Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.
Concept
Vector fields in cylindrical and spherical coordinates
Résumé
Note: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken in comparing different sources.
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:
or inversely by:
Any vector field can be written in terms of the unit vectors as:
The cylindrical unit vectors are related to the Cartesian unit vectors by:
Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.
To find out how the vector field A changes in time, the time derivatives should be calculated.
For this purpose Newton's notation will be used for the time derivative ().
In Cartesian coordinates this is simply:
However, in cylindrical coordinates this becomes:
The time derivatives of the unit vectors are needed.
They are given by:
So the time derivative simplifies to:
The second time derivative is of interest in physics, as it is found in equations of motion for classical mechanical systems.
The second time derivative of a vector field in cylindrical coordinates is given by:
To understand this expression, A is substituted for P, where P is the vector (ρ, φ, z).
This means that .
After substituting, the result is given:
In mechanics, the terms of this expression are called:
Centripetal forceAngular acceleration and Coriolis effect
Vectors are defined in spherical coordinates by (r, θ, φ), where
r is the length of the vector,
θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and
φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π).
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.