Question

In Exercises \(41-46,\) convert the point from cylindrical coordinates to spherical coordinates. $$ (4,-\pi / 6,6) $$

Short answer

The cylindrical coordinates \((4,-\pi/6,6)\) are equivalent to the spherical coordinates \((\sqrt{52}, -\pi/6, arctan(2/3))\)

Step-by-step solution

Compute ρ

Firstly, calculate the value of ρ using the given r and z in the cylindrical coordinates. \(ρ = \sqrt{r^2+z^2} = \sqrt{4^2+6^2} = \sqrt{52}\).

Set θ to the given θ

In both cylindrical and spherical coordinate systems, θ represents the same azimuthal angle. Therefore, the spherical coordinate θ will simply be equal to the given value, i.e., \(-\pi/6\).

Compute φ

Use the formula to find φ. \(\phi = arctan(\frac{r}{z}) = arctan(\frac{4}{6}) = arctan(\frac{2}{3})\) where arctan is the inverse tangent function.

Key concepts

Coordinate Systems

Understanding different coordinate systems is fundamental in mathematics, physics, and engineering. It's how we describe the position of points in various types of space using a set of numbers. The most common systems are Cartesian (rectangular), cylindrical, and spherical coordinates.

Each system is useful for different types of problems. Cartesian coordinates are perfect for simple, flat space problems—think of plotting graphs on a piece of paper. By contrast, cylindrical and spherical coordinate systems shine when dealing with round shapes like cylinders or spheres. They help simplify the mathematics for problems like analyzing waves, vibrations, heat flow, and many other scenarios in three-dimensional space.

To maneuver between these systems, one must understand how they relate to one another. Converting coordinates from one system to another can be crucial for solving complex problems and is a standard task in subjects like vector calculus and electromagnetism.

Spherical Coordinates

Spherical coordinates are akin to geographical longitude and latitude on a globe but extend the concept into three dimensions. In this system, a point in space is defined by three values: the radius (ρ), the inclination (φ), and the azimuthal angle (θ).

The radius represents the distance from the origin to the point. The inclination is the angle from the positive z-axis to the point, and the azimuthal angle is the same concept as longitude—the angle from the positive x-axis within the xy-plane.

Finding Spherical Coordinates

To find the spherical coordinates from cylindrical ones, you apply specific transformations. For the radius (ρ), you would use the Pythagorean theorem to calculate the distance from the origin. The azimuthal angle (θ) remains unchanged as it is the same in both systems. For the inclination (φ), you would calculate the angle using trigonometric functions such as the arctangent of the ratio between the cylindrical radius (r) and height (z).

Cylindrical Coordinates

Cylindrical coordinates offer an alternative way to describe the location of points in three-dimensional space, with emphasis on circular symmetry. This system uses a combination of radial distance, angle, and height to specify a point in space.

The components are radius (r), representing distance from the z-axis, azimuthal angle (θ), indicating the rotational position similar to an angle in polar coordinates, and height (z), which corresponds to the Cartesian z-coordinate.

Application in Conversions

When converting to spherical coordinates, the cylindrical radius (r) helps to determine the spherical radius (ρ) and the inclination angle (φ). The cylindrical system is often more intuitive for problems dealing with structures like pipes, wires, or anything with a circular base, since the coordinates wrap around in a cylinder shape. Calculating and converting these coordinates involves a blend of algebraic manipulation and basic trigonometry to translate the information into other coordinate systems.