I know that there are kinds of tensors: scalars, vectors and matrixes. I know they contain data in form of numbers, but what bugs me is what kind of data is represented in a tensor and what do we do to get that data? Its not just a collection some random numbers, but calculated ones are put in there. Out of what do we get those numbers?

Sorry if this may sound stupid...

  • $\begingroup$ Well, out of what do we get the scalar numbers (0-tensor), the numbers in a vector (1-tensor) and the numbers in a matrix (2-tensor)? $\endgroup$
    – md2perpe
    Aug 27 '18 at 15:29
  • $\begingroup$ @md2perpe IDK. Could you please explain? $\endgroup$ Aug 27 '18 at 16:49
  • $\begingroup$ Maybe this can help you, it helped me back then $\endgroup$ Aug 27 '18 at 16:56
  • $\begingroup$ In physics the numbers often come from measurements. $\endgroup$
    – md2perpe
    Aug 27 '18 at 20:10

In a geometrical perspective a 0-tensor is simply a number (a real scalar for example) and a 0-tensor field in a region $\Omega$ is a simply a (real) scalar-field $\Omega\to\Bbb R$.

A rank one tensor is an operator derivative $X$ which alows to speak of how a scalar field varies in direction of the $X$, being $X$ a linear combination $X=X^s\partial_s$ ot the basic derivatives $\partial_s=\frac{\partial}{\partial x^s}$. Since the components are function of the coordinates we should be speaking of rank one tensor fields

The metric tensor is a rank two tensor which generalize the analogous of an interior product on $\Bbb R^2$ and $\Bbb R^3$ and its uses there. It is defined as the bilinear map $g=g^{st}\partial_s\otimes\partial_t$ and the coefficients $g_{st}=\partial_s\cdot\partial_t$. In a flat euclidean space we have $g_{st}=\delta_{st}$, but in a space with curvilinear coordinates we would have $g_{s\mu}g^{\mu t}=\delta_s{}^t$.
This mechanism allows to model the geometrical distortions living on where the coordinates are employed.

With the aid of the low and raise indexation technique one can construct associated tensors for example $$A_i{}^{jk}=g_{is}A^{sjk}.$$ and with those one reduces the amount of data to manipulate any rank tensors.


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