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Confusion with Pseudo Scalars, Vectors, Tensors, and Tensor Densities


Keilman, Yuri




Mathematics and Applied Mathematics



Date Published:

April 11, 2014






In Geometry the true mathematical objects (scalars, vectors, and tensors), while being covariant with respect to arbitrary transformation of coordinates, are, at the same time, identity-invariant. By this term we mean that any contraction between them to a scalar gives us invariant. We call them absolute objects or primary objects to distinguish them from so called “pseudo” mathematical objects and tensor densities. If we are making an arbitrary transformation of coordinates then the only way to find the description of a primary object in a new coordinate system is to use its transformation law. A “Pseudo” mathematical object actually is not original (primary) object at all. A “pseudo” object is usually constructed from the components of the primary objects (like determinant of a second rank tensor aik is constructed from its components). There is a straight forward way to obtain this determinant in a new coordinates by obtaining transformed tensor aik' first and then calculating determinant. After that is done one can see that determinant in a new coordinates is equal to determinant in old coordinates multiplied by the Jacobian with the weight w=2. It seems that we found a new transformation law and call determinant by the name pseudo-scalar. But it is only illusion. It seems that it has its own transformation law but this law is not original.

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