Week 12: Integral Field Theorems

Finally,…, in place of our Programming Clinic, we conclude this first set of lectures with a quick, largely computational approach to the Field theorems of Continua. This, combined with our earlier treatment of differential calculus, completes the application of the results of calculus to tensor objects.

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Time did not permit us to look too deeply into some of these matters. Take this as a way to whet your appetite to a fruitful understanding of Continuum Mechanics. We can avoid further theoretical work and the covariant formulation by using Mathematica for our computations. We shall more of such in the next chapter where we shall begin to look at the geometric formulations of continua.

13 comments on “Week 12: Integral Field Theorems

  1. NWANKITI UGOCHUKWU says:

    Good day Sir,
    As regards the self test. Will the test affect our curative score as you said earlier, whereby the average will.be taken every Time we attempt the test?

    NWANKITI UGOCHUKWU
    Mechanical engineering
    160404017

  2. Eruse Oghenefega 160407016 says:

    In the table in page 11 of chapter 3, what is the difference between the “contraction product” operation and the other operations there

  3. oafak says:

    Yes. If you don’t want the tests to affect your grades, don’t take them. We will ensure your grades are not reduced by taking the test. If you take a test more than once, your average scores will be used.

  4. Temitayo omodehin 160404031 says:

    Good day sir,
    I don’t understand how this solution from chapter 2, question 2.12 works. In particular from line 2 downwards
    ()^ = (det())()−T
    = (^3 det())−1^(−)
    = (^2 det())^−T
    = ^2^c

    • oafak says:

      Hello Temitayo,
      There is an error in this question. It has been corrected and may have been posted by the time you read this. In order to follow the steps here, you will need to understand from 2.42 that \det\,\left(\alpha\bold{S}\right)=\alpha^3 \det\,\left(\bold{S}\right). Using this, beginning with the definition of the cofactor of \left(\alpha\bold{S}\right)^\textrm{c}on the first line, the rest of the proof is straightforward:

      (1)    \begin{align*} \left(\alpha\bold{S}\right)^\textrm{c}&=\det(\alpha\bold{S})\left(\alpha\bold{S}\right)^{-\textrm{T}}\\ &=\left(\alpha^3\det\bold{S}\right)\;\frac{1}{\alpha}\bold{S}^{-\textrm{T}}\\ &=\left(\alpha^2\det{\bold{S}}\right) \bold{S}^{-\textrm{T}}=\alpha^2 \bold{S}^\textrm{c}. \end{align*}

  5. Dada Victor 170404530 says:

    Good day sir,
    Can I call the curl of a vector, the surface base vector since it is the derivative of the position vector on the surface.

    • oafak says:

      Could you please indicate where you got that erroneous definition of the curl of a vector from? For the third-order Levi-Civita Tensor \mathcal{E}\equiv e_{ijk}\bold{e}_i\otimes\bold{e}_j\otimes\bold{e}_k S11.36 defines the curl of a vector as \textrm{curl}\bold{\,v}=\textrm{div}\left(\mathcal{E}\bold{v}\right). It is also easily shown that \textrm{curl}\bold{\,v}=-\textrm{div}\left(\bold{v\times}\right).

      • Dada Victor 170404530 says:

        Okay sir
        Tensor Analysis III slide 18

        • oafak says:

          Alas, you misunderstand the Mathematica code (in S12.18) you are referring to. There is no point in the code where the curl of a vector is defined. Instead, the code calls the in-built definition of the curl of a vector implemented by Mathematica. That definition conforms to the definition given in the slides and repeated here.
          It will be necessary for you to study and understand when a code defines and when it is a call to a defining function, as is the case here! These issues were taught in Programming Slides 1 PS1.7-11

  6. Dada Victor 170404530 says:

    Good day sir,
    Can I call the curl of a vector; surface base vectors since it is the derivative of the position vector on the integral surface.

  7. Fagoroye Ayomide 170407508 says:

    Good day sir, can we solve the disk with the stoke theorem by making the disk a surface?

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