Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to cylindrical coordinates

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \\(r^{2} + 4z^{2} = 10\\).

Step 1: Identify Cylindrical Coordinates

In cylindrical coordinates, we express points using \(r, \theta, z\), where \(r\) is the radial distance, \(\theta\) is the angle, and \(z\) remains the same as in Cartesian coordinates. We use the transformations \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Substitute Cylindrical Coordinates

Start by substituting the cylindrical coordinate expressions into the given equation: \((r \cos \theta)^{2} + (r \sin \theta)^{2} + 4z^{2}=10\).

Step 3: Simplify the Expression

Simplify the equation: \[ (r \cos \theta)^{2} + (r \sin \theta)^{2} = r^{2} \cos^{2} \theta + r^{2} \sin^{2} \theta = r^{2}(\cos^{2} \theta + \sin^{2} \theta) = r^{2} \] because \(\cos^{2} \theta + \sin^{2} \theta = 1\).

Step 4: Rewrite the Equation

Now the equation becomes \(r^{2} + 4z^{2} = 10\). This is the equation in cylindrical coordinates.

Key Concepts

Coordinate Transformation

Understanding coordinate transformation is essential in converting equations between different coordinate systems. When we talk about coordinate transformation, we mean changing from one type of coordinate representation to another, such as from Cartesian to cylindrical coordinates. These transformations help in solving mathematics problems that involve symmetries or geometrical shapes like cylinders or circles.

In the case of transforming Cartesian coordinates to cylindrical coordinates, we use the following transformations:

• Convert the horizontal plane components:
  • The radial distance \(r\) is calculated using \(x\) and \(y\) as \(r = \sqrt{x^2 + y^2}\).
  • The angle \(\theta\) is derived using \(\tan \theta = \frac{y}{x}\).
• The \(z\) coordinate remains the same as in Cartesian coordinates.

These transformations are crucial when dealing with equations involving rotational symmetry or axial structures. Transforming coordinates not only makes the mathematics more manageable but also provides a clearer visual and structural understanding of the problem.

Cartesian Coordinates

Cartesian coordinates are one of the most common ways to describe the position of a point in a plane, using two or three numerical values. In a two-dimensional plane, the Cartesian coordinate system uses two values (\(x, y\)), and in three-dimensional space, it uses three values (\(x, y, z\)).

The axes in Cartesian coordinates are perpendicular to each other. The values along these axes represent the point’s distance from the origin, which is the intersection point of the axes usually represented as (0,0,0) in a three-dimensional space. They provide simplicity when representing straight lines and angles. However, Cartesian coordinates can sometimes become complex when dealing with curves or circular shapes because these shapes are naturally non-linear in this system.

In problems that involve circular symmetry, such as cylinders, it is often advantageous to convert the Cartesian coordinates into cylindrical coordinates for simpler calculations. Understanding Cartesian coordinates fully serves as the foundation for understanding more complex systems.

Cylindrical Coordinates Conversion

Cylindrical coordinates combine the straightforward linearity of Cartesian coordinates with the circular aspect of polar coordinates. These are especially useful for problems involving circular symmetry around an axis, like the equation from the original exercise.

To convert Cartesian coordinates \((x, y, z)\) into cylindrical coordinates \((r, \theta, z)\):

• Calculate \(r\), the radial distance from the \(z\)-axis, as \(r = \sqrt{x^2 + y^2}\).
• Find \(\theta\), the angle in the \(xy\)-plane, using the relation \(\tan \theta = \frac{y}{x}\).
• Use the same \(z\) as in the Cartesian coordinate, as axial position remains unchanged.

These conversions make it simple to work with equations involving symmetry around an axis, as they naturally match the geometry of many physical systems. Successfully converting an equation means inserting these cylindrical terms into an original equation and simplifying it into a clearer form.

Equation Simplification

Simplifying equations is a fundamental skill in mathematics, making equations more concise and manageable to work with. The primary goal is to reduce complexity without altering the equation’s original meaning.

In the step-by-step solution provided, simplification was achieved by recognizing the trigonometric identity \(\cos^{2} \theta + \sin^{2} \theta = 1\). This allows the equation from the original exercise to be interpreted more visually and accurately in cylindrical coordinates.

Let's illustrate the transformation: When substituting \((x, y)\) in terms of \((r, \theta)\) results in:\
\[(r \cos \theta)^2 + (r \sin \theta)^2 + 4z^2 = 10\]
Then simplifying it by combining terms using \(\cos^{2} \theta + \sin^{2} \theta = 1\):
\[r^2 + 4z^2 = 10\]
This demonstrates the process of reducing terms to their simplest forms. Simplified notation offers not only elegance but also powerful insight into underlying physical phenomena or symmetries in problems.