Introductory Multi-Linear Algebra: Dual Spaces to Tensor Products - - PowerPoint PPT Presentation

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Aug 14, 2023 341 likes •387 views

Introductory Multi-Linear Algebra: Hayden Borg Introductory Multi-Linear Algebra: Dual Spaces to Tensor Products Hayden Borg University of Puget Sound May 2020 A Naive Introduction Introductory Multi-Linear Algebra: Tensors are a

SLIDE 1

Introductory Multi-Linear Algebra: Hayden Borg

Introductory Multi-Linear Algebra:

Dual Spaces to Tensor Products Hayden Borg

University of Puget Sound

May 2020

SLIDE 2

Introductory Multi-Linear Algebra: Hayden Borg

A Naive Introduction

Tensors are a generalization of vectors n components per basis tensor Rank n Independent of the choice of basis Transform linearly Examples Scalars: Tensors rank 0 Vectors: Tensors Rank 1 Matrix: Tensor Rank 2

SLIDE 3

Introductory Multi-Linear Algebra: Hayden Borg

A Naive Introduction: Stress Tensor

Consider the stress in a 3-dimension solid Convenient to represent cross-sectional area with a vector

rthogonal to its surface

There are three cross-sectional area for every point x, y, and z. For each cross-section, a force is described in three directions

SLIDE 4

Introductory Multi-Linear Algebra: Hayden Borg

Introductory Multi-Linear Algebra:

Dual Spaces to Tensor Products Hayden Borg

University of Puget Sound

May 2020

A Naive Introduction

Tensors are a generalization of vectors n components per basis tensor Rank n Independent of the choice of basis Transform linearly Examples Scalars: Tensors rank 0 Vectors: Tensors Rank 1 Matrix: Tensor Rank 2

A Naive Introduction: Stress Tensor

Consider the stress in a 3-dimension solid Convenient to represent cross-sectional area with a vector

There are three cross-sectional area for every point x, y, and z. For each cross-section, a force is described in three directions

A Naive Introduction: Stress Tensor

This information is most naturally encoded in a 3 × 3 array:   Fxx Fxy Fxz Fyx Fyx Fyz Fzx Fzy Fzz  