Chapter 14
Rotation of Tensors
Abstract This chapter is concerned with the active rotation of tensors. Firstly,
infinitesimal and finite rotations of vectors are described by second rank rotation
tensors, the connection with spherical components is pointed out. Secondly, the rota-
tion of second rank tensors is treated with the help of fourth rank projection tensors.
Thefourthrankrotationtensorisalinearcombinationoftheseprojectors.Thescheme
is generalized to the rotation of tensors of rank>2. Thirdly, the projection tensors
are applied to the solution of tensor equations. An example deals with the effect of
a magnetic field on the electrical conductivity.
Theactive rotation of a tensor, to be considered here, has to be distinguished from the
rotation of the coordinate system, see Sect.2.4. The passive rotation of the coordinate
system must not affect the physics. The rotation of a tensor, on the other hand,
describes physical changes. The rotation of vectors is discussed before second and
higher rank tensors are treated. This section follows and generalizes the material
given in the appendix of [42].
14.1 Rotation of Vectors.
14.1.1 Infinitesimal and Finite Rotation.
The rotation of a vectoraby the infinitesimal angleδφabout an axis, which is parallel
to the axial unit vectorh, generates a vectora′, whose components are given by
a′μ=aμ+δφ εμλνhλaν=aμ+δφHμνaν=(δμν+δφHμν)aν. (14.1)The antisymmetric second rank tensorHis defined by
Hμν≡εμλνhλ. (14.2)© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_14
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