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Concept
Vector fields in cylindrical and spherical coordinates
Summary
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).
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