Variety of Vector and Tensor Types

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The Variety of Vector and Tensor Types

What is a vector?

A three dimensional vector is represented in a particular coordinate system by a triplet of real numbers. But the vector is not that triplet of numbers, it is something whose representation as three numbers changes in a systematic way as the coordinate system changes.
What is ordinarily meant by the term vector is called a polar vector. There is also something called an axial vector, which is the vector (cross) product of two polar vectors. The difference between polar and axial vectors is revealed when we consider a transformation of the coordinate system that changes the handedness of the coordinate system.
Consider two vectors whose respresentation are:
A = (1, 2, 3) and B = (3, 4, 7).
Their cross product C=AxB is: (2, 2, -4).
If we invert the coordinate system; i.e.,

x → -x, y → -y, z → -z,

then the representation of A becomes
(-1, -2, -3) and likewise the representation of B becomes (-3, -4, -7). (If the original coordinate system was right-handed then the new coordinate system is left-handed.) If C were a true vector then its representation should go to (-2. -2, 4). But the cross product of the representations of A and B in the new coordinate system is (2, 2, -4). Thus representation of C = AxB did not go to (-2, -2, 4) as would be required if C were to be a true vector. For this reason C=AxB is called a pseudovector or axial vector.
An axial vector can also be considered a representation of a second order antisymmetric tensor; i.e.,

B₁

B₂

B₃

=

0 B₁ B₂

−B₁ 0 B₃

−B₂ −B₃ 0

The variety of vector and tensor concepts are shown below. There is even some variety of the scalar concept.

Types of Scalars, Vectors and Tensors
Name Origin Examples Transformation Law

Polar Vector Ordinary Vector Displacement,
Electric Field E'_i = Σl_irE_r

Axial Vector Cross product
of polar vectors Magnetic Field B'_i=
|det (l)|Σl_ir B_r

Polar Tensor Relation between
two vectors
of the same type Magnetic Permeability P'_i..j=
Σl_ir ...l_kwP_r..w

Axial Tensor Relation between
two vectors of
different types Optical Gyration Q'_i..j=|det(l)|Σl_ir ...l_kwQ_r..w

Psuedo-scalar polar vector
dotted with
axial vector Rotary Power of
Optically Active
Crystal

(To be continued.)

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Types of Scalars, Vectors and Tensors
Name	Origin	Examples	Transformation Law
Polar Vector	Ordinary Vector	Displacement, Electric Field	E'_i = Σl_irE_r
Axial Vector	Cross product of polar vectors	Magnetic Field	B'_i= \|det (l)\|Σl_ir B_r
Polar Tensor	Relation between two vectors of the same type	Magnetic Permeability	P'_i..j= Σl_ir ...l_kwP_r..w
Axial Tensor	Relation between two vectors of different types	Optical Gyration	Q'_i..j=\|det(l)\|Σl_ir ...l_kwQ_r..w
Psuedo-scalar	polar vector dotted with axial vector	Rotary Power of Optically Active Crystal