Why does a tensor have nine components

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What are tensors?

In chemical and physical experiments, substances and molecules are exposed to experimental quantities (mechanical pressure or tensile load, electric or magnetic field), which are described as vectors (cause vector). A variable is measured in each case (elastic deformation of the substance, electrical polarization of the substance or magnetic effects in molecules according to NMR spectroscopy), which is also described by a vector (action vector).

In these systems there is often a linear relationship between cause and effect vectors. If the examined substance is isotropic (e.g. liquids), the action vector has the same axis position as the cause vector. The simple relationship applies:

is a scalar quantity that characterizes the isotropic substance. Anisotropic substances (e.g. single crystals) behave differently: the linearity between cause and effect is retained, but usually show different directions. The relationship between and can be formulated in anisotropic systems with the help of linear systems of equations. To clarify the principle, we first do this in the coordinate system.

  • Cause vector in -direction: The cause vector generated an effects vector with a - and a - component, all of which are proportional to. The first letter of the proportionality constant refers to the relevant component of the effects vector, the second to the direction of the cause vector.
  • Cause vector in -direction: The cause vector generated an effects vector with a - and a - component, all of which are proportional to. It follows
  • The cause vector is the addition of both components in the / paragraph> direction: The components of the effects vector are then equal to the sum of the corresponding components in the two above cases. The following must apply: Matrix notation

In the three-dimensional case a matrix is ​​created, which of course also applies to.

Its nine real components characterize the substance that can be represented by the symbol. The system of coefficients is called a tensor. It is said that the coefficients form the tensor.

It is generally referred to as a 2nd order tensor or a dyad because it has two indices. Correspondingly, scalar (without an index) and vector quantities (with an index) from chemistry and physics represent tensors 0 and 1. They are characterized (in 3D space) by one, three or nine components. However, this “step” notation of the so-called tensor calculus is not easy to handle. For the description of physico-chemical laws it is more convenient and, for the rest, completely sufficient to speak of scalars, vectors and matrices. Thus tensors are 0., 1. resp. 2nd stage meant.

Note
The tensors from chemistry are determined by real, symmetrical matrices with respect to a given coordinate system (i.e.). However, such a matrix is ​​not assigned to every arbitrary tensor.