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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12 comments on “Chapter Two: Tensor Algebra

  1. Good morning sir. Trust this meets you well. I recollect this test question but I have not been able to understand how to solve it sir. It says ‘Given a and b are vectors, we should explain (a+b)x(a-b).(bxa) . I would bevery glad if you could explain it sir as I could not answer it in the test.

    • oafak says:

      Have you looked at Q1.16 in the course textbook, Chapter One? This question was based on it. It is a good idea to first look at the materials already given to you before asking a question. If after you have done that, you still have problems, I will be more than glad to help you.

  2. Chiedozie Chika-Umeh says:

    Good morning, sir. On this new chapter on tensor. What exactly is a Tensor? I understand it is a linear transformation of a vector to another vector. Does this mean that tensor is an operator? But there is still something like ‘second order tensor(vectors)’ Is there any other way to describe a tensor?

    Chiedozie Chika-Umeh 160404046

    • oafak says:

      When we say “Tensor” with no qualifications, we are talking about a second-order tensor. The slides as well as the Course text say that at the outset. If we want to talk about any other kind of tensor, we would say so explicitly. A second-order tensor is a linear transformation from one vector space to another vector space.
      We can write the tensor in an operator form, or in a functional form. For our use, it really does not make any difference. If you program in an C++, for example, you will see that the only real difference between an operator and a function is not what actually gets done, but rather the syntax of invocation.In the way most texts present it, a tensor is presented in the operator form. However, there is nothing preventing it from being presented in a functional form.
      After we have defined the tensor as a transformation, this week, we are exploring the implications of this definition in providing us induced attributes of a tensor. Those include its components, invariants, products (scalar products, compositions, tensor products and you already know the contraction with a vector which was the defining attribute. These together concretize what really a tensor is. Pay attention, read the materials you have. You appear to be a perceptive person who wants to learn. I think we can have a fair amount of discussion.

  3. Sumah Daniel Isunoya says:

    In the text, Vectors were differeciated from scalars by writing them in BOLD. In my note, I represent a vector ‘a’ by ‘a’ with a dash above it.

    On getting to Tensor components, there’s a need to differentiate the tensor symbol T from its scalar components Tij when writing it down.
    If I use a T with a dash above it, someone might take it as a Vector in a place where it is not. I know that a vector is a Tensor but I would like to know if there’s a generally acceptable way of writing the Tensor symbol?

    Daniel I. Sumah
    Mechanical Engineering.

    • oafak says:

      If you want to write your equations by hand, you could use the double underlining to represent tensors: \underline{\underline{T}} = T_{ij} \underbar{e}_i \otimes \underbar{e}_ j or, overlining, like this: \overline{\overline{T}} = T_{ij} \bar{e}_i \otimes \bar{e}_ j. This is the way people wrote before computers became common. But we are discussing in 2019! This is not 1974! Why on earth do you want to resort to the way we did things when I was an undergraduate? Why are you not finding out how I communicate with your class? It is NOT hard for you to learn the same tools that I use: Tex, Latex, Mathematica or even the Equations editor of Microsoft Word.
      Do you not prefer that the above expression looks like this: \bold{T} = T_{ij} \bold{e}_i \otimes \bold{e}_ j?
      Any of the software listed will give you the ability to write your equations with the same standard as your textbooks, my notes and slides! You can learn these things now rather than later!

  4. Ojewunmi Aanuoluwapo says:

    Good Evening Sir, please when would the self test for last week be open for us to attempt?

    • oafak says:

      The computer self test is still being programmed for a web server. I cannot guarantee when it will be ready. We are working hard on it. You can test yourself against the 100 problems that you have been given. Its essentially the same but in a different format.


    Good day sir
    Please I would like to ask, what is the spherical part in a tensor ?
    Since a tensor without the spherical part is a deviatoric tensor.
    Is it the spherical transformation?

    Mechanical Engineering

  6. Yekini-ajayi Hussein o. says:

    Good day sir
    Please can we get solutions to the test questions, i seem not to understand all.

    • oafak says:

      Test questions are selected from the one hundred solved problems in the course text that you have been given! What you faced as an individual is a random selection.

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