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18.104.22.168 Exponential Map
Conventionally, the rigid body rotation tensor corresponding to a rotation of angle about the unit vector may be expressed in terms of the vector as where is the third-order permutation pseudo-tensor with Cartesian components . Making use of the trigonometric identity, this expression may be rearranged as where we have made use of the identity Letting represent the antisymmetric tensor with axial vector , may now be represented as where is known as the exponential map. Thus, the exponential map provides the rotation tensor for a rotation about the unit vector . Note that , since .
Let be any orthogonal transformation, then where and its corresponding axial vector is , implying that . This property of the exponential map is used in the next derivation.
Consider a vector in the reference configuration of a rigid body. Upon rigid body rotation, this vector is currently at The corresponding axial vector of is . At a subsequent time , we would similarly have where here, is the incremental (finite) rotation from to . Alternatively, we may choose to write such that implying that Note from these relations that . Thus, is the material representation of the incremental rotation from to , while is the corresponding spatial representation.