Vector fields in cylindrical and spherical coordinates
Vector field representation in 3D curvilinear coordinate systems From Wikipedia, the free encyclopedia
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.[1]
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:
Second time derivative of a vector field
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:
The Cartesian unit vectors are thus related to the spherical unit vectors by:
Time derivative of a vector field
To find out how the vector field A changes in time, the time derivatives should be calculated.
In Cartesian coordinates this is simply:
However, in spherical coordinates this becomes:
The time derivatives of the unit vectors are needed. They are given by:
Thus the time derivative becomes: