# Tensor basic definition

DRAFT NOTES

# Easy but incomplete definition

## I - Tensor = multi-dimensional array of numbers

A tensor is often thought of as a generalized matrix. That is, it could be a 1-D matrix (i.e. a vector), a 3-D matrix (a cube of numbers), a 0-D matrix (a single number), or a higher dimensional structure that is harder to visualize.

The 'n' of n-D in this case is called the order, degree or rank of the tensor.

## Examples of rank 2 tensors

Electromagnetic tensor:

Stress tensor:

Rank 3, dimension - 3 (i.e 3 elements along the rows and columns)

Rank 4

## Index notation

Each element (number) position in the tensor can be described using indices:

- Rank-0: \( \tau \), a scalar
- Rank-1: \( \tau_i \), a vector
- Rank-2: \( \tau_{ij} \), a matrix (Latin letters for Dimension-2/3)

\( \tau_{\alpha\beta} \) (Greek letters for Dimension-4) - Rank-3: \( \tau_{ijk} \), a cube of matrices.
- Rank-4: \( \tau_{ijkl} \), an array of cubes
- ...

# II - Augmented array definition

Tensor = an object that is invariant under a change of coordinates and has components that change in a special, predictable way under a change of coordinates. Explanation:

Regardless of the coordinate system the pencil always points to the door with the same length. We can transform components with forward or backward transformations. (if the vector/tensor becomes null under specific coordinate systems then it means it's a pseudo-vector/tensor)

## III - Descriptive definition:

A rank-n tensor in m-dimensions is a mathematical object that has n indices and m^n components and obeys certain transformation rules.

or:

a tensor is a geometric object (vector, 3D reference frame) which is expressed and transformed according to a coordinate system defined by the user.

# IV- Alternate definitions

Tensor = a collection of vectors and covectors combined together using the tensor product. (kind of abstract/ recursive definition since you need to know about covectors and tensor product)

Alternate interpretation: Tensors as partial derivaties and gradients that transform with the Jacobian Matrix

# References

Good starter explanation

https://www.youtube.com/watch?v=bpG3gqDM80w

https://www.youtube.com/watch?v=TvxmkZmBa-k

Others:

https://www.youtube.com/watch?v=f5liqUk0ZTw&t=447s

https://www.youtube.com/watch?v=CliW7kSxxWU

https://open.oregonstate.education/intermediate-fluid-mechanics/chapter/chapter-1-mathematical-tools/

ebook

https://gumroad.com/l/ubSc (it has a chapter on tensors)

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