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This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
For a vector expressed in local unit cylindrical coordinates as = + +, where e a is the unit vector in direction a, the divergence is [1] = = + +. The use of local coordinates is vital for the validity of the expression.
The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero.
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane ...
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:
In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R N with r representing a positive real radius and θ an element of the unit sphere S N−1, = + + where Δ S N−1 is the Laplace–Beltrami operator on the (N − 1)-sphere, known as the spherical Laplacian.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point. The power of the del notation is shown by the following product rule: