202501040940
Status: #idea
Tags: Tensors, Physics, Tensor Calculus
State: #nascient
Tensors
Tensors are a physicists' answer to the problem of invariance of physical measures for complex physical phenomena.
For physics to be useful, we need it to behave the same and be consistent no matter the frame of reference in such a way that two individuals even from different species if intelligent enough to measure them according to their native measures would still reach the same constants when everything is accounted for.
The problem is that while this is relatively simple for simple concepts like velocity which is often introduced as a single quantity and is then shown to actually be a vector (rank 1 Tensor), this is not at all obvious when dealing with concepts which are inherently multidimensional like viscosity or whatnot and for which extension to higher dimensions would require much more than simple vectors.
So let's state it properly, a tensor is an invariant objects whose components change in a special and predictable way based on the coordinate system.
First thing to note, a tensor is NOT its components, this is how we can solve the problem of invariance considering that those components absolutely DO change under different coordinate systems. They change in a very specific way in such a way that through an operation called a change of basis it is trivial to take any insights from one coordinate system to another.
Under this definition, numbers, vectors, matrices, and beyond just become different ranks of tensors, respectively
There are three types of tensors:
- Covariant (Recto)
- Contravariant (Verso)
- Mixed (One example is the Metric Tensor which acts as the bridge between Covariant and Contravariant.)t