We hereby present tutorial three as a correction rearrangement, in slides of most of the Q&A in chapter 3 of Continuum Mechanics for Modeling Simulation and Design.
Obviously, these cannot be covered in a single tutorial class. We will select a number of these if a tutorial is needed before you semester exams.
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.
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.
We continue the application of differential calculus to tensor objects. This lecture begins with a review of limits and derivatives with its extension to large objects.
We note that online participation is improving. Keep it up!
We proceed to Differential and Integral Calculus wit tensor valued functions and tensor arguments. In the first set of slides, we extend out knowledge of scalar differentiation to larger objects. The examples we begin with are simple and easy to understand. The lecture concludes with the Gateaux extension to our elementary knowledge of differentials
In one more week, we shall cover the book up to the point of Integral Theorems.
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.
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