91Ó°ÊÓ

Rectangular coordinates, also known as Cartesian coordinates, are a fundamental system used in mathematics and physics to define the positions of points in a space. In this system, each point is specified by a set of numbers, typically denoted as

• \( (x, y, z) \) in three-dimensional space
• \( (x, y) \) in two-dimensional space.

Each coordinate represents a distance from the origin, measured along a perpendicular axis:

• The \(x\)-coordinate reflects how far a point is horizontally from the \(y\z\) plane.
• The \(y\)-coordinate indicates the distance from the \(x\z\) plane along the \(y\)-axis.
• The \(z\)-coordinate measures vertical displacement from the \(xy\) plane.

Rectangular coordinates are invaluable in calculating distances, analysing geometry, and converting complex shapes and surfaces from other coordinate systems. Understanding how to switch between systems, such as from cylindrical to rectangular as seen in this exercise, opens a world of mathematical possibilities.

Converting equations between coordinate systems is a crucial skill when dealing with complex surfaces. In the exercise, we start with the equation \( z = r^2 \cos^2 \theta \) in cylindrical coordinates and aim to transform it to rectangular coordinates. This involves:

• Recognizing that in cylindrical coordinates, \( r \) is the radial distance, \( \theta \) is the angle, and \( z \) represents height.
• Applying the relationships in rectangular coordinates: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r = \sqrt{x^2 + y^2} \).
• Transforming parts of the original equation using these relationships.

For example, substituting \( x = r \cos \theta \) yields \( \cos \theta = \frac{x}{r} \), and subsequently \( \cos^2 \theta = \left(\frac{x}{r}\right)^2 = \frac{x^2}{r^2} \).This substitution turns the cylindrical expression into a rectangular form, ultimately simplifying to \( z = x^2 \). Through this conversion process, we see how mathematical translation can lead to clearer visual and analytical understanding.

A parabolic cylinder is a fascinating geometric shape described by an equation that takes the form \( z = ax^2 \) in rectangular coordinates. Unlike a paraboloid, a parabolic cylinder extends infinitely along one axis—here, the \( y \)-axis. This creates a unique trough-like surface.In the context of this exercise, the rectangular equation \( z = x^2 \) defines a parabolic cylinder:

• All cross-sectional slices parallel to the \( yz \)-plane form identical parabolas that open upwards along the \( z \)-axis.
• The surface does not change as you move along the \( y \)-axis, demonstrating the 'cylinder' nature.

Graphing this equation results in a continuous, unbounded shape that provides valuable insights into both theory and real-world applications. Understanding parabolic cylinders enhances one's ability to model and interpret physical phenomena, such as the paths of projectiles or water waves in large channels.