There are lots of lectures, but that's the first. Make sure you find the replacement one at the end of the list — the last ten minutes of that video disappeared.
I really do "visualize" vector fields, work, and flux ... I visualize differential forms that come up in the context of differential geometry — to me the connection forms are extremely geometric (twisting of the frame, etc.).
Actually that's another question I have for you guys. How do you guys refresh your math skills. For example, I haven't worked with linear algebra for a couple years now and I might need to refresh. How would I do that? do you guys just reread books and redo a bunch of problems
well I remember two things: 1. scalar product of one-form with a vector is a scalar (by definition)
2. a p-form represents some "thing" you would integrate over, like a density
@TanMath if you already have done the material before and just need to brush up on stuff, I highly recommend doing an "exercises first" approach and then read back on chapters as you need to if you get stuck.
So just crack open a linear algebra book you have and do exercises from the start and move on as things come back to you slowly.
These two things you remember are not extremely helpful with regards to the exercise I gave you. I am not sure if that book of Penrose's gives you the actual definition of a manifold.
@anakhro I guess the general idea of it. The formal definition I don't remember. But there is an entire chapter dedicated to n-manifolds which is where the discussion on one-forms and p-forms started in the book.
@TanMath: My lectures cover multivariable analysis and linear algebra, including discussions of manifolds and different ways of thinking about them (not abstract ones, just ones in Euclidean space).
Yeah 34 chapters of that good good math and physics. Starts out with geometry, even hyperbolic geometry, talks about numbers, complex numbers, calculus, complex calculus, surfaces, hypercomplex numbers, manifolds, group theory, fibre bundles. then starts going into physics: SR, GR, EM, QM, particle physics, QFT, cosmology, supersymmetry, and quantum gravity theories like string theory, loop quantum gravity, and twistor theory.
Ok so, if $A$ a subspace of a metric space has no isolated points then its diameter is equal to the diameter of its interior: we obviously have $\delta(\text{Int}(A)) \le \delta(A)$ and if $x, y$ are two points of $A$, since they're no isolated points there exists $x', y'$ where $x'$ is within $\epsilon$ of $x$ and $y'$ likewise.
So we get $$d(x, y) \le d(x, x') + d(y, y') + d(x', y') \le 2 \epsilon + \delta(\text{Int}(A)))$$ Since there are open balls in $A$ containing $x'$ and $y'$. So $$\delta(A) \le 2 \epsilon + \delta(\text{Int}(A))), \ \text{for all} \ \epsilon \gt 0$$
By no means is it comprehensive though. for example it doesn't even have a formal definition of manifolds. I think Penrose it trying to focus on getting an intuition for these concepts, mainly a geometrical intuition. And I like it, but it starts getting a little confusing, given I am no math expert (have only gone till differential equations and some other math here and there)\