Coordinate Systems

A Cartesian coordinate system is defined by three right-handed mutually orthogonal axes, often labeled X, Y, and Z, that meet at the origin. The position of a point is given by its x, y, and z coordinates along the axes of the coordinate system. Vectors are described with respect to unit basis vectors (see Eqn. (5185)) aligned with the X, Y, and Z axes. The unit basis vectors that define a Cartesian coordinate system are often labeled i, j, and k.

Vectors and tensors are defined with respect to unit basis vectors aligned with the radial, tangential, and axial directions. The axial unit vector is fixed with respect to the reference coordinate system, whereas the radial and tangential unit vectors depend on the location within the reference system. The cylindrical unit basis vectors at a point $(x_R,y_R,z_R)$ are:

For physical domains with spherical symmetry, such as systems that are invariant for rotations about a point, it is convenient to define quantities with respect to a spherical coordinate system. In a spherical coordinate system, the position of a point is given by its radial, polar, and azimuthal coordinates $r$, $\theta$, and $\varphi$:

A spherical coordinate system is defined with respect to a reference Cartesian coordinate system, which can be either the Laboratory system, or a local Cartesian coordinate system. If $(x_R,y_R,z_R)$ are the coordinates of a point in the reference Cartesian coordinate system, the coordinates of the point in the spherical coordinate system are:

Vectors and tensors are defined with respect to unit basis vectors aligned with the radial, polar, and azimuthal directions. The spherical unit basis vectors at a point $(x_R,y_R,z_R)$ are: