
  • In the yz-plane \((x = 0)\): The resulting equation is \(z^2 - y^2 = 1\), showing a hyperbola.
  • In the xz-plane \((y = 0)\): Similarly, \(z^2 - x^2 = 1\) describes yet another hyperbola.
These traces on the coordinate planes help visualize how the surface appears when intersected by these planes, which can greatly aid in sketching and understanding the surface overall.
Hyperbola
A hyperbola is a type of conic section that is defined as the set of all points where the difference of the distances from two fixed points (foci) is constant. In a hyperboloid, the presence of hyperbolas in vertical traces reveals much about its shape.
In the case of the traces described in the equation \( z^2 - x^2 - y^2 = 1 \):
  • The yz-plane and xz-plane contain hyperbolas characterized by equations \(z^2 - y^2 = 1\) and \(z^2 - x^2 = 1\), respectively.
  • Hyperbolas signal that the hyperboloid extends infinitely along the axes perpendicular to the axis of symmetry.
  • These hyperbolas open along the z-axis since the z variable is positive in the given equation, dictating the main direction of extension.
By understanding the presence and form of hyperbolas, we can better infer the overall shape and spread of the hyperboloid.
Horizontal Cross-Sections
Horizontal cross-sections of a surface show how it behaves parallel to the xy-plane, giving insight into changes in profile as you move along the axis perpendicular to this plane.
For the hyperboloid \( z^2 - x^2 - y^2 = 1 \):When you set \( z = c \) and solve for \( x \) and \( y \):
  • If \(c^2 > 1\): The equation \(c^2 - x^2 - y^2 = 1\) results in a circle, suggesting the surface broadens out.
  • If \(c^2 = 1\): The equation simplifies to \(x^2 + y^2 = 0\), a degenerate circle, or essentially a single point.
  • If \(c^2 < 1\): There aren't real solutions, showing that the surface thins out, disappearing.
This analysis of cross-sections aids in visualizing how the surface's shape changes with height, illustrating the narrowing toward the center and subsequent widening as it extends.

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