**1.**

Angular velocity ? is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:.

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Angular velocity ? is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:.

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Knowing, we conclude that the radial component of the Angular velocity is given by, because is a radial unit vector; and the perpendicular component is given by because is a perpendicular unit vector.

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In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction.

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Components of the spin angular velocity pseudovector were first calculated by Leonhard Euler using his Euler angles and the use of an intermediate frame:.

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Angular velocity must be the same for the three vectors, so arranging the three vector equations into columns of a matrix, we have:.

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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.

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In three dimensions, angular velocity can be represented by a pseudovector because second rank tensors are dual to pseudovectors in three dimensions.

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Since the angular velocity tensor W = W is a skew-symmetric matrix:.

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Since the spin angular velocity tensor of a rigid body is a linear transformation that maps positions to velocities, it can be regarded as a constant vector field.

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In particular, the spin angular velocity is a Killing vector field belonging to an element of the Lie algebra SO of the 3-dimensional rotation group SO.

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We should prove that the spin angular velocity previously defined is independent of the choice of origin, which means that the spin angular velocity is an intrinsic property of the spinning rigid body.

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