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.
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
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.
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.
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.
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!