What is the physical meaning of the tensor


In physics, the term tensor (lat. tensio = Tension) first appeared in connection with mechanical tension in solids. To describe the mechanical tension, the current density of the pulse, the terms scalar and vector are no longer sufficient. In mathematics, the tensor is defined as a geometric object that is invariant under coordinate transformations. The transformation rule defines the level of the tensor. A scalar is a 0th order tensor, a vector is a 1st order tensor and the stress tensor is a 2nd order tensor.

A tensor (2nd level) can be represented in terms of a basis as a matrix (with real or complex numbers). The current density of a vector set or the gradient of a vector-valued field transform like a tensor. A tensor can be constructed from two vectors with the help of the tensor product.

Momentum current density

The current density of a scalar set is a vector. For example, the flow velocity describes the volume flow density of the fluid. The current density of the mass is then equal to density times volume flow density

[math] \ vec j_m = \ rho \ vec j_V = \ rho \ vec v [/ math]

or with the Einstein notation in terms of a coordinate system

[math] j_ {m_i} = \ rho v_i [/ ​​math]

If you transfer the idea that the current density can be written as density times volume flow density to the momentum, you get the relationship

[math] j _ {{pcon} _ij} = \ rho_ {p_i} v_j = \ rho v_i v_j [/ math]

The momentum current density, which can be defined as the momentum density times the flow velocity, is equal to the mass density times the tensor product of the velocity.

Now the impulse can not only be transported through movement, i.e. together with matter, but also through matter. In the first case and already treated above, one speaks of a convective transport, in the second of a [[conduction-like transport | conduction-like transport. The line-like impulse transport is known under the name of force flow and is described with the help of the stress tensor. Except for the sign and a transposition, the stress tensor corresponds to the momentum current density

[math] \ sigma_ {ji} = -j_ {p_ {ij}} [/ math]

The transposition has no effect because the stress tensor is symmetrical.

Transformation behavior