Problem 69 - Advanced Engineering Mathematics

Question

In Problems \(67-70\), convert the given equation to spherical coordinates. $$ z^{2}=3 x^{2}+3 y^{2} $$

Step-by-Step Solution

Answer

The equation in spherical coordinates is \( \phi = \frac{\pi}{6} \) or \( \phi = \frac{5\pi}{6} \).

Step 1: Identify the relationship

In spherical coordinates, the variables are expressed as \( x = \rho \sin(\phi) \cos(\theta) \), \( y = \rho \sin(\phi) \sin(\theta) \), and \( z = \rho \cos(\phi) \). We will use these formulations to transform the given equation.

Step 2: Substitute spherical coordinates into the equation

Substitute \( x = \rho \sin(\phi) \cos(\theta) \), \( y = \rho \sin(\phi) \sin(\theta) \), and \( z = \rho \cos(\phi) \) into the given equation: \( z^2 = 3x^2 + 3y^2 \).

Step 3: Simplify the equation

Substitute the expressions for \( x, y, z \):\( (\rho \cos(\phi))^2 = 3(\rho \sin(\phi) \cos(\theta))^2 + 3(\rho \sin(\phi) \sin(\theta))^2 \).Simplify to:\( \rho^2 \cos^2(\phi) = 3 \rho^2 \sin^2(\phi)(\cos^2(\theta) + \sin^2(\theta)) \).

Step 4: Use trigonometric identities

Apply the Pythagorean identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \) to simplify:\( \rho^2 \cos^2(\phi) = 3 \rho^2 \sin^2(\phi) \).

Step 5: Solve the simplified equation

Divide both sides by \( \rho^2 \) (assuming \( \rho eq 0 \)) to get:\( \cos^2(\phi) = 3 \sin^2(\phi) \).Now use the identity \( \cos^2(\phi) + \sin^2(\phi) = 1 \) to further relate \( \cos^2(\phi) \) and \( \sin^2(\phi) \).

Step 6: Further simplify and solve for \( \phi \)

Divide the equation \( \cos^2(\phi) = 3 \sin^2(\phi) \) by \( \sin^2(\phi) \) to obtain:\[ \frac{\cos^2(\phi)}{\sin^2(\phi)} = 3 \]which simplifies to \( \cot^2(\phi) = 3 \), giving \( \phi = \tan^{-1}(\sqrt{3}) \), which corresponds to \( \phi = \frac{\pi}{6} \) (approximately 30 degrees) and \( \phi = \frac{5\pi}{6} \) (approximately 150 degrees).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics that help simplify complex equations and expressions, particularly when converting between different coordinate systems. They're particularly useful in the realm of spherical coordinates transformation, as seen in our exercise.

One key identity often used is the Pythagorean identity, which states:

• \( \cos^2(\theta) + \sin^2(\theta) = 1 \)

This identity forms the basis for many simplifications, allowing us to reduce expressions to more manageable forms. For example, in our exercise, applying this identity to \( \cos^2(\theta) + \sin^2(\theta) \) yields 1, which is pivotal in simplifying the expression for \( z^2 = 3x^2 + 3y^2 \), resulting in a straightforward spherical coordinates transformation.

Understanding and using trigonometric identities doesn't stop at simplifying equations. They also allow us to find relationships between different trigonometric functions, critical in solving equations and transformations, as shown with the identity \( \cos^2(\phi) + \sin^2(\phi) = 1 \). It served as a bridge to further simplify and eventually solve the expression \( \cos^2(\phi) = 3 \sin^2(\phi) \).

Spherical Coordinates

Spherical coordinates provide a way to represent points in three dimensions using radii and angles, differing from traditional Cartesian coordinates. In this system, any point in space is described with three parameters:

• \( \rho \): the radial distance from the origin.
• \( \phi \): the polar angle, equivalent to the angle measured from the positive z-axis.
• \( \theta \): the azimuthal angle, measuring the angle in the x-y plane from the positive x-axis.

This coordinate system is particularly advantageous in situations with spherical symmetry and is widely used in physics, engineering, and computer graphics.

In our exercise, the conversion to spherical coordinates involved expressing x, y, and z of the original equation in terms of \( \rho \), \( \phi \), and \( \theta \). Each Cartesian coordinate has its form in spherical coordinates:

• \( x = \rho \sin(\phi) \cos(\theta) \)
• \( y = \rho \sin(\phi) \sin(\theta) \)
• \( z = \rho \cos(\phi) \)

By substituting these into the original equation, the transition from Cartesian to spherical form is effectively initiated, allowing further simplification using trigonometric identities.

Equation Transformation

Transforming an equation from Cartesian to spherical coordinates involves substitution and simplification leveraging trigonometric concepts. Let's break down this process in our exercise example.

Initially, substituting the spherical coordinate expressions for x, y, and z into the equation \( z^2 = 3x^2 + 3y^2 \) sets the foundation. The focus shifts to algebraically manipulating the equation using identities, like the Pythagorean identity. Here, after substitution:

• \( (\rho \cos(\phi))^2 = 3(\rho \sin(\phi) \cos(\theta))^2 + 3(\rho \sin(\phi) \sin(\theta))^2 \)

Upon applying trigonometric identities, the equation elegantly simplifies to:

• \( \rho^2 \cos^2(\phi) = 3 \rho^2 \sin^2(\phi) \)

Next, by dividing through by \( \rho^2 \), provided \( \rho eq 0 \), the equation becomes:

• \( \cos^2(\phi) = 3 \sin^2(\phi) \)

This is further simplified using the identity \( \cos^2(\phi) + \sin^2(\phi) = 1 \), effectively extracting angles that satisfy the condition. Consequently, knowing the trigonometric solutions leads to angles such as \( \phi = \frac{\pi}{6} \) and \( \phi = \frac{5\pi}{6} \), illustrating a successful transformation.

The transformation process shows how powerful these mathematical techniques are in regulating complex problems into comprehendible solutions.