Tensors

What is a tensor?

its a mathematical object that generalizes scalars, vectors, and matrices to an arbitrary dimensions. Think of it as a multi-dimensional array.

Scalars, Vectors, and Matrices:

• A scalar is a single number (0-dimensional).  
• A vector is a 1-dimensional array of numbers.  
• A matrix is a 2-dimensional array of numbers.  
• A tensor extends this concept to any number of dimensions.

Multidimensional Arrays:

• Tensors can be visualized as grids of numbers with any number of axes. For example:
  • A 3-dimensional tensor could be thought of as a cube of numbers.
  • A 4-dimensional tensor, and higher dimensions, become harder to visualize, but the concept of a multi-dimensional array still applies.

Transformation Properties:

• Beyond just being multi-dimensional arrays, tensors also have specific rules for how their components change when the coordinate system is changed. This is a crucial aspect of their mathematical definition.

Applications:

• Physics: Tensors are used to describe physical quantities that have multiple components and transform in specific ways, such as stress, strain, and electromagnetic fields.
  • $F_{\mu \nu} = \partial_{\mu} A_\nu - \partial_{\nu} A_\mu$, where $A$ is the four-potential.
  • $s_{ij} = \sigma_{ij} - \frac{\sigma_{kk}}{3}\delta_{ij}$, where $\sigma$ is the Cauchy stress tensor and $s_{ij}$ is the deviatoric stress tensor  
• Machine Learning: Tensors are fundamental to deep learning. Frameworks like TensorFlow are built around the concept of tensors, as they provide an efficient way to represent and manipulate the large amounts of data used in neural networks. For example, images can be represented as 3-dimensional tensors (height, width, color channels).

More about tensors

Notation of tensors: [1]

• Coordinates of a point in $\mathbf{R}^n$ is denoted as $(x^1, x^2, x^3, …, x^n)$
• rectangular coordinate system: $(x^i)$
• curvilinear coordinates $\bar{x}^i$ based on the following transformation

• $\mathscr{T}$ is a bijective map called coordinate transformation
• 3 common curvilinear-coordinate system
  1. polar coordinates : \(\mathscr{T}:\left\{\begin{array}{l} \bar{x}^1=x^1 \cos x^2 \\ \bar{x}^2=x^1 \sin x^2 \end{array}\right.\)
  2. cylindrical coordinates : \(\mathscr{T}:\left\{\begin{array}{l} \bar{x}^1=x^1 \cos x^2 \\ \bar{x}^2=x^1 \sin x^2 \\ \bar{x}^3=x^3 \end{array}\right.\)
  3. spherical coordinates : \(\mathscr{T}:\left\{\begin{array}{l} \bar{x}^1=x^1 \sin x^2 \cos x^3 \\ \bar{x}^2=x^1 \sin x^2 \sin x^3 \\ \bar{x}^3=x^1 \cos x^2 \end{array}\right.\)
• partial derivatives $\partial \bar{x}^i/\partial x^j$ from the coordinate transformation arranged in a square ($n\times n$) matrix is called the Jacobian matrix, its determinant is called the Jacobian.

References

1. Kay, D. C. Schaum’s outline of theory and problems of tensor calculus.
2. Flügge, W. (1972). Tensor analysis and continuum mechanics (Vol. 4, No. 1). New York: Springer-Verlag.