What are some reliable sites to learn it from, someone on YouTube or a course on coursera or anything? Also, would I be expected to know how to solve double differential equations already in case of vector cal?
https://imgur.com/K5imAEQ Here I have drawn out (the red lines) how the vector e_phi changes as theta changes. Is this correct?
If so, can someone help me in understanding as to how the vector d(e_phi) points in the direction of e_theta?
I tried applying this vector subtraction rule https://imgur.com/a/FRNstkP, but when I do this I get that d(e_phi) points in the direction of e_phi, which is not correct.
Hello all,
I’m an undergrad studying mathematics currently as at the junior level. In my free time I enjoy reading various mathematics book. I’ve come across tensors and tensor calculus and I’m having a difficult time fully understanding the concept. Can anyone here explain what a tensor is to someone who understands math but not yet on a deeper level? TIA!
Does anyone know any good resources (textbooks, videos, etc.) to learn tensor albebra and/or tensor calculus? My university doesn't offer any formal course in this subject and I think it would be a good asset in my courses to come (especially GR).
So I want to practice tensors and I want to know some booklets, textbooks, or worksheets that have worked out problems so I can attempt them and then check myself. I want a LOT of practice so I can master it. I have reviewed the concept but without practice I doubt I'll be able to understand them deeper.
I also need it for relativistic quantum mechanics.
Hi, so the title is pretty much it. I'd like to have some book that complements, those two YouTube playlists. As for my level, I am a junior at an engineering college and my linear algebra is a bit rusty, but I can pick it up pretty fast (I just fell out of practice). Here's the two playlists for your reference.
https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx
https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG
Thanks!
Please correct me if I'm using wrong terminology, I'm a CS student.
In deep learning libraries such as pytorch or tensorflow, one can define multi-dimensional matrices, e.g A in R^{n x m x p}
I believe the correct terminology for these are tensors?
These deep learning libraries allow multiplication of such tensors, e.g B in R^{p x q} => (A*B) in R^{n x m x q}
I've have an equation written down in terms of such tensors and would like to solve it for a variable. However applying matrix calculus on it leads to inconsistencies with respect to the dimensionalities. I've tried looking up properties on basic operations (like the ones available for matrices), but I've only come across pages similar to this which confuse me and I am not sure if the mathematical tensor is the same as those defined in the deep learning libraries. My question are as follows:
- Are the tensors contained in these libraries the same as the mathematical tensors?
- Is there resources similar to the page I linked for matrices which explains in simpler terms the properties of tensor operations? What I need is a less abstract definition and something more concrete.
Thank you!
Hi,
I’m a mechanical engineering undergrad student. I love the maths and physics part of my programme.
I can’t believe i’ll never use it again after I graduate and start to work in some mechanical engineering field. I mostly see design jobs which involves lots of CAD / documenting... I also like doing CAD but I love the construction stuff more. Like doing FEA or Calculating stress/strain/torque on all kinds of mechanical parts.
Heavy machinery design, using dynamics and statics to determine the parameters of the machine design
Are there jobs which require alot of the maths and physics that you’ve taken in your bachelor’s / master’s degree?
I love doing these calculations by hand and I also am willing to use (FEA) software for more advanced stuff.
The reason I got into engineering was the whole maths and physics stuff. Doing problems, lots of them. And i’d like to have a job that requires me to do all that stuff.
Even in my free time I am doing mathematics and physics to learn more about their fields. Maybe I should go into physics?
I am interested in heavy offshore constructions, maritime constructions (vessels) heavy machinery for production plants.
Forgive me for any mistakes that I made. English is not my native language. I am a student from the Netherlands (Europe).
If anyone could chime in that would be great!
Thanks in advance!
“In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita it was used by Albert Einstein to develop his theory of general relativity. Contrasted with the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold. Tensor calculus has many real-life applications in physics and engineering, including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity) and quantum field theory.” -Wikipedia
Prerequisites:
Books:
-
Schutz - Geometrical Methods of Mathematical Physics 1st Edition
-
Frankel - The Geometry of Physics: An Introduction 3rd Edition
-
Bishop & Goldberg - Tensor Analysis on Manifolds (Dover Books on Mathematics)
Articles:
-
[TU](https://www.win.tue.nl/~lflorack/Extensions/2WAH
I am currently a junior and finished Calc 3 and linear algebra this year. I'm not super interested in DEs and I just found out I only have a few days and I need to develop a curriculum for an independent study in mathematics for next year. I want to generally design a curriculum on advanced calculus- tensors, manifolds, differential forms i.e. there are so many texts online and so many different approaches to learning the material so can someone help me create a structured order to learning tensor calculus, differential forms and calculus on manifolds, that could include some preliminary info from other sources at the beginning to cover up any holes I might have? Thanks!
Hi all, I’m trying to bump up my math abilities and I’m looking for resources that cover these topics. I have textbooks that are helping but I was wondering if anyone knew of solid YouTube channels/videos that cover these? I know Andrew Dotson has a series in tensor calculus but I’m looking for another past that one. Thanks in advance!
Hello all, I just began working through Synge and Schild's Tensor Calculus to fill my free time, and can't figure out an exercise in the first chapter. Help would be appreciated!
Let \phi = a_{rs} x^r x^s ; Show \frac{\partial \phi}{\partial x^r} = (a_{rs}+a_{sr}) x^s.
I cannot for the life of me figure out how the indices switch in the RHS... For context, I have a degree in physics but never encountered much index notation.
"Boy! Johnny has really taken an interest in Tensor Analysis. My son must be smartest 12 year old in the city!"
I've been testing myself on the topic of tensor analysis by deriving the Navier Stokes equation in cylindrical coordinates. I am having a problem with the Laplacian, and I'm not sure what I did wrong. I asked the question in Stack Exchange as well, and my process is written there:
Derivation of Vector Laplacian in Cylindrical Coordinates through Tensor Analysis
Any ideas on where I went wrong?
EDIT :: I found the issue! I have answered my own question on Stack Exchange detailing the issue. In short, I was treating v^i _{,j} as a vector, when in fact it is a tensor. This changes the calculation of its covariant derivative. For more details, you can look at my solution here.
When the tensors are mixed.
I'm attaching at the bottom an example of a derivation for momentum continuity to illustrate what I'm talking about. It seems like eventually you get to a point in the multivariable algebra and calculus where the rules and notation are less standardized. I feel like I'm missing a standard set of rules for how to deal with keeping track of tensor order and rank, and things like when are some operations commutative/distributive or not. Frequently I find myself having to scour wikipedia for obscure rules and identities that I've never heard of or seen before. For example, today in this derivation I needed to look up what a dyad was (the multiplication of u_vec with itself as an outer product) and how to take a divergence of a dyad. I get the vague impression that inner products reduce the order of the tensor product and outer products increase the order of the tensor product, but I'm not entirely sure about how this stuff generalizes. I'm also starting to work with optimization problems and needing to get way more familiar with generalized multivariable operations that map vector spaces to vector spaces, and generalized derivatives and gradients where you specify what you are taking it with respect to.
My general question is, "Is there a good resource that covers generalized rules for multivariable algebra/calculus that handles tensor stuff in a standardized way?"
https://preview.redd.it/wemgua80doz21.png?width=2180&format=png&auto=webp&s=545e83d29d5de5ec721f39df6c2cedc6daa946bc
https://preview.redd.it/j0aou4y2doz21.png?width=2144&format=png&auto=webp&s=9b1c1c16f84753e765a860c6dcf2bad1956a2a22
T is a 3x3 tensor (cauchy stress tensor in my application) and u is a velocity vector.
I want to evaluate the quantity ∇∙u∙∇T. At the end I should get a vector. In terms of dimension ∇T is 3x3x3, u∙∇T should have 9 numbers, but in what "dimensions"? If I represent by its "column vectors" T:=[t1,t2,t3] does u∙∇T=[u∙∇t1,u∙∇t2,u∙∇t3]? After all this, in what direction does the final divergence "compress" our array?
My physicist friend says that I can just apply shit to the rows and its fine, but my geometry friend said at the end I should either get a covector or a "tower vector".
I am a CS graduate that has been working for a have a few publications and I am quite comfortable around topology and information theory out of life living it's course I have to now look into tensor calculus. I know this is math / physics undergraduate material but I would like to know some materials for going into it. My linear algebra is OK in terms of using it but I don't know all the proofs anymore.
Resources from ages tutorials or books on tensor calculus appreciated.
Thank you all in advance.
Okay, so it would be a stretch to try and list everything I currently know, but I just graduated high school a couple of months ago and have done a lot of external reading in my free time. I've been told that I will need to polish up my vector calculus and learn some differential geometry before attempting tensor calc, but I'm still unsure of where/ how to learn either of these three things.
Can someone please help me out?
Hi, So this is a long shot, I'll be honest. Not because I think this community isn't capable, but I'm not sure that this question is within its scope.
I need a "kind-of" first principles calculus-based derivation of the strain tensor. If any of you are familiar with Landau and Lifshitz's Theory of Elasticity, what I'm looking for is something that starts about one step back from their treatment, i.e. starting with finite quantities rather than differentials, so that the strain tensor is acquired in a limit.
I'm sure lots of you know what the strain tensor is, just not sure if anyone has dealt with it at this level, so asking this here might be dumb.
Cause, for example, you guys might actually be useful with this stuff, rather than knowing in painful detail esoteric and totally useless theory (which is where I am).
Anyway, just let me know if you want to take a stab at it. Thanks.
P.S. why am I not going to the math or physics subs? Because I've dealt with them before and I A) don't really like them and B) found them to be mostly useless.
P.P.S. In the meantime I will be writing up a document to give a precise picture of what I'm asking for. I've been intentionally general with the above text.
Here is a link to my treatment of the "derivation" L&L do. What I'm looking for is
-
A) A step backward from the L&L treatment, starting with finites then transitioning to differentials in a limiting approach.
-
B) A more direct path to the strain tensor, I don't understand why they deal entirely with magnitude (dl and dl' squared) and that, for me, totally obscures the greater point.
Tomorrow, when I get to work, I'll work on writing up the starting point for my approach.
So I want to practice tensors and I want to know some booklets, textbooks, or worksheets that have worked out problems so I can attempt them and then check myself. I want a LOT of practice so I can master it. I have reviewed the concept but without practice I doubt I'll be able to understand them deeper.
I'm a beginner at them so I don't understand the operational basics, so a lot of these problems woud come in handy ,a long with some more advanced ones.
“In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita it was used by Albert Einstein to develop his theory of general relativity. Contrasted with the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold. Tensor calculus has many real-life applications in physics and engineering, including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity) and quantum field theory.” -Wikipedia
Prerequisites:
Books:
-
Schutz - Geometrical Methods of Mathematical Physics 1st Edition
-
Frankel - The Geometry of Physics: An Introduction 3rd Edition
-
Bishop & Goldberg - Tensor Analysis on Manifolds (Dover Books on Mathematics)
Articles:
-
[TU](https://www.win.tue.nl/~lflorack/Extensions/2WAH
