Recent content by Mozibur Rahman Ullah

Nice phrase - I'll have to use it somewhere :-).

I've noticed. But this thread began as a reply to a request for advice on self-studying physics beyond high school physics and I mentioned differential geometry and it went downhill from there ;-).

Not quite. Usually k-multivectors (or k-polyvectors or k-vectors) are wedge products of k vectors rather than the tensor product of k vectors. But given that there is no specialised term for them, I think it's no great stretch to use the same term for them.

Ah, I thought you were a 13 old year boy! A gender assumption on my part! I'm glad that what I wrote was taken in by you and understood. Richard Feynman once said that you only understand a concept if you can explain it to a student - so I guess I can say I understand the exterior covariant...

I've edited it to remove the offending word!

I meant to say at this stage of your education! Good catch.

Great, I'm glad you feel its worth learning. But I'd hold your horses about learning this stuff seriously at this stage of your education. This is advanced stuff and the exterior covariant derivative is really advanced stuff! Typically, you will learn vector analysis in undergraduate courses and...

We only need partitions of unity for smooth manifolds so we can develop a reasonable coordinate-free theory of integration. For topological manifolds, we don't have such a theory so we don't need partitions of unity. Nevertheless the condition here of second countability is enough to prove that...

The chain rule is best understood geometrically. To avoid thinking about higher dimensions, we will just consider 2d surfaces. However, the reasoning that follows goes through with minor adjustments for higher dimensions. Assume we have a surface ##M##, then we can attach to it all its tangent...

The most general definition of a manifold is that of a topological manifold. Now these are defined as Hausdorff and second countable topological spaces that are locally Euclidean. Now if we drop second countability and Hausdorff condition from the definition of a topological manifold we get...

To highlight the difference between differential geometry and vector analysis, it's worth noting that in vector analysis we have the four equations of Maxwell's equations (in Maxwell's original description there were in the order of twenty equations). In differential geometry there are two...

I agree that the internet can be dangerous but I think it can also be valuable. I learnt a lot from Cohl Furey's set of short videos on the octonions and their use in studying the structure of the standard model, I also learnt a lot about spinors from Eigenchris's set of video lectures on...

> What - really? Yeah, it was a surprise to me too! If you're familiar with tensors then you should know that they're classified as covariant and contravariant tensors as well as into symmetric and antisymmetric tensors. We also differentiate between tensors and tensor fields in the same way...

It's probably worth pointing out - if it isn't obvious to you - that the calculus of differential forms that you will come across in any differential geometry course is basically tensor calculus. Given your handle, I thought it important to point this out ;-). The course that I like is John...

I'm surprised that nobody has suggested Feynman's Lectures on Physics. There are 3 volumes. They are designed for a university undergraduate curriculum. But a bright teenager should be able to get something out them. Also try out his popular books - it was one of those that inspired me to study...