Problem 43 - Advanced Engineering Mathematics

Question

Conven the given equation to cylindrical codrdinates. $$ x^{2}+y^{2}+z^{2}=25 $$

Step-by-Step Solution

Answer

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

Step 1: Identify Cartesian to Cylindrical conversion formulas

To convert from Cartesian coordinates \(x, y, z\) to cylindrical coordinates \(r, \theta, z\), we use the formulas: \(r = \sqrt{x^2 + y^2}\), \(\theta = \tan^{-1}(\frac{y}{x})\), and \(z = z\).

Step 2: Apply conversion for \\(x^2 + y^2\\)

Substitute the expression \(x^2 + y^2\) in Cartesian coordinates with \(r^2\) in cylindrical coordinates, since \(r^2 = x^2 + y^2\).

Step 3: Rewrite the equation in cylindrical coordinates

Replace \(x^2 + y^2\) with \(r^2\) in the given equation. Therefore, the equation \(x^2 + y^2 + z^2 = 25\) becomes \(r^2 + z^2 = 25\).

Key Concepts

Coordinate Transformation

Changing coordinates from one system to another is often necessary in math and physics. This process is known as coordinate transformation. It allows us to reinterpret geometric shapes and equations in terms that might be more appropriate for the specific problem we're solving.

• A common transformation is from Cartesian coordinates to cylindrical coordinates.
• This involves using a set of equations to change how we express points and distances.
• The purpose is often to make calculations simpler or to align more closely with a physical system.

Understanding coordinate transformations requires familiarity with both the initial and the target coordinate systems. This means knowing how to express one coordinate system in terms of another. By getting comfortable with these transformations, we can easily switch perspectives depending on what suits our analysis best.

Cartesian Coordinates

Cartesian coordinates are one of the most straightforward ways to describe points in space. They rely on orthogonal axes—usually labeled as \(x, y,\) and \(z\)—that intersect at a right angle.

• This system is easy to understand and use because it's intuitive and maps well onto our usual sense of direction.
• Each point is described as a set of three values \((x, y, z)\), representing distances from the planes formed by the axes.
• It's ideal for solving many types of problems, especially when dealing with straight-line distances and right angles.

However, this isn’t always the most efficient system for certain shapes and problems, particularly those involving symmetry around an axis.

Cylindrical Coordinate System

The cylindrical coordinate system provides a great alternative for problems involving circular symmetry or when working with rotations. It blends features of both Cartesian and polar coordinate systems.

• Here, each point is described using \(r, \theta, \) and \(z\).
• \(r\) represents the radial distance from the origin to the point's projection in the \(xy\)-plane.
• \(\theta\) is the angle made with the positive \(x\)-axis, offering a rotational component not found in Cartesian coordinates.
• \(z\) remains the same as in Cartesian coordinates, describing the height above or below the \(xy\)-plane.

This system is particularly useful when dealing with problems involving cylinders, as it mirrors the geometry of the shapes. It directly relates to the convertibility provided by formulae like \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\), making it a versatile and highly useful coordinate system.