Problem 8 - Calculus: Early Transcendentals

Question

Find an equation for the plane \(z=1\) in spherical coordinates.

Step-by-Step Solution

Answer

The equation in spherical coordinates is \(\rho = \frac{1}{\cos \phi}\), with \(\phi\) and \(\theta\) free.

Step 1: Understand Spherical Coordinates

Spherical coordinates are defined with three parameters: the radial distance \(\rho\), the polar angle \(\phi\), and the azimuthal angle \(\theta\). They relate to Cartesian coordinates \((x, y, z)\) as follows: \(x = \rho \sin\phi \cos\theta\), \(y = \rho \sin\phi \sin\theta\), \(z = \rho \cos\phi\).

Step 2: Set Cartesian Equation for the Plane

The plane is given by the equation \(z = 1\) in Cartesian coordinates. This means that the \(z\)-coordinate of any point on this plane is 1.

Step 3: Substitute into Spherical Coordinates Equation

Substitute \(z = \rho \cos \phi = 1\) from spherical coordinates into the given plane equation. This gives the relationship \(\rho \cos \phi = 1\).

Step 4: Express \(\rho\) in terms of \(\phi\)

Rearrange the equation from Step 3 to express the radial distance \(\rho\) as a function of the polar angle \(\phi\): \(\rho = \frac{1}{\cos \phi}\).

Step 5: Formulate the Equation in Spherical Coordinates

The equation for the plane \(z = 1\) in spherical coordinates becomes defined by the relationship \(\rho = \frac{1}{\cos \phi}\), with \(\phi\) and \(\theta\) varying freely.

Key Concepts

Coordinate Conversion

Coordinate conversion is a key process when dealing with different geometric systems like Cartesian and spherical coordinates. The main idea is to translate a point's location from one system to another. This becomes necessary when different systems provide simpler mathematical expressions for a problem. For instance, in spherical coordinates, which are particularly useful for problems with symmetry around a point, every point in space is described by its radial distance from the origin, a polar angle from the vertical axis, and an azimuthal angle around a horizontal plane.

• The Cartesian coordinates \(x, y, z\) use straight-line distances along each axis.
• Spherical coordinates \(\rho, \phi, \theta\) provide an angle-based description.

For conversion between them:

• The \(x\) coordinate is given by \(x = \rho \sin\phi \cos\theta\).
• The \(y\) coordinate is expressed as \(y = \rho \sin\phi \sin\theta\).
• The \(z\) coordinate becomes \(z = \rho \cos\phi\).

Using these formulas, you can translate any spherical system point into its Cartesian counterpart.

Radial Distance

The radial distance, often symbolized as \(\rho\), is an essential part of spherical coordinates. It represents how far the point is from a fixed central point, usually the origin. Imagine drawing a straight line between the point and the origin — the length of this line is the radial distance.

• \(\rho\) is always positive since it measures distance.
• For the given problem, we observed that \(\rho = \frac{1}{\cos \phi}\) to express the plane \(z = 1\).

By knowing this distance, you can describe the possible location of a point in 3-dimensional space, provided the angles are also known. In the case of the plane \(z = 1\), \(\rho\) adjusts in response to changes in the polar angle \(\phi\), dictating a dynamic relationship between spot height and orientation.

Polar Angle

The polar angle, denoted as \(\phi\), plays a crucial role in spherical coordinates. This angle indicates how far a point's line of sight is from the positive z-axis.

• The range of \(\phi\) is typically between \(0\) and \(\pi\) radians.
• When \(\phi = 0\), the point is directly above the origin along the z-axis.
• When \(\phi = \frac{\pi}{2}\), the point lies on the xy-plane.

In the context of the plane \(z = 1\), this angle helps determine the relation \(\rho = \frac{1}{\cos \phi}\), ensuring that the plane's height remains constant. As \(\phi\) changes, the radial distance \(\rho\) responds to maintain the plane's equation in spherical terms, with no restriction on \(\phi\) beyond its natural range.

Azimuthal Angle

The azimuthal angle, indicated by \(\theta\), describes how far around the equatorial plane a point is rotated from a reference direction, commonly the positive x-axis. It allows us to pinpoint a point's horizontal placement.

• \(\theta\) typically ranges from \(0\) to \(2\pi\) radians.
• It is analogous to longitude in geographic coordinate systems.
• In the plane \(z = 1\), \(\theta\) can be any value within its range as it does not affect the z-level.

This flexibility makes \(\theta\) essential for understanding circular or rotational symmetry in problems. While it doesn't directly influence the equation \(z = 1\) in spherical terms, it completes the positioning by detailing the plane's aspects that are orthogonal to height.