Question

Explain the difference between the roles \(r,\) in cylindrical coordinates, and \(\rho\), in spherical coordinates, play in determining the location of a point.

Short answer

\( r \) measures distance to the z-axis on the xy-plane, \( \rho \) measures full distance from origin to the point.

Step-by-step solution

Understanding Cylindrical Coordinates

Cylindrical coordinates is a system where the location of a point in three-dimensional space is determined by three variables: \( r \), the radial distance from the z-axis; \( \theta \), the azimuthal angle in the xy-plane from the positive x-axis; and \( z \), the height above the xy-plane. Here, \( r \) represents the distance from the point to the z-axis, making it similar to the radius in polar coordinates, projecting on the xy-plane.

Understanding Spherical Coordinates

In spherical coordinates, a point's position is determined by three variables: \( \rho \), the radial distance from the origin to the point; \( \theta \), the azimuthal angle similar to that in cylindrical coordinates; and \( \phi \), the polar angle measured from the positive z-axis. The role of \( \rho \) is different from \( r \) in that it measures the full distance from the origin to the point, regardless of its direction.

Comparing \( r \) and \( \rho \)

While both \( r \) and \( \rho \) are radial distances, their references differ: \( r \) measures distance perpendicular to the z-axis, effectively on the xy-plane, while \( \rho \) measures the direct, straight-line distance from the origin to the point in space.

Key concepts

Spherical coordinates

Spherical coordinates offer a comprehensive way to determine the position of a point in three-dimensional space. Unlike a rectangular or Cartesian coordinate system, which uses \(x, y, z\) to locate a point, the spherical system employs three variables: \(\rho\), \(\theta\), and \(\phi\). Each variable provides unique information about the point's location relative to the origin.
The main advantages of spherical coordinates lie in their suitability for scenarios involving spherical symmetry, such as physics problems related to gravitational fields or radio waves. These coordinates tend to simplify the mathematics involved, particularly when dealing with angles and distances in three-dimensional spheres.

Radial distance

The radial distance in spherical coordinates is represented by \(\rho\). This is a crucial component as it measures the straight-line distance from the origin to the point in question. \(\rho\) is analogous to how you might use a ruler to measure the length between two points directly, bypassing any intermediary space.
This direct measurement makes \(\rho\) fundamentally different from the radial distance \(r\) in cylindrical coordinates, where \(r\) measures distance only within the xy-plane, perpendicular to the z-axis. In summary, \(\rho\) provides the full three-dimensional radial distance.

Azimuthal angle

The azimuthal angle, denoted as \(\theta\), is an integral part of both spherical and cylindrical coordinate systems. It represents the angle around the z-axis, starting from the positive x-axis. You can think of it as a "compass" direction on the horizontal plane.

• In spherical coordinates, \(\theta\) takes on the same role as in cylindrical coordinates, providing the rotational component around the z-axis.
• To visualize this, imagine rotating a point 360 degrees around the z-axis; \(\theta\) tells you exactly where that point lands in relation to the x-axis.

This angle is essential for pinpointing a point's location, especially when combined with the other two variables, \(\rho\) and \(\phi\), in spherical coordinates.

Polar angle

The polar angle in spherical coordinates is represented as \(\phi\). This angle is measured from the positive z-axis to the point in question, effectively describing how far "up" or "down" a point is in relation to a sphere's surface.

• In a way, \(\phi\) helps differentiate points on the same radial distance \(\rho\) by providing a vertical component.
• Think of \(\phi\) as an elevator guide, indicating at which "level" a point is situated above or below the xy-plane.

Together with the azimuthal angle \(\theta\), the polar angle \(\phi\) plays a crucial role in accurately determining a point's position in three dimensions, rounding out the information that spherical coordinates provide.