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Chapter 16: Problem 36

Find a parametrization of the hyperboloid of two sheets \(\left(z^{2} / c^{2}\right)-\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\)

Short Answer

Expert verified
The parametrization is: \( x = a \sinh(u) \sin(v) \), \( y = b \sinh(u) \cos(v) \), \( z = c \cosh(u) \).

Step by step solution

01

Identify the Hyperboloid Equation

The given equation is \( \frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This represents a hyperboloid of two sheets, as the \( z^2 \) term is positive, and the equation equals 1.
02

Express in Terms of Hyperbolic Functions

Notice the form of the equation resembles the hyperbolic identity \( \cosh^2(t) - \sinh^2(t) = 1 \). We can use \( z = c \cosh(u) \), where \( u \) is a parameter.
03

Rewriting for the xy-plane

Given \( z = c \cosh(u) \), the equation shows \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = \sinh^2(u) \). Let \( x = a \sinh(u) \sin(v) \) and \( y = b \sinh(u) \cos(v) \), incorporating a new parameter \( v \).
04

Parameterization Result

The parameterization of the hyperboloid is: \( x(u,v) = a \sinh(u) \sin(v) \), \( y(u,v) = b \sinh(u) \cos(v) \), and \( z(u) = c \cosh(u) \). Here, \( u \geq 0 \) (since \( \cosh(u) \geq 1 \)) and \( v \in [0, 2\pi) \).

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Key Concepts

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

Hyperboloid of Two Sheets
Hyperboloids are fascinating shapes often discussed in multivariable calculus and geometry. Specifically, the hyperboloid of two sheets can be visualized as two separate, mirror-like surfaces or shells. They do not connect at any point, except perhaps at infinity in some geometric considerations.

The standard equation for a hyperboloid of two sheets is:
  • \[ \frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
This equation tells us several things:
  • First, it indicates that the shape spreads out infinitely in the z-direction, because the z-component, represented by \( z^2/c^2 \), is positive. This also explains why there are two separate sheets 鈥� one for positive z-values and one for negative.
  • The equation's equality to 1 and the presence of subtraction in the x and y terms are key aspects that distinguish it as a hyperboloid of two sheets.
Visualizing this shape within a standard coordinate system can help students better consider surface plotting and properties of 3D geometry.
Parametrization
Parametrization is a method we use in mathematics to express complex surfaces or shapes using simpler, manipulatable equations. This is particularly helpful in calculus for integration, differentiation, and graphical representation of surfaces or curves.

In this exercise, the focus is on parametrizing the hyperboloid of two sheets. An intuitive way to do this is to express x, y, and z in terms of two parameters, usually noted as u and v. Here is how it's done:
  • The z-component is managed through the hyperbolic cosine function: \( z = c \cosh(u) \). By letting \( u \) act as a parameter, we explore different levels of z values across the surface.
  • For the x and y components, parameter v comes into use. The expressions are: \( x(u,v) = a \sinh(u) \sin(v) \) and \( y(u,v) = b \sinh(u) \cos(v) \). These incorporate trigonometric functions to account for the circular symmetry in the x-y plane as influenced by the hyperbolic sine.
This setup describes all possible points on the hyperboloid surface by varying parameters \( u \) and \( v \). Parametrization thus simplifies the understanding and manipulation of the surface in 3-dimensional space.
Hyperbolic Functions
Hyperbolic functions are analogs to trigonometric functions and are crucial in many fields of mathematics, including calculus and geometry. Understanding these functions aids in the visualization and manipulation of hyperbolas and related shapes like hyperboloids.

Key hyperbolic functions:
  • The hyperbolic cosine function, \( \cosh(t) \), is defined as \( \cosh(t) = \frac{e^t + e^{-t}}{2} \). In our parametrization, it helps express the z-component, stretching the hyperboloid along the z-axis.
  • Conversely, \( \sinh(t) \) or hyperbolic sine is expressed as \( \sinh(t) = \frac{e^t - e^{-t}}{2} \). This function is employed in parametrizing the x and y components, bringing in a radial stretch component that augments the x-y plane for our shape.
These functions hold the identity \( \cosh^2(t) - \sinh^2(t) = 1 \), much like their trigonometric counterparts in unit circles where \( \cos^2(\theta) + \sin^2(\theta) = 1 \). Understanding this identity is fundamental in solving and manipulating hyperbolic equations, such as the hyperboloid discussed.

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Most popular questions from this chapter

Maximum flux Among all rectangular solids defined by the inequalities \(0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq 1,\) find the one for which the total flux of \(\mathbf{F}=\left(-x^{2}-4 x y\right) \mathbf{i}-6 y z \mathbf{j}+12 z \mathbf{k}\) outward through the six sides is greatest. What is the greatest flux?

a. A torus of revolution (doughnut) is obtained by rotating a circle \(C\) in the \(x z\) -plane about the \(z\) -axis in space. (See the accompanying figure.) If \(C\) has radius \(r>0\) and center \((R, 0,0),\) show that a parametrization of the torus is $$\begin{aligned}\mathbf{r}(u, v)=&((R+r \cos u) \cos v) \mathbf{i} \\\&+((R+r \cos u) \sin v) \mathbf{j}+(r \sin u) \mathbf{k}\end{aligned}$$ where \(0 \leq u \leq 2 \pi\) and \(0 \leq v \leq 2 \pi\) are the angles in the figure.b. Show that the surface area of the torus is \(A=4 \pi^{2} R r\)

Apply Green's Theorem to evaluate the integrals. \(\oint_{C}\left(2 x+y^{2}\right) d x+(2 x y+3 y) d y\) C: Any simple closed curve in the plane for which Green's Theorem holds

Let \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} .\) Show that the clockwise circulation of the field \(\mathbf{F}=\nabla f\) around the circle \(x^{2}+y^{2}=a^{2}\) in the \(x y\) -plane is zero a. by taking \(\mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi,\) and integrating \(\mathbf{F} \cdot d \mathbf{r}\) over the circle. b. by applying Stokes' Theorem.

a. Show that the outward flux of the position vector field \(\mathbf{F}=\) \(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) through a smooth closed surface \(S\) is three times the volume of the region enclosed by the surface. b. Let \(n\) be the outward unit normal vector field on \(S\). Show that it is not possible for \(\mathbf{F}\) to be orthogonal to \(\mathbf{n}\) at every point of \(S\)
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