In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2. Tensor contraction can be seen as a generalization of the trace. (Wikipedia).
What is length contraction? Length contraction gives the second piece (along with time dilation) of the puzzle that allows us to reconcile the fact that the speed of light is constant in all reference frames.
From playlist Relativity
Special Relativity C2 Length Contraction
Relativistic length contraction.
From playlist Physics - Special Relativity
What is a Tensor? Lesson 12 (redux): Contraction and index gymnastics
What is a Tensor? Lesson 12 (redux): Contraction and index gymnastics I have redone the index gymnastics lecture to try and fill in the details regarding contractions. I will keep them both in the playlist for now.
From playlist What is a Tensor?
What is a Tensor? Lesson 29: Transformations of tensors and p-forms (part review)
What is a Tensor? Lesson 29: Tensor and N-form Transformations This long lesson begins with a review of tensor product spaces and the relationship between coordinate transformations on spacetime and basis transformations of tensor fields. Then we do a full example to introduce the idea th
From playlist What is a Tensor?
Special Relativity C3 Length Contraction
Relativistic length contraction.
From playlist Physics - Special Relativity
Visualization of tensors - part 1
This video visualizes tensors. It shows some introduction to tensor theory and demonstrates it with the Cauchy stress tensor. Future parts of this series will show more theory and more examples. It talks about the term 'tensor' as used in physics and math. In the field of AI the term 'te
From playlist Animated Physics Simulations
Calculus 3: Tensors (1 of 28) What is a Tensor?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is a tensor. A tensor is a mathematical representation of a scalar (tensor of rank 0), a vector (tensor of rank 1), a dyad (tensor of rank 2), a triad (tensor or rank 3). Next video in t
From playlist CALCULUS 3 CH 10 TENSORS
What Is A Tensor Lesson #1: Elementary vector spaces
We define a vector space and lay the foundation of a solid understanding of tensors.
From playlist What is a Tensor?
What is General Relativity? Lesson 68: The Einstein Tensor
What is General Relativity? Lesson 68: The Einstein Tensor The Einstein tensor defined! Using the Ricci tensor and the curvature scalar we can calculate the curvature scalar of a slice of a manifold using the Einstein tensor. Please consider supporting this channel via Patreon: https:/
From playlist What is General Relativity?
What is General Relativity? Lesson 14: The covariant derivative of a covector
We start by demonstrating that contraction commutes with directional covariant derivative and then derive the CFREE and COMP expressions for the covariant derivative of a covector.
From playlist What is General Relativity?
Johnnie Gray: "Hyper-optimized tensor network contraction - simplifications, applications & appr..."
Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021 Workshop I: Tensor Methods and their Applications in the Physical and Data Sciences "Hyper-optimized tensor network contraction - simplifications, applications and approximations" Johnnie Gray - California Ins
From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021
What is General Relativity? Lesson 15 The covariant derivative of a (p,q)-rank tensor
In this lesson we review all the CFREE algebraic rules and the COMP conversions and then demonstrate the CFREE and COMP formulas for the covariant derivative of an arbitrary tensor.
From playlist What is General Relativity?
Live CEOing Ep 80: Language Design for Wolfram Language
Watch Stephen Wolfram and teams of developers in a live, working, language design meeting. This episode is about Language Design in the Wolfram Language.
From playlist Behind the Scenes in Real-Life Software Design
Garnet Chan - Arithmetic tensor networks and integration - IPAM at UCLA
Recorded 26 January 2022. Garnet Chan of the California Institute of Technology presents "Arithmetic tensor networks and integration" at IPAM's Quantum Numerical Linear Algebra Workshop. Abstract: I will discuss how to perform arithmetic with tensor networks and the consequences for the in
From playlist Quantum Numerical Linear Algebra - Jan. 24 - 27, 2022
Tensor Calculus Lecture 6c: The Covariant Derivative 2
This course will eventually continue on Patreon at http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Te
From playlist Introduction to Tensor Calculus
Lei Wang: "Tropical Tensor Networks"
Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021 Workshop I: Tensor Methods and their Applications in the Physical and Data Sciences "Tropical Tensor Networks" Lei Wang - Chinese Academy of Sciences Abstract: I will present a unified exact tensor network ap
From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021
An Introduction to Tensor Renormalization Group (Lecture 3) by Daisuke Kadoh
PROGRAM NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (II
From playlist NUMSTRING 2022
What is a Tensor? Lesson 31: Tensor Densities (Part 2 of Tensor Transformations)
This video is about What is a Lesson 31: Tensor Densities (Part 2 of Tensor Transformations) We introduce the *classical* definition of a tensor density and connect that definition to our more robust approach associated with vector spaces and their associated bases. I will demonstrate som
From playlist What is a Tensor?
Philippe CORBOZ - Simulation of 2D strongly correlated systems...
Simulation of 2D strongly correlated systems with infinite projected entangled-pair states https://indico.math.cnrs.fr/event/2435/
From playlist Workshop “Hamiltonian methods in strongly coupled Quantum Field Theory”