Tensor Properties III: The Eigenvalue Problem

In this last week of lectures before your second test, we shall be looking at the Eigenvalue problem you have encountered already in your Engineering Mathematics. There is an equipollency between the tensor eigenvalues and those of matrices. The issues here are therefore familiar. The tensor form has specific physical implications in our engineering courses especially as it relates to the Principal invariants of tensors and the spectral form using eigenbases.
We shall also have two tutorial classes on the worked examples to help prepare you for the test. Remember that the test is a paraphrase of the same problems that are solved for you in the Q&A at the end of each chapter. EVERYTHING in the Q&A is part of the questions populating the database from where your questions are drawn.

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Tensor Properties II: Decompositions & Orthogonal Tensors

This week we continue our study of tensor properties with additive and spectral decompositions of tensors. We shall also look at orthogonal tensors. The slides are presented in the video

Here are the actual slides we will use in class. It begins with a repeat of some of last week’s outstanding issues that you ought to have gone through on your own. These are: Products of determinants, trace of compositions and scalar product of tensors. You must be current on these in order to understand this weeks menu.
The five topics covered here include:
1. The tensor set as a Euclidean Vector Space,
2. Additive Decompositions
3. The Cofactor Tensor, its geometric interpretation
4. Orthogonal Tensors
5. The Axial Vector
These are vocalized in the above Vimeo video and the downloadable slides are here

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Chapter Two: Tensor Algebra

Chapter two of the course text is posted here for your use. It is our hope that you will make use of this material given to you this time. Most students ignored the materials, turned deaf ears to all warnings and reaped the expected results in the test! The second chapter will be no different. Here, as before, there are 100 solved problems that are being turned into Multiple-Choice Questions. You will be examined on them within four weeks of today. We are working on the Web server that will allow you to also self test.

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We will cover some outstanding materials, hopefully, during the Programming class. We also include some guidance on the Q&A problems that are relevant to the week’s slides. If you work through them weekly, it will make the rush to cover the 100 questions unnecessary when the time for testing is near. Remember, these are mere guides. They are not to constrain you.

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The same set of slides can be seen in the following video:

As in the past, we urge you to make your comments on these notes. Include information on your department and Matric number because we will grade your participation on the website as part of your Continuous assessment. The full details for the course grading are as follows:

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Last Lecture Chapter One

Today is the last class on chapter one. I will still answer questions online on this chapter. Test is tomorrow, 11-1pm. Next week, we begin Chapter Two, Tensor Algebra. The goal of all you have learned in chapter one is to put you on a sound footing to learn tensor analysis. If you do chapter one well, the next chapter will be about the same or even easier. If you find the next chapter harder, it is likely coming from insufficient mastery of the material under vector analysis. Tensor theory is an extension of vectors.

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We have added some of the materials that should have been covered in the Programming practice of yesterday.

Programming Clinic II

The programming examples here and subsequently are given to explain certain concepts on Vectors, tensors, continuum mechanics or show you how to use specific Mathematica functions. The following steps should be followed:
1. Type in the codes and ensure they run error-free.
2. Try to understand the working program so that you can use similar constructs in your own coding. Make use of the help system in Mathematica to understand other ways you may be able to invoke the same functions with other argument sets and options. Sometimes the codes involve technical issues or a deeper understanding of vectors, geometry and transformations. Do not rush over those issues. A good programmer learns to address such issues and increases in knowledge.
3. Write your own programs by modifying and adding to the code you have been given to do similar or other things.
These steps will help you become more confident on your ability to create your own codes.
Note that turning your understanding of a concept to computer program that works is one of the best ways to master concepts. There is no way to forget such things that you have turned into codes by yourself!

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Note that all the codes included in this practical class are either given in the Continuum Mechanics main text that you should have downloaded by now, or in the appropriate lecture slides.
We remind you again that the questions on the test on Friday are essentially paraphrases, in objective form, of the same examples that have been solved for you in the book. You are to further remember that you may be able to get further assistance from us by posting questions and comments on the same web page.
Please post questions ONLY after availing yourself of all the other facilities already available. In particular, do not ask questions that have been adequately answered in the book unless you specifically want further elucidation after a honest attempt at the one given in the book.

Vector Space

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There are a number of examples in the text that today’s lecture expects you to type in and practice. If in future, you do not know how to program, it will be because you were lazy in simply typing in code and running them. “The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings.” A word enough for the wise!
There is a test next week on Chapter One. Those who refuse to practise are going to have a cold shower on a harmattan morning!