Problem 43 - Advanced Engineering Mathematics

Question

In Problems \(43-46\), convert the given equation to cylindrical coardinates. $$ x^{2}+y^{2}+z^{2}=25 $$

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \\(r^2 + z^2 = 25\\).

Step 1: Understand the Relation between Cartesian and Cylindrical Coordinates

In Cartesian coordinates, a point in three-dimensional space is given by \(x, y, z\). In cylindrical coordinates, a point is given by \(r, \theta, z\) where: - \(r\) is the radial distance from the z-axis, - \(\theta\) is the angle in the xy-plane from the positive x-axis, and - \(z\) is the same as in Cartesian coordinates.

Step 2: Convert Cartesian Coordinates to Cylindrical Coordinates

Using the conversion formulas: - \(r = \sqrt{x^2 + y^2}\),- \(\theta = \tan^{-1}(\frac{y}{x})\), and- \(z = z\).We can express the equation \(x^2 + y^2 + z^2 = 25\) in terms of cylindrical coordinates.

Step 3: Substitute Cartesian Expressions for \(x^2 + y^2\) into Cylindrical Coordinates

Since \(r = \sqrt{x^2 + y^2}\), the expression for \(x^2 + y^2\) becomes \(r^2\) in cylindrical coordinates. So, substitute this into the equation to get: \(r^2 + z^2 = 25\).

Step 4: Write the Final Equation in Cylindrical Coordinates

The equation \(r^2 + z^2 = 25\) is the representation of the original Cartesian equation in cylindrical coordinates. Therefore, the expression of the given equation in cylindrical coordinates is: \(r^2 + z^2 = 25\).

Key Concepts

Coordinate Conversion

Coordinate conversion is like translating words from one language to another. In geometry, it's changing how we express a point's location. This is important because different problems are easier to solve in different coordinate systems. While Cartesian coordinates use a grid with straight-line measurements along the x, y, and z axes, cylindrical coordinates streamline circular symmetry problems.

The conversion between these two includes:

• Finding the radial distance, denoted by \(r\), using the formula \(r = \sqrt{x^2 + y^2}\)
• Determining the angle, \(\theta\), in the xy-plane from the positive x-axis with \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)
• Keeping the height \(z\) the same as it is already in Cartesian coordinates.

These formulas help us switch from the rectangular grid of Cartesian coordinates to the circular view of cylindrical coordinates, forming the basis for converting equations like \(x^2 + y^2 + z^2 = 25\) into a format that's often easier to understand and manipulate for certain types of problems.

Cartesian Coordinates

Cartesian coordinates are a staple tool for describing locations in three-dimensional space, using three perpendicular axes labeled x, y, and z. Think of these as directions: x is left-right, y is forward-backward, and z is up-down.

When describing a sphere centered at the origin, an equation like \(x^2 + y^2 + z^2 = 25\) captures all points that are 5 units away from the center, forming a perfect sphere.

It is important in geometric contexts because many actions or forces (like gravity, light, and sound) naturally exhibit symmetry around a central point. This makes Cartesian coordinates especially effective for visualizing and calculating problems where three-dimensional layers interact. For example, complex intersections and angles are often easier to outline in this straightforward system.

Equation Transformation

Equation transformation involves rewriting equations to fit different systems or simplify calculations. In this context, it's moving an equation from Cartesian coordinates to cylindrical coordinates. This isn't just changing the format—it's adapting the equation so it represents the same reality from a fresh perspective.

Take \(x^2 + y^2 + z^2 = 25\) as an example. In cylindrical coordinates:

• The term \(x^2 + y^2\) is replaced by \(r^2\), reflecting the radial component.
• This changes the equation to \(r^2 + z^2 = 25\) in cylindrical form.

Transforming equations is crucial. It reveals new properties or simplifies the problem, depending on the coordinate system's suitability. Each system highlights different aspects of physical phenomena, and this flexibility is the real power behind mathematical analysis.