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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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Related concepts (5)
Vector fields in cylindrical and spherical coordinates
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.
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form.
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In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region.
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