Question

Elliptic cylinder coordinates \(u, v, z:\) $$ \begin{aligned} &x=a \cosh u \cos v \\ &y=a \sinh u \sin v \\ &z=z \end{aligned} $$

Short answer

Transform Cartesian coordinates using given elliptic cylinder relations using \(x = a \cosh u \cos v\), \(y = a \sinh u \sin v\) and \(z=z\), ensuring correct value ranges.

Step-by-step solution

Identify the Coordinate Transformation Equations

The given coordinate transformation equations are: \( x = a \cosh u \cos v \) \( y = a \sinh u \sin v \) \( z = z \)

Understand Hyperbolic Functions

Recall the definitions of the hyperbolic functions: \( \cosh u = \frac{e^u + e^{-u}}{2} \) \( \sinh u = \frac{e^u - e^{-u}}{2} \).

Express Elliptic Cylinder Coordinates in Terms of Cartesian Coordinates

For each set of coordinates, express the Cartesian coordinates as a function of elliptic cylinder coordinates. This helps in understanding the mapping between the two coordinate systems.

Understand the Nature of Functions

Function \( \cosh u \) is always greater than or equal to 1, and \( \sinh u \) can take any real value. While \( \cos v \) and \( \sin v \) correspond to the standard trigonometric functions, periodic in nature.

Check for Range and Constraints

Ensure the range of values for \( u \), \( v \), and \( z \) fits the physical space described by the elliptic cylinder coordinates: likely \( u \geq 0 \), \( 0 \leq v < 2\pi \), and \( z\) can be any real value

Visual Representation

Plot the mapping on graph paper or visualization software, showing how elliptic cylinder coordinates transform into Cartesian coordinates for different values of \( u \), \( v \), and \( z \).

Key concepts

Coordinate Transformation

Coordinate transformations are essential for converting between different coordinate systems. In this exercise, we look at elliptic cylinder coordinates \(u, v, z\) and convert them to Cartesian coordinates through these equations:

\(x = a \cosh u \cos v\)
\(y = a \sinh u \sin v\)
\(z = z\).

This transformation is handy when dealing with problems involving elliptical cylinders. The given set of equations helps translate points from elliptic cylinder coordinates to the more familiar Cartesian coordinates. This makes complex geometry problems easier to solve.

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas instead of circles. The key hyperbolic functions used here are:

\( \cosh u = \frac{e^u + e^{-u}}{2} \)
\( \sinh u = \frac{e^u - e^{-u}}{2} \)

These equations are integral to transforming from elliptic cylinder coordinates to Cartesian coordinates. Hyperbolic functions grow exponentially, unlike their trigonometric counterparts that oscillate. Understanding these functions is crucial in visualizing and working with elliptic cylinder coordinates effectively.

Cartesian Coordinates

Cartesian coordinates \((x, y, z)\) are the most commonly used coordinate system. They consist of three axes: the x-axis, y-axis, and z-axis, all perpendicular to each other. In this exercise, we transformed elliptic cylinder coordinates to Cartesian coordinates to simplify our geometric understanding.

Using the transformations:
\(x = a \cosh u \cos v\)
\(y = a \sinh u \sin v\)
\(z = z\),
we see how each elliptic cylinder coordinate directly converts to its Cartesian counterpart.
This transformation is significant for solving more complex geometric problems involving shapes like ellipses and hyperbolas.

Range and Constraints

Understanding the range and constraints of elliptic cylinder coordinates is vital. For the given transformations:

\(u \geq 0\): ensures \cosh u\ remains greater than or equal to 1.
\(0 \leq v \textless 2\pi\): keeps the values of \sin v\ and \cos v\ within their standard periodic ranges.
\(z\) is unrestricted and can represent any real number.

These constraints ensure that the transformations map correctly to valid Cartesian coordinates within the context of an elliptic cylinder.

Graphical Visualization

Visualizing the transformation between elliptic cylinder coordinates and Cartesian coordinates is a great way to understand their spatial relationships. Using graphing tools, you can plot how points in elliptic cylinder coordinates \((u, v, z)\) translate to Cartesian coordinates.

For instance:
When \u = 0, v = 0\, and a = 1\, the conversion gives

\(x = \cosh(0) \cos(0) = 1\)
\(y = \sinh (0) \sin (0) = 0\)
\(z = 0\).

This kind of plotting helps to visually comprehend the effect of different values of u, v, and z, providing a clearer understanding of the shape and dimension of an elliptic cylinder.