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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.
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.
Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field v {\displaystyle \mathbf {v} } (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as ∇ ⊗ v {\displaystyle \nabla \otimes \mathbf {v} } , where ⊗ {\displaystyle \otimes ...
Del in cylindrical and spherical coordinates – Mathematical gradient operator in certain coordinate systems; Differentiation rules – Rules for computing derivatives of functions; Exterior calculus identities; Exterior derivative – Operation on differential forms; List of limits
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:
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
Harris English lines up a putt on the fourth hole on the South Course at Torrey Pines during the third round of the Farmers Insurance Open golf tournament Friday, Jan. 24, 2025, in San Diego.
Another way to write the d'Alembertian in flat standard coordinates is . This notation is used extensively in quantum field theory , where partial derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian.