Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Make the required change in the given equation. \(\rho \sin \phi=1\) to Cartesian coordinates

Step-by-Step Solution

Answer

The equation in Cartesian coordinates is \(y = 1\).

Step 1: Identify the Conversion Formulas

When converting from cylindrical ( \(\rho, \phi\) ) to Cartesian coordinates (\(x, y\)), we use the following relations:\[ x = \rho \cos \phi \]\[ y = \rho \sin \phi \]

Step 2: Substitute the Expression for \(y\)

Given that \(\rho \sin \phi = 1\), we can substitute the equation in terms of \(y\) since \(y = \rho \sin \phi\). Thus the equation becomes:\[ y = 1 \]

Step 3: Write the Equation in Cartesian Coordinates

Since we have determined that \(y = 1\), and there is no restriction on \(x\), the equation in Cartesian coordinates is simply:\[ y = 1 \]

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are particularly useful in situations where there is symmetry around an axis, much like polar coordinates but extendable into three dimensions. They are expressed as \((\rho, \phi, z)\), where:

• \(\rho\) represents the radial distance from the origin to the point in the \(xy\)-plane.
• \(\phi\) is the angular coordinate, often measured from the positive x-axis.
• \(z\) is the height or elevation above the \(xy\)-plane.

Using these coordinates can simplify the analysis of problems with circular or cylindrical symmetry, such as many in electromagnetism and fluid dynamics.Understanding this coordinate system is crucial for performing conversions to Cartesian coordinates, a topic we'll delve into next.

Coordinate Conversion

Coordinate conversion is a key mathematical process that involves changing from one coordinate system to another. For example, converting from cylindrical coordinates \((\rho, \phi)\) to Cartesian coordinates \((x, y)\), we use the equations:

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

The above formulas allow us to seamlessly transition between the two systems. This conversion is necessary in many applications like solving mathematical problems or computing geographic data. Knowing how to navigate between coordinate systems ensures a broad understanding of spatial relationships in various contexts.

Equation Transformation

Equation transformation is the process of rewriting an equation in one coordinate system into an equivalent form in another. This is necessary when working with different mathematical models or when simplifying complex equations.In the given exercise, the equation \(\rho \sin \phi = 1\) was transformed from cylindrical to Cartesian coordinates using the relationship \(y = \rho \sin \phi\). By substituting \(\rho \sin \phi = 1\) with \(y = 1\), the transformation resulted in \(y = 1\) in Cartesian coordinates.Such transformations are vital for analysis and visualization, making them indispensable in applied mathematics and engineering. They help in understanding the geometry and shape of plots in various simulated or real-world models. By practicing equation transformations, one can gain deeper insights into the structure and implications of mathematical models.