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Vector And Tensor Analysis


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Vector And Tensor Analysis


Scalar multiplication is equivalent to switching the order of vectors forming a tensor (likewise for scalar subtraction). Therefore, tensor structure is preserved by combined mathematical operations such as addition, subtraction, multiplication, division, etc.


Scalar products (scalar products of tensors) take into account the combined effect of the tensor components. For example: the cartesian component of a vector is obtained by multiplying that vector with a tensor.


Tensors made of two vectors are called traces. Tensors of rank 2 are called symmetric or second-order tensors. This description has a particular significance in mechanics. For example, the stress-strain tensor in the elastic limit of a solid expands like a "second-order tensor". This fact was known not only to Maupertuis, but also to Henry Cavendish who related it to the deformation of the liquid surface made by the vibrating elastic solid.


All substances increase the tensorial rank by the same factor, and inversely, the torsion tensor of a material is the tensor of maximum rank, i.e., the tensor of lowest rank is the dissipative torsion tensor (greek typical word).


The concept of invariance, which is the basis of tensor analysis, is applicable to all physical quantities including the inertial forces and noninvariant quantities such as temperature to a certain degree. In this way, tensor analysis may be used more definitely in the theory of diffraction and the paradox of relativity. d2c66b5586






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