Maths - Tensor Applications

In order to be a tensor, a quantity needs to be a linear function. It also needs to change, according to certain rules, in response to a change of frame.

Geometry Tensors

Projection Tensor

Pe =
cos α
sin α
cos α sin α
=
cos2α sin α cos α
sin α cos α sin2 α

Rotation Tensor

Pe =
   
   

Reflection Tensor

Pe =
   
   

Mechanics Tensors

Inertial Moment Tensor

This relates the torque bivector to the angular acceleration bivector of a solid body. This is a property of the solid body. In three dimensions it is defined by a 3×3 matrix, the components of it depends on its shape and in particular its mass distribution, however the way that these components vary with a change in reference frame is a tensor property.

I = ∫ r2 dm =
Ix -Pxy -Pxz
-Pxy Iy -Pyz
-Pxz -Pyz Iz

The Inertia Tensor in 'n' dimensions is discussed on this page.

Stress Tensor

Pe = =
   
   

Strain Tensor

Pe = =
   
   

Elastic Tensor

Pe = =
   
   

 

 


metadata block
see also:
Correspondence about this page The Inertia Tensor in 'n' dimensions is discussed on this page.

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 

cover us uk de jp fr ca Schaum's Outline of Theory and Problems of Tensor Calculus - I'm finding this hard going, it starts off with as review of linear algebra, matrix notation, etc. It redefines a lot of conventions which are hard to relearn, such as superscrips instead of subscripts to identify elements, and a summation convention, then it goes into coordinate transformations. I cant find any definition of what a tensor is. I think this book is aimed at people who already have some knowledge of the subject.

I can't find any mention in this book of the term hyper-matrix.

Terminology and Notation

Specific to this page here:

 

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.