Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-30, 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: Recall the Conversion Formulas

To convert from Cartesian coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\), use the formulas: - \(x = r \cos \theta\)- \(y = r \sin \theta\)- \(z = z\) (stays the same) - \(r^2 = x^2 + y^2\).

Step 2: Substitute Cartesian to Cylindrical

In the given equation \(x^{2}+y^{2}+4z^{2}=10\), substitute the equivalent cylindrical coordinates:- \(x^2 + y^2 = r^2\)- The equation becomes \(r^2 + 4z^2 = 10\).

Step 3: Simplify the Equation

The cylindrical form of the equation is \(r^2 + 4z^2 = 10\). This expresses the equation in terms of \((r, \theta, z)\), where \(\theta\) is not present. This is a simplified representation.

Key Concepts

Conversion Formulas

When working with different coordinate systems, conversion formulas are essential. These formulas help us shift from one form to another. In the realm of 3D coordinate systems, cylindrical coordinates are often used, especially when dealing with problems that have rotational symmetry.
To move from Cartesian coordinates \(x, y, z\) to cylindrical coordinates \(r, \theta, z\):

• \(x = r \cos \theta\) describes the relationship of the x-coordinate in terms of \(r\) and \(\theta\).
• \(y = r \sin \theta\) expresses the y-coordinate similarly.
• \(z = z\) remains unchanged, hinting that in cylindrical coordinates, the z-dimension shares the same form as in Cartesian coordinates.
• \(r^2 = x^2 + y^2\) provides the radial distance in terms of x and y.

These formulas help transition from a rectangular space, allowing one to explore the geometry in a more intuitive axis-point angle model.

Cylindrical Form

Understanding the cylindrical form of an equation means recognizing how shapes and spaces can look different depending on the coordinate system utilized.
Cylindrical coordinates \(r, \theta, z\) are particularly useful because:

• \(r\) represents the distance from the z-axis, showing how far out the point is radially.
• \(\theta\) indicates the angle that the projection of the point makes with the positive x-axis, essentially capturing the rotational component.
• \(z\) remains the same as in Cartesian coordinates and gives the height or depth from a fixed reference plane.

For the given equation \(x^2 + y^2 + 4z^2 = 10\), replacing \(x^2 + y^2\) with \(r^2\) shifts it into cylindrical form: \(r^2 + 4z^2 = 10\). This highlights the role \(r\) and \(z\) play within the spatial structure, with \(\theta\) absent as the equation is rotationally symmetric.

Simplifying Equations

Simultaneously transforming from Cartesian to cylindrical coordinates can simplify equations by reducing complexity and highlighting symmetry.
The original equation, \(x^2 + y^2 + 4z^2 = 10\), when translated to cylindrical coordinates becomes \(r^2 + 4z^2 = 10\). This reveals a clearer picture of the spatial configuration.
Eliminating \(\theta\) in the transition highlights:

• The symmetry around the z-axis, as the equation does not vary with changes in \(\theta\).
• Allows for focusing analysis on radial distance \(r\) and vertical displacement \(z\) exclusively.

This is a key reason mathematicians and engineers favor cylindrical coordinates in contexts such as electromagnetism and fluid dynamics, where rotational symmetry abounds. The act of simplification is more than reduction; it is about unveiling the essence of the structure within the coordinate framework.