Tensor is a number/collection of numbers that maintains its meaning under transformations
More complex, but correct definition - An nth-rank tensor in m-dimensional space is a mathematical object that has n indices and m^n components and obeys certain transformation rules.

Each index of a tensor ranges over the number of dimensions of space. However, the dimension of the space is largely irrelevant in most tensor equations (with the notable exception of the contracted Kronecker delta).

  1. Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices.
  2. Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity.

Understanding Covariance & Contravariance

A vector can be expressed in terms of its contravariant or covariant components

CASE 1: Expressing in terms of contravariant components → vector described in terms of components with basis vectors

We need 1 index for each of the components as there is only 1 directional indicator per component. 1 index, 1 basis vector, per component

If we decrease the length of the “basic vectors” (in a vector space), then the number of components required to make up a vector increases. Because the index & the number of components change contrary to each other, they are known as contravariant components of a vector Describing a vector in terms of these contravariant components (eg : 2 i + 3 j + 4 k) is how we usually describe a vectors

CASE 2: Expressing in terms of covariant components → vector described in terms of dot-product with basis vectorsIf we decrease the length of the basis vectors, then the dot-product decreases & vice versa. Since these properties are varying in the same way, we say that these are covariant components of a vector.

Specifying the notation for covariant & contravariant components

Now, say we take 2 vectors V & P

Index (for this rank 2 tensor, i.e. a vector) = unique combination of basis vectors

CASE 1: We multiply the contravariant values of V & P. On considering all possible ways, we get a matrix:

$$ \begin{bmatrix}   V^{1} P^{1} & V^{1} P^{2} &   V^{1} P^{3} \\ V^{2} P^{1} & V^{2} P^{2} &   V^{2} P^{3} \\ V^{3} P^{1} & V^{3} P^{2} &   V^{3} P^{3}    \end{bmatrix} $$

This will give us a tensor of rank 2 with 2 contra-variant index values. Here, each T value is called an element of this tensor. And since for each element, 2 indices (2 directional indicators are required) eg: Understanding by example, say we have a combination of 3 area vectors & 3 force vectors. To get the combination of all the forces on all the area vectors, for each such combination, we require 2 indices (i.e. 2 directional indicators → 1 for the force & 1 for the area). We get 9 components, each with 2 indices (referring to 2 basis vectors)

$$ \begin{bmatrix}   T^{11} & T^{12} &   T^{13}\\ T^{21} & T^{22} &   T^{23}\\ T^{31} & T^{32} &   T^{33}\end{bmatrix} $$

CASE 2: We multiply the covariant components of P with contravariant values of V . On considering all possible ways, we get a matrix:

$$ \begin{bmatrix}   V_{1} P^{1} & V_{1} P^{2} &   V_{1} P^{3} \\ V_{2} P^{1} & V_{2} P^{2} &   V_{2} P^{3} \\ V_{3} P^{1} & V_{3} P^{2} &   V_{3} P^{3}    \end{bmatrix} $$

This will give us a tensor of rank 1 co-variant index value & 1 contra-variant index value

$$ \begin{bmatrix}   T_{1}^{1} & T_{1}^{2} &   T_{1}^{3}\\ T_{2}^{1} & T_{2}^{2} &   T_{2}^{3}\\ T_{3}^{1} & T_{3}^{2} &   T_{3}^{3}\end{bmatrix} $$

CASE 3: We multiply the covariant values of V & P. On considering all possible ways, we get a matrix:

$$ \begin{bmatrix}   V_{1} P_{1} & V_{1} P_{2} &   V_{1} P_{3} \\ V_{2} P_{1} & V_{2} P_{2} &   V_{2} P_{3} \\ V_{3} P_{1} & V_{3} P_{2} &   V_{3} P_{3}    \end{bmatrix} $$

This will give us a tensor of rank 2 with 2 contra-variant index values

$$ \begin{bmatrix}   T_{11} & T_{12} &   T_{13}\\ T_{21} & T_{22} &   T_{23}\\ T_{31} & T_{32} &   T_{33}\end{bmatrix} $$

What makes a tensor a tensor, is that, when the basis vectors change, the components of the tensor would change in the same manner as they would in one of these objects → thus keeping the overall combination the same

What is it about the combination of components & basis vectors that makes tensors so powerful → all observers in all reference frames agree, not on the components or the basis vectors → but on their combinations, because they are the same

Index & Rank of tensors

Rank = the total number of indices = total number of coordinate systems (we need to have knowledge of to understand the location of the tensor) = total number of unique basis vector combinations = total number of unique linearly independent combinations = The total number of contravariant and covariant indices of a tensor.

The rank R of a tensor is independent of the number of dimensions N of the underlying space.

A tensor does not necessarily have to be created from vector components as is shown in these examples

A tensor of rank 1 is a “vector” and has a number associated with each of the basis vectors. Only 1 index i.e. 1 combo of basis vectors is required to know the location

Here, \($V^1 = 5, V^2 = 3, V^3 = 2$\)

A tensor of rank 2 - we associate a number with every possible combination of 2 basis vectors

In a tensor of rank 3 - we associate a number with every possible combination of 3 basis vectors - composed of the components of 3 basis vectors. We can create different descriptions of this tensor by using different contravariant & covariant components of the basis vectors

More mathematical facts

  1. While the distinction between covariant and contravariant indices must be made for general tensors, the two are equivalent for tensors in three-dimensional Euclidean space (Euclidean space is the fundamental space of geometry, intended to represent physical space) , and such tensors are known as Cartesian tensors.
  2. Objects that transform like zeroth-rank tensors are called scalars, those that transform like first-rank tensors are called vectors, and those that transform like second-rank tensors are called matrices. In tensor notation, a vector v would be written $v_i$, where i=1, ..., m, and matrix is a tensor of type (1,1), which would be written $a_i^j$ in tensor notation.
  3. Tensors may be operated on by other tensors (such as metric tensors, the permutation tensor, or the Kronecker delta) or by tensor operators (such as the covariant derivative). The manipulation of tensor indices to produce identities or to simplify expressions is known as index gymnastics, which includes index lowering and index raising as special cases. These can be achieved through multiplication by a so-called metric tensor $g_{ij}, g^{ij}, g_i^j$, etc., e.g.,
  4. Tensor notation can provide a very concise way of writing vector and more general identities. For example, in tensor notation, the dot product u·v is simply written u·v = $u_iv^i$ If two tensors A and B have the same rank and the same covariant and contravariant indices, then they can be added in the obvious way.
  5. The generalization of the dot product applied to tensors is called tensor contraction, and consists of setting two unlike indices equal to each other and then summing using the Einstein summation convention. Various types of derivatives can be taken of tensors, the most common being the comma derivative and covariant derivative.
  6. If the components of any tensor of any tensor rank vanish in one particular coordinate system, they vanish in all coordinate systems. A transformation of the variables of a tensor changes the tensor into another whose components are linear homogeneous functions of the components of the original tensor.

Tensors in the context of machine learning

The property of tensors to maintains its meaning under transformations is used also by machine learning, where real world data is transformed into corresponding tensors & then heavy calculations are done on these tensors to get valuable insights.

Reference videos

  1. What is a tensor? - Link
  2. Covariant & Contravariant Matrices - Link