Problem 70 - Advanced Engineering Mathematics

Question

In Problems \(67-70\), convert the given equation to spherical coordinates. $$ -x^{2}-y^{2}+z^{2}=1 $$

Step-by-Step Solution

Answer

The equation in spherical coordinates is \( \rho = \sqrt{\frac{1}{\cos(2\phi)}} \).

Step 1: Recall the Spherical Coordinates Formulas

Spherical coordinates are defined as: - \( x = \rho \sin \phi \cos \theta \)- \( y = \rho \sin \phi \sin \theta \)- \( z = \rho \cos \phi \)where \( \rho \) is the radial distance, \( \phi \) is the polar angle, and \( \theta \) is the azimuthal angle.

Step 2: Substitute into the Given Equation

We substitute the expressions for \( x \), \( y \), and \( z \) into the equation \( -x^2 - y^2 + z^2 = 1 \). This gives us:\[ -(\rho \sin \phi \cos \theta)^2 - (\rho \sin \phi \sin \theta)^2 + (\rho \cos \phi)^2 = 1 \]

Step 3: Simplify the Expression

Simplifying, we have:\[ -\rho^2 \sin^2 \phi (\cos^2 \theta + \sin^2 \theta) + \rho^2 \cos^2 \phi = 1 \]Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), the equation becomes:\[ -\rho^2 \sin^2 \phi + \rho^2 \cos^2 \phi = 1 \]

Step 4: Factor Out Common Terms

Factor out \( \rho^2 \) from the equation:\[ \rho^2 (\cos^2 \phi - \sin^2 \phi) = 1 \]This simplifies to: \[ \rho^2 \cos(2\phi) = 1 \]

Step 5: Solve for \( \rho \)

Divide both sides by \( \cos(2\phi) \) assuming it's not zero:\[ \rho^2 = \frac{1}{\cos(2\phi)} \]Thus, \( \rho = \pm \sqrt{\frac{1}{\cos(2\phi)}} \). Typically, \( \rho \) is taken to be positive, so:\[ \rho = \sqrt{\frac{1}{\cos(2\phi)}}\]

Step 6: Final Conversion

The equation in spherical coordinates is \( \rho = \sqrt{\frac{1}{\cos(2\phi)}} \).

Key Concepts

Coordinate Conversion

Coordinate conversion is fundamental in mathematics, particularly when working with systems like cartesian, cylindrical, or spherical coordinates. In many cases, you need to convert equations or points from one coordinate system to another to simplify the problem or match the conditions provided. For example, Cartesian coordinates (x,y,z) can be transformed into spherical coordinates (\(\rho, \phi, \theta\)) using specific formulas.

• Cartesian to Spherical Conversion:
  • \( x = \rho \sin \phi \cos \theta \)
  • \( y = \rho \sin \phi \sin \theta \)
  • \( z = \rho \cos \phi \)

To convert an equation like \(-x^{2}-y^{2}+z^{2}=1\), substitute the expressions for \( x, y, \) and \( z \) from spherical coordinates into this equation. The conversion reveals the geometric nature of the equation more intuitively in the new coordinate system. This mapping between systems is essential for solving complex problems and understanding different perspectives in mathematical scenarios.

Pythagorean Identity

The Pythagorean identity is one of the most crucial trigonometric identities. It tells us that for any angle \(\theta\), the sum of the squares of sine and cosine is always 1:\( \cos^2 \theta + \sin^2 \theta = 1 \)This identity helps in simplifying and solving various trigonometric expressions or equations. In the context of spherical coordinates, it's particularly useful. When substituting spherical coordinates into cartesian equations, terms like \(\cos^2 \theta + \sin^2 \theta\) often appear, allowing us to replace it with 1. For the exercise, this simplification is made right after translating the given equation into spherical terms:\[ -\rho^2 \sin^2 \phi (\cos^2 \theta + \sin^2 \theta) + \rho^2 \cos^2 \phi = 1 \quad \Rightarrow \quad -\rho^2 \sin^2 \phi + \rho^2 \cos^2 \phi = 1 \]This way, the Pythagorean identity aids in unveiling the hidden simplicity within seemingly complicated mathematical expressions.

Spherical Coordinate System

The spherical coordinate system is an extension of polar coordinates into three dimensions. It's particularly useful for scenarios where points are described by their distance from a central point (origin), an inclination angle (angle down from the z-axis), and an azimuthal angle (angle on the x-y plane from the x-axis).

• Components of Spherical Coordinates:
  • \( \rho \) (radial distance): Distance from the origin to the point.
  • \( \phi \) (polar angle): Angle between the positive z-axis and the line connecting the origin to the point.
  • \( \theta \) (azimuthal angle): Angle between the positive x-axis and the projection of the line connecting the origin to the point on the x-y plane.

Unlike Cartesian coordinates, which can become cumbersome with complex boundaries or rotations, spherical coordinates streamline calculations in modeling radial phenomena. For example, relations in physics concerning spheres or on the surface of spheres are much more manageable within this system.

Mathematics Problem Solving

Mathematics problem-solving is a strategic process that draws upon an understanding of mathematical concepts and the ability to manipulate equations and expressions. The goal is to reach solutions efficiently and accurately.In this particular exercise, solving \(-x^{2}-y^{2}+z^{2}=1\) using spherical coordinates falls into several key steps:

• Understand the Problem: Recognize what the equation represents and the desired outcome in spherical coordinates.
• Formulate a Plan: Use the known spherical coordinate transformations to substitute into the original equation.
• Carry Out the Plan: Execute the transformation, apply identities like the Pythagorean identity to simplify terms, and solve for \( \rho \).
• Interpret the Outcome: The final equation in spherical form describes the original surface differently, enhancing understanding of its properties within the spherical context.

This systematic approach is crucial across various branches of mathematics and helps in tackling both theoretical and practical problems.