Multilinear algebra | Tensors | Differential geometry | Differential topology

Tensor field

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. Many mathematical structures called "tensors" are tensor fields. For example, the Riemann curvature tensor is not a tensor, as the name implies, but a tensor field: it is named after Bernhard Riemann, and associates a tensor to each point of a Riemannian manifold, which is a topological space. (Wikipedia).

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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?

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What is a Tensor? Lesson 16: The metric tensor field

What is a Tensor? Lesson 16: The metric tensor field

From playlist What is a Tensor?

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What is a Tensor? Lesson 29: Transformations of tensors and p-forms (part review)

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From playlist What is a Tensor?

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Visualization of tensors - part 1

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What is a Tensor 10: Metric spaces

What is a Tensor 10: Metric spaces

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What is a Tensor? Lesson 11: The metric tensor

What is a Tensor 11: The Metric Tensor

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What is a Tensor 3: Dual Spaces

What is a Tensor? Dual Spaces

From playlist What is a Tensor?

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Lek-Heng Lim: "What is a tensor? (Part 1/2)"

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From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021

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What is a Tensor? Lesson 18: The covariant derivative continued

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From playlist What is a Tensor?

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An Introduction to Tensor Renormalization Group (Lecture 2) by Daisuke Kadoh

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Tensor Network Methods in Four Dimensional Field Theory by Daisuke Kadoh

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The Einstein Field Equations | Tensor Calc Finale

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An Introduction to Tensor Renormalization Group by Daisuke Kadoh

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What is General Relativity? Lesson 39: The curvature - formal introduction

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What is a Tensor? Lesson 20: Algebraic Structures II - Modules to Algebras

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From playlist What is a Tensor?

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What is a Tensor 5: Tensor Products

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