MTH623 Lectures 1-57

Introduction to Tensors

Introduction

Why we need tensors?

What tensors are actually?

The origin of tensors

The word tensor has its root tensus in Latin, meaning stretch or tension. J. W. Gibbs in (1884) developed the concept of dyadics (object with two parts or elements) and polyadics (an object comprising of many elements) which are actually tensors in modern sense but only different by names.

The origin of tensors

W. Voigt in (1898) introduced the term tensor in his study of stress and strain of crystals. Later on many physicists including Albert Einstein and Marcel Grossmann in (1913) used the concepts with the name tensors instead of systems.

Why we need tensors?

A scalar is a physical quantity that is represented by a number at a particular point in space. For example, mass, density and temperature.

A vector is a tool to keep track of two pieces of information, typically magnitude and direction, for a physical quantity. For example position, force and velocity.

Why we need tensors?

What happens when we need to keep track of three pieces of information for a given physical quantity? Say, three pieces of information are magnitude, direction and plane. For example stress, strain and inertia tensors.

Why we need tensors?

In short: A scalar is a quantity whose specification requires just one number. On the other hand, a vector is a quantity whose specification requires three numbers, namely its components with respect to some basis. Scalars and vectors are both special cases of a more general term called a tensor.

What is a tensor?

Tensor can be regarded as a general name of multi-way array data. In an m dimensional space, a tensor of rank n is a mathematical object that has n indices, components and obey certain transformation rule.

Rank of a tensor

The rank of a tensor also known as order or degree of tensor is the total number of free indices in a tensor.

1)

2)

Higher Dimensional Spaces

Dimensions:

The dimensions of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

2-Dimensional space

A coordinate system in which the exact position of a point is determined with two numerical values, is called the two dimensional space.

In Euclidean geometry, two perpendicular number-lines are intersected at their middle points for splitting the whole space into four equal parts. The number lines are used as base line to measure the distance of every point in two perpendicular directions from their bisection point.

2-Dimensional Space

There are four fundamental concepts related to two dimensional Cartesian coordinate system: Coordinate axes, origin, coordinates of a point and quadrants.

3-Dimensional space

We can expand our 2-dimensional (xy) coordinate system into a 3-dimensional coordinate system, using x, y, and z-axes. In 3dimensional space the exact position of a point is determined with three numerical values, called the coordinates.

3-Dimensional space

The xy-plane is horizontal in diagram below and shaded green. It can also be described using the equation z = 0, The xz-plane is vertical and shaded pink above. This plane can be described using the equation y=0. The yz-plane is also vertical and shaded blue. The yz-plane can be described using the equation x=0.

Higher Dimensional Spaces

More than three dimensional spaces are treated as higher dimensional spaces. For example, we can think of dimensions as ways in which we can move up and down, left and right, forward and backwards, which leads us to believe we live in a three-dimensional world. Every point in this world is defined by three coordinates. We can't imagine living in a four-dimensional world because we simply can't conceive of another direction to move in.

Higher Dimensional Spaces

However, now imagine you are arranging to meet someone. Not only do you need to give them the three coordinates of the place you are meeting, you also need to tell them the time you are meeting. Thus, time and space together form a four-dimensional space: you need four numbers to specify a point.

Generally, a point in an n-dimensional space is given by n dimensional spaces and objects within them as geometric objects. Algebra gives the tools to deal with these objects. Tensors have a wide range of applications in different fields. Following are different fields of Mathematics and Physics where tensors are used. Mechanics Stress tensor, Strain tensor, Elasticity tensor, Inertia tensor Electromagnetics Maxwell equations and related tensors Special theory of relativity Metric tensor, four vector formalism General theory of relativity Riemannian tensor, Ricci tensor, Einstein tensor Following are some applications of tensors in Engineering To represent mechanical stress as the stress tensor in continuum mechanics To measure deformations through deformation tensor and strain via strain tensor in continuum mechanics To represent diffusion as a tensor in diffusion tensor imaging. To describe moment of inertia as inertia tensor.

Tensor Flow

Tensor flow is an open-source end-to-end platform for creating machine learning applications. It is a symbolic math library that uses dataflow and differentiable programming to perform various tasks focused on training and inference of deep neural networks.

How Tensor Flow Works

Tensor Flow enables you to build dataflow graphs and structures to define how data moves through a graph by taking inputs as a multi-dimensional array called Tensor. It allows you to construct a flowchart of operations that can be performed on these inputs, which goes at one end and comes at the other end as output.

Applications of tensor flow

Currently, tensor flow is used in a variety of Google applications like Google photos, Google translate, Gmail, search engine, translation, image captioning or recommendations.

Tensor Notation:

Tensor notation provides a convenient and unified system for describing physical quantities. In the most general representation, a tensor is denoted by a symbol followed by a collection of subscripts (or superscripts).

i ij l klm

X · U · B · Subscript and Superscript:

A subscript is a character (such as a number or letter) that is written slightly below and a superscript is written above the normal line of type. It is usually smaller than the rest of the text. Subscripts appear below the baseline, while superscripts are above.

These subscripts and superscripts are also called indices. iA j

u

k ij

B · C · subscripts · superscripts · Indices:

Indices assume values from an index set. They successively assume each value from the given index set. For example, if takes the values from 1 to 4, then denotes the set and we say that the index runs from 1 to 4.

Example:

1,2,3 0,1,2,3 j k a j b k