Question

In Exercises \(55-60\), convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates $$ x^{2}-y^{2}=9 $$

Short answer

The equation \(x^{2}-y^{2}=9\) in cylindrical coordinates is \(r^{2}\cos(2\theta) = 9\), and in spherical coordinates is \(\rho^{2}\sin^{2}\phi (\cos^{2}\theta - \sin^{2}\theta) = 9\).

Step-by-step solution

Transform to Cylindrical Coordinates

To convert the rectangular coordinates equation \(x^{2} - y^{2} = 9\) into cylindrical coordinates, substitute \(x = r\cos\theta\) and \(y = r\sin\theta\). This gives \((r\cos\theta)^{2} - (r\sin\theta)^{2} = 9\). So, after simplifying, we have \(r^{2} (\cos^{2}\theta - \sin^{2}\theta) = 9\).

Transform to Spherical Coordinates

To convert the same equation into spherical coordinates, substitute \(x = \rho\sin\phi\cos\theta\) and \(y = \rho\sin\phi\sin\theta\). This gives \((\rho\sin\phi\cos\theta)^{2} - (\rho\sin\phi\sin\theta)^{2} = 9\). Simplifying, now we have \(\rho^{2}\sin^{2}\phi (\cos^{2}\theta - \sin^{2}\theta)= 9\).

Final Simplification

The equation in Step 1 can be further simplified since \(\cos^{2}\theta - \sin^{2}\theta\) is the double angle formula for cosine, and hence, \(r^{2}\cos(2\theta) = 9\). Notice that Step 2 can't be simplified further as it relates both distance and angle in spherical coordinates.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates provide a way of describing the position of a point in three-dimensional space using three numbers: the radial distance, the angle around a central axis, and the height along that axis.

In more technical terms, a point P in cylindrical coordinates is represented by \( (r, \theta, z) \), where \( r \) is the distance from the point to the z-axis, \( \theta \) is the angle measured counterclockwise from the positive x-axis to the point's projection onto the xy-plane (also known as the azimuth angle), and \( z \) refers to the point's height above the xy-plane.

This system is particularly useful when dealing with problems involving symmetry around a central axis, such as in cylindrical objects or when analyzing circular motion.

Spherical Coordinates

Spherical coordinates represent points in three-dimensional space with a distance and two angles. A point P is denoted as \( (\rho, \phi, \theta) \), where:\

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• \(\rho\) is the radial distance from the origin to the point,\
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• \(\phi\) (often called the polar angle) is the angle between the positive z-axis and the line segment from the origin to the point, and\
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• \(\theta\) is the angle (azimuth) from the the positive x-axis to the projection of the point in the xy-plane.\
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Spherical coordinates are valuable when problems exhibit radial symmetry, such as those involving celestial bodies or radial fields in physics.

Rectangular to Cylindrical Conversion

Converting rectangular coordinates (x, y, z) to cylindrical coordinates (r, \theta, z) involves finding the radius and angle in the xy-plane while retaining the z-height. For the radius, we use the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \). The angle \theta is found by using trigonometric ratios: if x > 0, then \( \theta = \arctan(\frac{y}{x}) \); otherwise, adjust \theta to point to the correct quadrant. The height z remains the same.

The conversion is useful, for instance, when we want to simplify equations or explore symmetrical systems, as cylindrical coordinates can make circular patterns more evident.

Rectangular to Spherical Conversion

The transformation from rectangular coordinates (x, y, z) to spherical coordinates (\rho, \phi, \theta) is slightly more involved. The radial distance \(\rho\) is determined by the equation \(\rho = \sqrt{x^2 + y^2 + z^2}\). The polar angle \phi requires an understanding of the spatial relationship among the axes, and is computed as \(\phi = \arccos(\frac{z}{\rho})\) when \(\rhoeq 0\). Lastly, the azimuthal angle \theta mirrors the one used in cylindrical coordinates.

This conversion process can often reveal underlying structures in physical phenomena, especially when sources or fields spread out in all directions from a point, or when describing the location of objects in astronomy.