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The Tensor Confusion: Why So Many Definitions?

math_is_fun

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Computer scientists say tensors are multidimensional arrays.

Physicists say tensors are geometric objects that obey transformation laws.

Mathematicians say tensors are multilinear maps.

How can all of these definitions be talking about the same thing?

In this video we begin a journey through the different ways tensors are understood across mathematics, physics, engineering and computer science.

We start with tensors as multidimensional arrays, explore the physicist's viewpoint based on transformation laws, and then develop the modern mathematical interpretation of tensors as multilinear maps.

Topics covered:

• Tensors as multidimensional arrays
• Covariant and contravariant components
• Tensor transformation laws
• Scalars, vectors and covectors
• Dual spaces
• Multilinear maps
• Tensor notation and components
• Why different communities define tensors differently

This is Part 1 of a series. In the next videos we will develop the tensor product construction, quotient spaces, the Universal Property, and see how these viewpoints are ultimately unified.

Chapters:

00:00 Introduction
01:21 Tensors as Arrays
03:25 Physicist's Viewpoint & Essentials Background
10:12 How Different Objects Transform
21:20 Tensor Transformation Laws
23:40 Mathematician's Viewpoint: Multilinear Maps
30:04 Tensor Transformation Laws from Multilinear Maps

#Tensor #TensorCalculus #LinearAlgebra #Mathematics #Physics #MultilinearAlgebra #MathEducation #TensorProduct #DifferentialGeometry #STEM #Engineering #Manim

10 Comments

@MinhAnh-n1w-h9b 3 weeks ago

The best explanation of tensor

@Sergio_M97 1 month ago

A tensor can be understood as a multilinear function, an element of a tensor product, or a multidimensional array, due to the isomorphisms between these spaces. This is precisely why the tensor product is ultimately defined through its universal property.

@Rick-h4q9s 1 month ago

Great vid. Is it possible to get one on tensor products?

@DR-tx3ix 4 weeks ago

The first half was fine, I could follow along keeping the (x,y) basis vs (r, theta) basis in mind. But after around 21:00 it became mostly textbook equations. (nothing wrong with that but it didn't do me any good).

@harrywoodell7008 3 weeks ago

I walk up a hill, and as the crow flies, I walked way more. The metric tensor. Want skew with that? Walk a back and forth path up a hill to keep the slope and grip on the surface manageable. So we get skew. Walking in Appalachia is a metric tensor. QED.

@fabiobrainer573 8 days ago

Em alguns casos se pode dizer que vetores são tensores e tensores são vetores, se colocados juntos, isso parece não fazer sentido

@fabiobrainer573 8 days ago (edited)

Eu: venho pela definição formal matemática
Ele: "vou ignorar a definição formal neste vídeo!"
🤨 Estou sendo sabotado...

@TonyIta-w1p 1 month ago

ALL THREE DEFINITIONS ARE RIGHT, IF YOU KNOW THAT A JACOBIAN IS ANOTHER NAME FOR A MULTILINEAR MAP AND A MULTILINEAR MAP IS JUST ANOTHER NAME FOR A MULTIPLE ARRAY. THE DIMENSIONS OF THE JACOBIAN = SAME DIMENSION OF THE MULTILINEAR MAP = SAME DIMENSION OF THE MULTIPLE ARRAY. THEY ARE POINTING TO THE EXACT SAME PHYSICAL OBJECT IN A GIVEN-DIMENSIONAL HILBERT SPACE. THE PHYSICISTS OFTEN STOP AT FOUR DIMENSIONS, WHILST THE COMPUTER SCIENTISTS & MATHEMATICIANS MAY DEAL WITH TENSORS OF ANY DIMENSIONS. THE EXPRESSIVE POWER OF TENSORS IN COMPUTATION IS PHENOMENAL. E.G. COMPUTER GAMES & LLMs.