Problem 74 - Advanced Engineering Mathematics

Question

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \rho \sin ^{2} \phi=\cos \phi $$

Step-by-Step Solution

Answer

The equation in rectangular coordinates is \( x^2 + y^2 = z \).

Step 1: Understand the Given Equation

The equation \( \rho \sin^2 \phi = \cos \phi \) is given in spherical coordinates. Spherical coordinates consist of \( \rho \) (the radial distance), \( \phi \) (the polar angle), and \( \theta \) (the azimuthal angle). Our task is to convert this equation into rectangular coordinates \( (x, y, z) \).

Step 2: Recall Relationships Between Spherical and Rectangular Coordinates

In spherical coordinates, \( \rho = \sqrt{x^2 + y^2 + z^2} \), \( \sin \phi = \frac{r}{\rho} \) where \( r = \sqrt{x^2 + y^2} \), and \( \cos \phi = \frac{z}{\rho} \). We'll use these relationships for the conversion.

Step 3: Substitute Spherical Coordinate Expressions

Substitute \( \sin \phi = \frac{r}{\rho} \) and \( \cos \phi = \frac{z}{\rho} \) into the equation. Therefore, \( \rho \left( \frac{r}{\rho} \right)^2 = \frac{z}{\rho} \), simplifying to \( \frac{r^2}{\rho} = \frac{z}{\rho} \).

Step 4: Remove \( \rho \) from the Equation

We can cancel \( \rho \) from both sides (assuming \( \rho eq 0 \)), resulting in \( r^2 = z \). Recall that \( r^2 = x^2 + y^2 \), giving us \( x^2 + y^2 = z \).

Step 5: Write the Equation in Rectangular Coordinates

The equation in rectangular coordinates is \( x^2 + y^2 = z \).

Key Concepts

Spherical Coordinates

Spherical coordinates offer a convenient way to describe the position of a point in three-dimensional space. Unlike rectangular coordinates, which rely on the orthogonal x, y, and z axes, spherical coordinates use three values:

• \( \rho \): The radial distance from the origin to the point.
• \( \phi \): The polar angle measured from the positive z-axis.
• \( \theta \): The azimuthal angle, akin to longitude, measured in the xy-plane from the positive x-axis.

These coordinates are particularly useful when dealing with symmetrical objects like spheres or circles in three-dimensional space, as they simplify the equations governing their geometry. In this exercise, we begin with an equation in spherical coordinates and aim to express it using the more familiar rectangular coordinate system.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, employ a perpendicular triple-axis system (x, y, z) to specify a point's location. This method breaks down space into three dimensions, making it intuitive for many geometry and algebra problems.

• x-axis: Measures horizontal distance.
• y-axis: Measures vertical distance perpendicular to the x-axis.
• z-axis: Extends into the third dimension, perpendicular to both x and y axes.

For conversion tasks, rectangular coordinates might provide a straightforward framework, especially in problems requiring linear or quadratic expressions. In this exercise, the spherical equation is transformed into an equation of a surface with coordinates defined as \(x^2 + y^2 = z\). This indicates a paraboloid in R3, a typical shape dealt with in geometry.

Coordinate Transformation

Coordinate transformation involves changing the way a point's position is described. Moving from spherical to rectangular coordinates requires specific relationships between these two systems:

• Radial distance \(\rho\) is equal to the three-dimensional vector length, \(\sqrt{x^2 + y^2 + z^2}\).
• Polar angle \(\phi\) relates to the vertical elevation, expressed as \(\cos \phi = \frac{z}{\rho}\), and \(\sin \phi = \frac{\sqrt{x^2 + y^2}}{\rho}\).
• Azimuthal angle \(\theta\) has no direct influence in this problem since the original equation does not include \(\theta\).

By understanding these relationships, we can substitute the spherical values with the corresponding rectangular expressions. Thus, from \(\rho \sin^2 \phi = \cos \phi\), using the transformations, we derive \(x^2 + y^2 = z\). This systematic approach helps simplify multi-dimensional geometric queries.

Mathematical Problem Solving

Solving mathematical problems, particularly those involving coordinate transformations, requires a clear understanding of the relationships between different systems. Initially, it's crucial to comprehend the question and system you're working in. Retain fundamental relations and properties of each coordinate system at your fingertips.

• Identify the Coordinates: Recognize spherical components \(\rho, \phi, \theta\).
• Apply Known Formulas: Use relationships like \(\sin \phi\) and \(\cos \phi\) for transformations.
• Transform Methodically: Carefully substitute and simplify to switch to rectangular form.
• Check Your Result: Transformations should maintain the essence of the original equation in a new form.

In this exercise, these steps led us from a spherical equation to a rectangular equation, representing a parabolic surface. Keeping problem-solving strategies in mind ensures precision and accuracy in any coordinate transformation task.