Problem 69 - Advanced Engineering Mathematics

Question

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 \( \cot(\phi) = \sqrt{3} \), with \(\phi = \frac{\pi}{6}\) or \(\phi = \frac{5\pi}{6}\).

Step 1: Introduce Spherical Coordinates

In spherical coordinates, a point in space is described by three parameters: radius \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \). The conversion from Cartesian to spherical coordinates follows these relationships: \[ x = r \sin(\phi) \cos(\theta) \]\[ y = r \sin(\phi) \sin(\theta) \]\[ z = r \cos(\phi) \].We will use these relationships to convert the Cartesian equation to spherical coordinates.

Step 2: Substitute x, y, z in the Equation

Replace \(x\), \(y\), and \(z\) in the original equation with their spherical equivalents:\[ z^2 = 3x^2 + 3y^2 \Rightarrow (r \cos(\phi))^2 = 3(r \sin(\phi) \cos(\theta))^2 + 3(r \sin(\phi) \sin(\theta))^2 \].

Step 3: Simplify the Equation

Let's simplify the right-hand side first:\[(r \cos(\phi))^2 = 3(r^2 \sin^2(\phi) \cos^2(\theta)) + 3(r^2 \sin^2(\phi) \sin^2(\theta)) \].Factoring out terms:\[ r^2 \cos^2(\phi) = 3r^2 \sin^2(\phi)(\cos^2(\theta) + \sin^2(\theta)) \].Since \(\cos^2(\theta) + \sin^2(\theta) = 1\), we have:\[ r^2 \cos^2(\phi) = 3r^2 \sin^2(\phi) \].

Step 4: Solve for Spherical Coordinates

Cancel \( r^2 \) from both sides of the equation (assuming \(r eq 0\)):\[\cos^2(\phi) = 3\sin^2(\phi) \].Divide both sides by \( \sin^2(\phi) \):\[\frac{\cos^2(\phi)}{\sin^2(\phi)} = 3 \Rightarrow \cot^2(\phi) = 3 \].Taking the square root of both sides, we obtain:\[ \cot(\phi) = \sqrt{3} \].

Step 5: Solve for φ (Phi)

Solve \( \cot(\phi) = \sqrt{3} \) to find the possible values for \(\phi\). Recall that \( \cot(\phi) = \frac{1}{\tan(\phi)} \), and that \(\phi = \frac{\pi}{6}\) or \(\phi = \frac{5\pi}{6}\) are angles whose cotangent equals \(\sqrt{3}\). Hence, \(\phi = \frac{\pi}{6}\) or \(\phi = \frac{5\pi}{6}\).

Key Concepts

Coordinate Transformation

Understanding the concept of coordinate transformation is essential when moving from one coordinate system to another, like Cartesian to spherical coordinates. This method involves using specific formulas to translate points, making calculations and interpretations more manageable in different contexts. In the exercise, we transform a Cartesian equation to spherical coordinates.
Here's a quick reminder on how this translation occurs:

• The Cartesian coordinate system is characterized by three axes: x, y, and z.
• In spherical coordinates, a point is defined by a radius \( r \), a polar angle \( \theta \), and an azimuthal angle \( \phi \).
• The relationships are defined as:
  • \( x = r \sin(\phi) \cos(\theta) \)
  • \( y = r \sin(\phi) \sin(\theta) \)
  • \( z = r \cos(\phi) \)

By substituting Cartesian variables with these expressions, we begin transforming the original equation into an equivalent spherical form.

Trigonometric Identities

Trigonometric identities play a pivotal role when simplifying equations, especially those involving spherical coordinates. They help in reducing the complexity of the expressions.
In the original solution, one of the most fundamental trigonometric identities is used: \( \cos^2(\theta) + \sin^2(\theta) = 1 \). This identity simplifies the expression significantly by combining terms under a single trigonometric function.
Consider the simplification process from the exercise:

• Initially, the expression \( 3(r^2 \sin^2(\phi) \cos^2(\theta)) + 3(r^2 \sin^2(\phi) \sin^2(\theta)) \) might look complex.
• Using \( \cos^2(\theta) + \sin^2(\theta) = 1 \), this simplifies to \( 3r^2 \sin^2(\phi) \), hence reducing the equation significantly.

After this simplification using trigonometric identities, the equation becomes more manageable, guiding us towards finding the solutions for \( \phi \).

Polar Angles

Understanding the significance of polar angles within spherical coordinates is crucial for geometric interpretations. The polar angle \( \phi \) measures the angle from the positive z-axis to the point, providing a clear positional relationship in 3D space.
In our problem, solving for the angles \( \phi \) involves interpreting the cotangent function. Here are the key takeaways:

• The relationship \( \cot^2(\phi) = 3 \) simplifies to \( \cot(\phi) = \sqrt{3} \), which helps determine the possible angles.
• Recognizing that \( \cot(\phi) = \sqrt{3} \) leads us to specific angles: \( \phi = \frac{\pi}{6} \) and \( \phi = \frac{5\pi}{6} \), which align with common angles on the unit circle.

These angles describe how our 3D point is oriented relative to the z-axis, offering keen insight into its spatial position. Polar angles help in visualizing the entire structure of spherical coordinates, making such problems not only solvable but also intuitive.