Del in cylindrical and spherical coordinates

This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. 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. The azimuthal angle is denoted by : it is the angle between the x-axis and the projection of the radial vector onto the xy-plane. The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and . The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π]. CAUTION: the operation must be interpreted as the two-argument inverse tangent, atan2. This page uses for the polar angle and for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses for the azimuthal angle and for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch and in the formulae shown in the table above. (Lagrange's formula for del) The expressions for and are found in the same way. The unit vector of a coordinate parameter u is defined in such a way that a small positive change in u causes the position vector to change in direction. Therefore, where s is the arc length parameter. For two sets of coordinate systems and , according to chain rule, Now, we isolate the th component. For , let .

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Del in cylindrical and spherical coordinates

Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

Cylindrical coordinate system

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 containing the purple section). The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

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