Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-30, make the required change in the given equation. \(\rho=2 \cos \phi\) to cylindrical coordinates

Step-by-Step Solution

Answer

In cylindrical coordinates, the equation is \( r = 2 \) and \( z = 2 \).

Step 1: Understand Spherical Coordinates

The equation given is in spherical coordinates, where \( \rho \) is the radial distance, and \( \phi \) is the angle made with the positive z-axis. Our objective is to convert this equation to cylindrical coordinates.

Step 2: Identify Conversion Formulas

To convert from spherical to cylindrical coordinates, remember the relationships: \( \rho = \sqrt{r^2 + z^2} \), \( x = r \cos \theta \), \( y = r \sin \theta \), and \( z = \rho \cos \phi \).

Step 3: Substitute Values in Terms of Cylindrical Coordinates

Since \( \rho = 2 \cos \phi \), we use \( \rho = \sqrt{r^2 + z^2} \) and \( z = \rho \cos \phi \), which implies \( z = 2 \). Substitute these into the expression and solve for \( r \).

Step 4: Solve the Equation

From the conversion formulas, substitute \( \rho = \sqrt{r^2 + z^2} \) and set it equal to \( 2 \cos \phi = 2 \frac{z}{\rho} \). This gives us \( \sqrt{r^2 + z^2} = 2\frac{z}{\sqrt{r^2 + z^2}} \).

Step 5: Simplify to Find r

Square both sides to eliminate the square root: \( r^2 + z^2 = \left( \frac{2z}{\sqrt{r^2 + z^2}} \right)^2 \). Simplifying, \( r^2 + z^2 = \frac{4z^2}{r^2 + z^2} \). Multiply both sides by \( r^2 + z^2 \) to clear the fraction, giving \((r^2 + z^2)^2 = 4z^2 \).

Step 6: Express the Equation in Cylindrical Form

Continue simplifying: \( r^2 + z^2 = 2z \) (by taking the positive square root and since \( \rho \) represents a distance, it should be nonnegative), where \( z = 2 \). The equation becomes \( r^2 = 4 \), leading to \( r = 2 \). Hence, in cylindrical coordinates, the equation is \( r = 2, z = 2 \).

Key Concepts

Spherical Coordinates

Spherical coordinates offer a three-dimensional system that simplifies the representation of points in space where symmetry is about the origin. The system is defined using three components:

• \( \rho \) (rho) - the radial distance from the origin.
• \( \phi \) (phi) - the angle between the point and the positive z-axis.
• \( \theta \) (theta) - the azimuthal angle in the xy-plane from the positive x-axis.

This coordinate system is particularly useful when dealing with geometrical problems that naturally align with a central point, like spheres and cones. It translates complex xyz calculations into more manageable trigonometric equations. Each point is expressed as \((\rho, \phi, \theta)\), making it easier to describe the position of an object relative to a centric source. Understanding the components and their realizations is key to leveraging spherical coordinates in math and physics.
Imagine locating a point by tilting from the z-axis and swinging around the x-axis - that's the crux of spherical coordinates. They are widely used in electromagnetic theory, quantum mechanics, and various fields dealing with spherical symmetries.

Coordinate Conversion

Knowing how to move between coordinate systems is crucial in mathematics and physics, especially when calculations in one system become cumbersome. **Coordinate conversion** involves transforming equations and points between different coordinate systems - spherical, cylindrical, and Cartesian, to name a few.
For instance, converting from spherical to cylindrical coordinates requires substituting spherical variables \(\rho\) and \(\phi\) in terms of cylindrical coordinates \(r, \theta,\) and \(z\). The relationships include:

• \(\rho = \sqrt{r^2 + z^2}\)
• \(z = \rho \cos \phi\)
• \(r = \rho \sin \phi\)

These equations ensure the transformation preserves the spatial description from one form to another seamlessly. Converting coordinates simplifies solving integrals and differential equations, and provides clearer spatial interpretations depending on the problem at hand. It's like translating between languages - knowing the rules allows for fluency and understanding regardless of the "language" being used.

Equation Transformation

Equation transformation is similar to converting language in mathematics. It involves reformulating equations to fit a different system or condition better suited for analysis or computation. This is particularly beneficial in engineering, physics, and geometry where varying equations provide clearer problem-solving pathways.
In converting the equation \(\rho = 2 \cos \phi\) from spherical to cylindrical coordinates, the transformation follows systematic algebraic steps to replace \(\rho\) and \(\phi\) with corresponding cylindrical variables \(r\) and \(z\). Starting from the relationships:

• \(\rho = \sqrt{r^2+z^2}\)
• \(z = \rho \cos \phi\)

You substitute these into the equation, simplifying and rearranging terms to express it fully in cylindrical form.
The meticulous process results in an equation in cylindrical coordinates \(r = 2, z = 2\), which helps visualize geometry such as cylinders in a symmetric form. Transforming equations changes their appearance but not their geometric or physical meaning, allowing simpler or more intuitive analysis.