Lecture 27. Properties of tensor products
0:00 Use properties of tensor products to effectively think about them! 0:50 Tensor product is symmetric 1:17 Tensor product is associative 1:42 Tensor product is additive 21:40 Corollaries 24:03 Generators in a tensor product 25:30 Tensor product of f.g. modules is itself f.g. 32:05 Tenso
From playlist Abstract Algebra 2
A Concrete Introduction to Tensor Products
The tensor product of vector spaces (or modules over a ring) can be difficult to understand at first because it's not obvious how calculations can be done with the elements of a tensor product. In this video we give an explanation of an explicit construction of the tensor product and work
From playlist Tensor Products
What is a Tensor 5: Tensor Products
What is a Tensor 5: Tensor Products Errata: At 22:00 I write down "T_00 e^0 @ e^1" and the correct expression is "T_00 e^0 @ e^0"
From playlist What is a Tensor?
Complete Derivation: Universal Property of the Tensor Product
Previous tensor product video: https://youtu.be/KnSZBjnd_74 The universal property of the tensor product is one of the most important tools for handling tensor products. It gives us a way to define functions on the tensor product using bilinear maps. However, the statement of the universa
From playlist Tensor Products
Proof: Uniqueness of the Tensor Product
Universal property introduction: https://youtu.be/vZzZhdLC_YQ This video proves the uniqueness of the tensor product of vector spaces (or modules over a commutative ring). This uses the universal property of the tensor product to prove the existence of an isomorphism (linear bijection) be
From playlist Tensor Products
Tensors Explained Intuitively: Covariant, Contravariant, Rank
Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. My Patreon page is at https://www.patreon.com/EugeneK
From playlist Physics
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?
Physics - Advanced E&M: Ch 1 Math Concepts (7 of 55) What is the Vector Product?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain and develop the vector product of 2 vectors. Next video in this series can be seen at:
From playlist PHYSICS 67 ADVANCED ELECTRICITY & MAGNETISM
There are two types of vector multiplication. In this tutorial we take a look at the vector dot product, also known as the vector inner product. The result of a vector inner product is a scalar. There are two ways to calculate this scalar, which can help us to determine the angle betwee
From playlist Introducing linear algebra
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?
Tensor Calculus 6b: The Covariant Derivative
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
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?
What is General Relativity? Lesson 13 Some important CFREE relations
We prove some critical CFREE expressions required for the derivation of the metric connection. Errata: At 46:00 the argument should have the vector "Z" not the vector "X". X is part of the example tensor and Z is being fed to the tensor.
From playlist What is General Relativity?
What is a Tensor? Lesson 18: The covariant derivative continued
What is a Tensor? Lesson 18: The covariant derivative continued This lesson covers some of the "coordinate free" language used to describe the covariant derivative. As a whole this lecture is optional. However, becoming comfortable with coordinate free language is probably a good idea. I
From playlist What is a Tensor?
Lec 4 - Phys 237: Gravitational Waves with Kip Thorne
Watch the rest of the lectures on http://www.cosmolearning.com/courses/overview-of-gravitational-wave-science-400/ Redistributed with permission. This video is taken from a 2002 Caltech on-line course on "Gravitational Waves", organized and designed by Kip S. Thorne, Mihai Bondarescu and
From playlist Caltech: Gravitational Waves with Kip Thorne - CosmoLearning.com Physics
What is General Relativity? Lesson 48: Ricci tensor and conformal transformations
What is General Relativity? Lesson 48: Ricci tensor and conformal transformations We introduce the Ricci tensor, curvature scalar, and begin the difficult derivation of the Weyl tensor. Please consider supporting this channel via Patreon: https://www.patreon.com/XYLYXYLYX and discussin
From playlist What is General Relativity?
Lec 4b - Phys 237: Gravitational Waves with Kip Thorne
Watch the rest of the lectures on http://www.cosmolearning.com/courses/overview-of-gravitational-wave-science-400/ Redistributed with permission. This video is taken from a 2002 Caltech on-line course on "Gravitational Waves", organized and designed by Kip S. Thorne, Mihai Bondarescu and
From playlist Caltech: Gravitational Waves with Kip Thorne - CosmoLearning.com Physics
What is General Relativity? Lesson 39: The curvature - formal introduction
What is General Relativity? Lesson 39: The curvature - formal introduction The Riemann Curvature Tensor is presented as a strictly formal object. Take note of an error captured by viewer "Endevor" in the comments. I may redo this video soon to fix it! Please consider supporting this chan
From playlist What is General Relativity?
The vector cross-product is another form of vector multiplication and results in another vector. In this tutorial I show you a simple way of calculating the cross product of two vectors.
From playlist Introducing linear algebra
Tensor Calculus Lecture 7d: The Voss-Weyl Formula
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