# Tensor Analysis with Applications in Mechanics

Tensor Analysis with Applications in Mechanics by Leonid P. Lebedev, Michael J. Cloud, and Victor A. Eremeyev

## Contents Tensor Analysis with Applications in Mechanics

• Preliminaries
• Transformations and Vectors
• Tensors
• Tensor Fields
• Elements of Differential Geometry
• Applications in Mechanics
• Linear Elasticity
• Linear Elastic Shells

## Preface to Tensor Analysis with Applications in Mechanics

The first edition of this book was written for students, engineers, and physicists who must employ tensor techniques.

We did not present the material in complete generality for the case of n-dimensional space, but rather presented a three-dimensional version (which is easy to extend to n dimensions);

Hence we could assume a background consisting only of standard calculus and linear algebra.

We have decided to extend the book in a natural direction, adding two chapters on applications for which tensor analysis is the principal tool.

One chapter is on linear elasticity and the other is on the theory of shells and plates.

We present complete derivations of the equations in these theories, formulate boundary value problems,

And discuss the problem of uniqueness of solutions, Lagrange’s variational principle, and some problems on vibration.

Space restrictions prohibited us from presenting an entire course on mechanics; we had to select those questions in elasticity where the role of tensor analysis is most crucial.

We should mention the essential nature of tensors in elasticity and shell theory.

Of course, to solve a certain engineering problem, one should write things out in component form; sometimes this takes a few pages.

The corresponding formulas in tensor notation are quite simple, allowing us to grasp the underlying ideas and perform manipulations with relative ease.

Because tensor representation leads quickly and painlessly to component-wise representation, this technique is ideal for presenting continuum theories to students.

The first five chapters are largely unmodified, aside from some new problem sets and material on tensorial functions needed for the chapters on elasticity.

The end-of-chapter problems are supplementary, whereas the integrated exercises are required for a proper understanding of the text. In the first edition we used the term rank instead of order.

This was common in the older literature. In the newer literature, the term “rank” is often assigned a different meaning.

Because the book is largely self-contained, we make no attempt at a comprehensive reference list.

We merely list certain books that cover similar material, that extend the treatment slightly, or that may be otherwise useful to the reader.

We are deeply grateful to our World Scientific editor, Mr. Tjan Kwang Wei, for his encouragement and support!

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