Vector fields in cylindrical and spherical coordinates

Note: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken in comparing different sources. Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by: or inversely by: Any vector field can be written in terms of the unit vectors as: The cylindrical unit vectors are related to the Cartesian unit vectors by: Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. To find out how the vector field A changes in time, the time derivatives should be calculated. For this purpose Newton's notation will be used for the time derivative (). In Cartesian coordinates this is simply: However, in cylindrical coordinates this becomes: The time derivatives of the unit vectors are needed. They are given by: So the time derivative simplifies to: The second time derivative is of interest in physics, as it is found in equations of motion for classical mechanical systems. The second time derivative of a vector field in cylindrical coordinates is given by: To understand this expression, A is substituted for P, where P is the vector (ρ, φ, z). This means that . After substituting, the result is given: In mechanics, the terms of this expression are called: Centripetal forceAngular acceleration and Coriolis effect Vectors are defined in spherical coordinates by (r, θ, φ), where r is the length of the vector, θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π).