Chapter 5: Problem 17

\(x=a \cosh u \cos v\) \(y=a\) sinh \(u \sin v, \quad(u\) and \(v\) are called elliptic cylinder coordinates).

Short Answer

Expert verified

Given \( x = a \cosh u \cos v\) and \( y = a \sinh u \sin v\), these equations describe the coordinate transformations in elliptic cylinder coordinates.

Step by step solution

Introduction to Elliptic Cylinder Coordinates

Elliptic cylinder coordinates \(u \) and \(v \) are used to describe points in a plane, with parameters related to hyperbolic and trigonometric functions. Let's solve the given expressions.

Understanding x-coordinate

The x-coordinate is given by the expression \(x = a \cosh u \cos v\). This implies that \(x \) is a product of \(a \cosh u\) and \ \cos v\. Here, \ \cosh u\ is the hyperbolic cosine of \ u\, and \ \cos v\ is the trigonometric cosine of \ v\.

Analyzing y-coordinate

The y-coordinate is described by the equation \(y = a \sinh u \sin v\). This means \(y \) is a product of \ a \sinh u\ and \ \sin v\, where \ \sinh u\ is the hyperbolic sine of \ u\, and \ \sin v\ is the trigonometric sine of \ v\.

General Solution Interpretation

Given the equations for \(x \) and \(y \), substitute different values for \(a, u, \) and \(v\) to describe points in the elliptic cylinder coordinate system. These coordinates relate to transformations of the hyperbolic trigonometric functions to describe points in a two-dimensional plane.

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Key Concepts

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

Hyperbolic Functions

Hyperbolic functions are analogous to trigonometric functions but for a hyperbola, rather than a circle. They include hyperbolic sine (\text{sinh}), hyperbolic cosine (\text{cosh}), and hyperbolic tangent (\text{tanh}):

The hyperbolic sine is defined as \text{sinh}\((u) = \frac{e^u - e^{-u}}{2}\).
The hyperbolic cosine is defined as \text{cosh}\((u) = \frac{e^u + e^{-u}}{2}\).
Both functions are vital in describing curves and surfaces that appear in various coordinate systems.

In the given problem, \text{cosh}\((u)\) and \text{sinh}\((u)\) transform the coordinates in a way that resembles an ellipse in the elliptic cylinder coordinate system.
These functions help move between linear and curved spaces efficiently.

Trigonometric Functions

Trigonometric functions are fundamental in relating the angles of triangles to the lengths of their sides. Familiar ones include sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}):

The sine function is given by \text{sin}\((v) = \frac{\text{opposite}}{\text{hypotenuse}}\).
The cosine function is \text{cos}\((v) = \frac{\text{adjacent}}{\text{hypotenuse}}\).

In the exercise, these functions \text{cos}\((v)\) and \text{sin}\((v)\) are used to scale the hyperbolic functions to describe coordinates \text{x} and \text{y}.
Combining hyperbolic and trigonometric functions gives a robust framework for expressing complex moves in different geometric settings.

Coordinate Systems

Coordinate systems are used to uniquely determine the position of a point or other geometric elements in space. Elliptic cylinder coordinates are one such system, defined by parameters \(u\) and \(v\):

The coordinate \text{x} is defined as \(x = a \text{cosh} u \text{cos} v\).
The coordinate \text{y} is defined as \(y = a \text{sinh} u \text{sin} v\).

Here, \(a\) is a constant that can stretch or shrink these coordinates.
This setup allows mapping points in space more intuitively for shapes that bear elliptical features.

Two-Dimensional Plane

A two-dimensional plane is a flat surface extending infinitely in two dimensions (length and width). Points on this plane are specified by pairs of numbers, such as \text{x} and \text{y}.
In the elliptic cylinder coordinate system, the plane is described with transformed coordinates \((x, y)\) based on parameters \((u, v)\).

Each point in the 2D plane can be located using transformations involving both hyperbolic and trigonometric functions.
This allows for versatile representations that can describe more complex geometries, such as elliptical shapes.

Understanding how different coordinates map onto this plane helps in visualizing and solving geometrical problems effectively.