Particle in two dimensions
In the simplest case of circular motion at radius , with position given by the angular displacement from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: .

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Given a rotating frame of three unit coordinate vectors, all the three must have the same angular speed at each instant. In such a frame, each vector may be considered as a moving particle with constant scalar radius.
The rotating frame

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The angular velocity vector defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) W = W(t) defined by:
This is an

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In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor, which is a second rank skew-symmetric tensor.
This tensor W will have n(n−1)/2 independent components, which is the dimension of the

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• Angular acceleration
• Angular frequency
• Angular momentum

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• A college text-book of physics By Arthur Lalanne Kimball (Angular Velocity of a particle)
• Pickering, Steve (2009). "ω Speed of Rotation [Angular Velocity]". Sixty Symbols. Brady Haran for the University of Nottingham.

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The same equations for the angular speed can be obtained reasoning over a rotating rigid body. Here is not assumed that the rigid body rotates around the origin. Instead, it can be supposed rotating around an arbitrary point that is moving with a linear velocity V(t)

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