1.1. Basis Vectors

We have drawn the z=0 plane in Figure 1b. There is a baseline or reference line relative to which angle θ is measured. The length of the line joining the center and the projected point is r while the angle that it makes from the baseline is θ.

In cylindrical coordinates, we start by observing the z=0 plane. This is the horizontal plane, where we can visualize the setup more clearly. The angle θ is measured from a reference line emanating from the center of the cylindrical coordinate system. The distance from this center to any point on the plane is denoted by r. This setup is crucial because it helps us understand how points in cylindrical coordinates are defined and measured.

Think of a round table where you measure the distance to a guest seated at the table. The distance from the center of the table is r, and the angle from a reference line (like a pointing direction) to the guest is θ.

Two of the basis vectors lie in this plane: e_r points radially outward from the center while e_θ points in the direction of increasing θ and is perpendicular to e_r. The third basis vector e_z lies along the axis of the cylinder.

In a cylindrical coordinate system, basis vectors are unit vectors that define directions in the system. e_r points outward from the center of the cylinder, representing the radial direction. e_θ points tangentially around the cylinder and is perpendicular to e_r, indicating the angular direction of increasing θ. Finally, e_z points vertically along the height of the cylinder, completing our set of three basis vectors, essential for describing any motion or force in this space.

Imagine a cylindrical can. If you were to point from its center to the outer edge, that’s e_r. If you then rotated your finger around the edge of the can, that would be direction e_θ. Lastly, if you were to point straight up, that is the direction of e_z.

Thus, the basis vectors for a cylindrical coordinate system are (e_r, e_θ, e_z) whereas in the Cartesian coordinate system, the basis vectors are (e_1, e_2, e_3). Although both sets of basis vectors are orthonormal, there is a big difference in their properties: the basis vectors of the Cartesian coordinate system are fixed in direction and do not change from one point to the other but in cylindrical coordinate systems, two of the basis vectors (e_r, e_θ) change when θ coordinate of a point changes.

In Cartesian coordinates, the basis vectors e_1, e_2, and e_3 point in fixed directions along the x, y, and z axes, respectively. They remain constant regardless of the position of the point in space. In contrast, in a cylindrical coordinate system, the basis vectors e_r and e_θ change their orientations depending on the angle θ. This makes cylindrical coordinates particularly useful for problems involving circular or rotational symmetries, as it allows greater flexibility in describing motion.

Consider how a camera positioned at a fixed point will always point in the same direction (like Cartesian coordinates). In contrast, if you were to rotate around a circular track, your direction of view would change as you moved along the track (like the basis vectors in cylindrical coordinates).

This has been illustrated in Figure 2a where the basis vectors at the two points are differently oriented although only the θ coordinate of the two points is different. The third basis vector e_z remains fixed at every point though. One thing to note here is that e_r and e_θ change only when the θ coordinate changes. If only the radial coordinate or/and z-coordinate of a point is changed, the basis vectors do not change.

Figure 2a demonstrates that as you move around the cylindrical coordinate system, the orientations of e_r and e_θ vary based on the value of θ. However, e_z remains constant and does not change as you move up or down along the cylinder's height. This is an important property in understanding how forces and velocity vectors behave in cylindrical coordinates, as it indicates that radial and angular displacements affect the basis vectors, but vertical displacements do not.

Think of a bicycle wheel. As the wheel spins, the spokes (which we can think of as similar to the basis vectors e_r and e_θ) change direction depending on where they are positioned on the wheel, but the vertical rim (akin to e_z) stays the same regardless of how the wheel rotates or moves up and down.

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