This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes 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 can be used instead of the mathematical function owing to its domain and image. The classical arctan function has an image of , whereas atan2 is defined to have an image of . Coordinate conversions CAUTION: the operation must be interpreted as the two-argument inverse tangent, atan2. Unit vector conversions Del formula 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. Calculation rules (Lagrange's formula for del) Cartesian derivation The expressions for and are found in the same way. Cylindrical derivation Spherical derivation Unit vector conversion formula 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 is the arc length parameter. For two sets of coordinate systems and , according to chain rule, Now, we isolate the th component. For , let . Then divide on both sides by to get: See also Del Orthogonal coordinates Curvilinear coordinates Vector fields in cylindrical and spherical coordinates References External links Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates. Vector calculus Coordinate systems