Introduction to vector and tensor analysis

in modern vector analysis to define the cross product through the determinant,
Eq. (2.20) og equivalently through the Levi-Civita symbol, Eq. (2.21). In
replacing the geometric definition, Eq. (2.5) which pressumes an absolute notion
of handedness, with an algebraic one the direction of the cross-product will
simply be determined by what-ever convention of handedness that has been
adopted with the ortonormal coordinate system in the first place.
Physically, the arbitrariness in the choise of handedness implies that the
orientation of the cross product of two ordinary vectors is not a physical objective
quantity, in contrast to vectors representing displacements, velocities,
forces etc. One therefore distinguishes between proper or polar vectors whose
direction are independent on the choise of handedness and axial or pseudovectors
whose directions depend on the choise of handedness. The distinction between
the two types of vectors becomes important when one considers transformations
between left- and right-handed coordinate systems.
Dimensionality
For all vector operations the only explicit reference to dimension of the physical
space appears in the cross product. This can be seen from the definition of
the ɛ-symbol, which explicitly has three indices running from 1 to 3. All other
vector operations are valid in arbitrary dimensions.
The generalization of the cross product to any dimension d is obtained by
constructing a Levi-Civita symbol having d indices each running from 1 to d
and being totally antisymmetric.
For instance for d = 2 :
ɛ12 = 1
ɛ21 = −1
ɛ11 = ɛ22 = 0
Here, we get the “cross-product”, â, of a vector a = (a1, a2)E by
or in doublet notation
[â]j =
i
ɛjiai
â = (â1, â2)E = (a2, −a1)E,
(2.25)
which is the familiar expression in d = 2 for obtaining a vector orthogonal to a
given vector. Generally, in d > 1 dimensions the cross product will take d − 1
vectors as arguments.
2.5 Ordinary derivatives and integrals of vectors
Let c(t) be a vector depending on a single scalar variable t. In this section we
will consider how to define derivatives and integrals of c with respect to t.
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