Here is the addition to the slides you have starting from page 73 of the slides you have on Tensor Calculus. It served as our introduction to the lecture this morning.
[gview file=”http://oafak.com/wp-content/uploads/2014/08/Executive-Summary1.pdf”]
I hope the class today was a bit clearer. Let me have your comments here.
Yes Sir, today’s class was somewhat clearer. From the injection you gave us at the early hours of the class, I was able to see the mechanics involved in finding the gateaux differential of a function of a scalar, vector and tensor w.r.t which one that is of interest. I saw clearly that the Grad, also known as the frechete derivative is borne out of the gateaux. Interestingly, I now ve a clearer picture of a differential and a derivative. Before today, I thought it they mearnt the same thing(though with some doubt). I think where I need to do a bit of work is to carefully look at the note on Gauss Divergence Theorem so that it becomes clearer and familiar.
A question for you Sir, since you said the christofel symbol behaves like a tensor most of the time. How many components does it have ? or does the components size vary depending on the order of operation.
As for components, the Christoffel symbols has 27 components just like any tensor with the same number of indices. The Christoffel symbol of the first kind has only covariant indices while the second kind has one raised index making it a tensor-like object with mixed indices: one up, two down. In Cartesian coordinates, all the components vanish. In the coordinate systems of importance such as the cylindrical and spherical, many of the components vanish but there are significant ones that remain non-zero.
We have code that compute these easily. Once you see the codes, you will find it easy to compute them yourself.
Sir. can i call the divergence of a Gateaux Differential a Frechet Differential ?
You can talk about the divergence of a vector or that of a tensor. The “Divergence of a Gateaux” is not a specific thing! A Gateaux of a scalar function is the differential that you are used to in what you used to call “Advanced Calculus” – which I can assume you can now see to be rather elementary. The Gateaux of a vector or a tensor function is a differential of either function. You have to be specific about the entity you are dealing with before starting to take its divergence. By definition, to obtain a divergence of any entity, you first find its gradient – a Frechet derivative. Then you do a contraction of its last (differential) base. The process is straightforward. I will soon post fifty solved problems in Tensor Calculus. There are plenty of examples there for you to see.
The class was clearer expecially the frechet derivative.It was ok to understand.Thanks for the way the class went sir
Sir, i really enjoyed both the Vector in Linear Spaces , and Tensor Algebra lectures, but still having a bit difficulties as to assimilate the Tensor Calculus because i missed a class. I have also checked the 50 questions. I believed they will be helpful.
Thanks.
Sir, i really enjoyed both the Vector in Linear Spaces , and Tensor Algebra lectures, but still having a bit difficulties as to assimilate the Tensor Calculus because i missed a class. I have also checked the 50 questions. I believe they will be helpful.
Thanks.
I forgot to mention Levi-Civita in the list of “Tormentors”!