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  2. Quaternions and spatial rotation - Wikipedia

    en.wikipedia.org/wiki/Quaternions_and_spatial...

    3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]

  3. Quaternion - Wikipedia

    en.wikipedia.org/wiki/Quaternion

    [8] The unit quaternions give a group structure on the 3-sphere S 3 isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). The positive and negative basis vectors form the eight-element quaternion group.

  4. Versor - Wikipedia

    en.wikipedia.org/wiki/Versor

    In mathematics, a versor is a quaternion of norm one (a unit quaternion).Each versor has the form = ⁡ = ⁡ + ⁡, =, [,], where the r 2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions).

  5. Tensor operator - Wikipedia

    en.wikipedia.org/wiki/Tensor_operator

    In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation.

  6. Rodrigues' rotation formula - Wikipedia

    en.wikipedia.org/wiki/Rodrigues'_rotation_formula

    In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...

  7. Levi-Civita connection - Wikipedia

    en.wikipedia.org/wiki/Levi-Civita_connection

    The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-)inner product of X and Y. In the latter case, the inner product of X p, Y p is taken at all points p on the manifold so that g(X, Y) defines a smooth function on M. Vector fields act (by definition) as differential ...

  8. List of formulas in Riemannian geometry - Wikipedia

    en.wikipedia.org/wiki/List_of_formulas_in...

    6.4 Scalar curvature. 6.5 Traceless Ricci tensor. 6.6 ... The variation formula computations above define the principal symbol of the mapping which sends a pseudo ...

  9. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field T {\displaystyle \mathbf {T} } of non-zero order k is written as div ⁡ ( T ) = ∇ ⋅ T {\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} } , a contraction of a tensor field ...