San José State University |
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The Variety of Vector and Tensor Types |
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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=A×B is: (2, 2, -4).

If we invert the coordinate system; i.e.,

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.,
**

B_{1} |

B_{2} |

B_{3} |

→ |

0 | B_{1} | B_{2} |

−B_{1} | 0 | B_{3} |

−B_{2} | −B_{3} | 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 | |||
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Name | Origin | Examples | Transformation Law |

Polar Vector | Ordinary Vector | Displacement, Electric Field |
E'_{i} = Σl_{ir}E_{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}=Σ l_{ir}
...l_{kw}P_{r..w} |

Axial Tensor | Relation between two vectors of different types | Optical Gyration | Q'_{i..j}=|det(l)|Σl_{ir}
...l_{kw}Q_{r..w} |

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

(To be continued.)

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