Utilisateur:Gigowatts/Brouillon opérateur nabla en coordonnées cylindriques et sphériques

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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 θ: 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 ϕ: 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 image. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].

Formulae

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Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates
Operation Cartesian coordinates (x, y, z) Cylindrical coordinates (ρ, ϕ, z) Spherical coordinates (r, θ, ϕ) Parabolic cylindrical coordinates (σ, τ, z)
Definition
of
coordinates
Definition
of
unit
vectors
A vector field
scalar field Gradient
Divergence
Curl
Laplace operator
Vector Laplacian Modèle:Collapsible section Modèle:Collapsible section
Material derivative[1]

Modèle:Collapsible section Modèle:Collapsible section
Differential displacement
Differential normal area
Differential volume
Non-trivial calculation rules:
  1. (Lagrange's formula for del)

See also

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References

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  1. Weisstein, Eric W., « Convective Operator », Mathworld (consulté le )
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[[Category:Vector calculus]] [[Category:Coordinate systems]]