Recent content by Lucas SV

Yes. Unless of course you decide to change domains to complex numbers.

My bad, I've made a mistake. Corrected it.

Not correct. Rather, ##f^{-1}(D) = [-\sqrt{2}, \sqrt{2}]##, since ##f^{-1}(D)## is defined as the set of all ##x## such that ##f(x) \in D##. Also you can use ##D## and the inverse image is always well defined, even if ##D## is not completely contained in the range ##f(\mathbb{R})##. In fact...

Why a class though? Since such metrics are written as ##d: \mathbb{R}\times\mathbb{R} \rightarrow [0,\infty)##, which can be thought as a subset of ##\mathbb{R} \times \mathbb{R} \times \mathbb{R}##, so the set of all metrics equivalent to the Euclidean metric would be well defined in set theory...

Yes, I've seen the metrizability concept before. But my question is not so much on the existence of a metric function which generates the topology, but given you already know it exists, for instance the Euclidean topology, is there some structure to the set of all such metric functions. For...

Given a topological space ##(\chi, \tau)##, do mathematicians study the set of all metric functions ##d: \chi\times\chi \rightarrow [0,\infty)## that generate the topology ##\tau##? Maybe they would endow this set with additional structure too. Are there resources on this? Thanks

Good luck! I hope you also learn some good skills in your job to pay him up.

Is that not true in the U.S? What are generically the requirements to start a PhD in the U.S?

Why can you only start at 27? is it financial issues? Also where would you be willing to live? Are you willing to relocate? If not can be done about getting a PhD earlier, what would you do in those two years? Interesting that you'd find university research suffocating. By the way, if you are...

There are no operators. ##\phi## is a real valued function, called the field configuration, and the path integral is taken over all such configurations. Operators fields are only introduced in the canonical formulation. ##\phi## and ##\hat{\phi}## are related, but not the same thing.

Because it is the generator of green's functions (also called n-point correlation functions). See the first section of https://arxiv.org/abs/0712.0689 for definitions. c-number. Yes, it does make sense to talk about the partition function of a classical field theory. c-number.

That is nice, it is a huge area. My suggestion is to continue learning calculus, aiming for functions of multiple variables. Meanwhile read about thermodynamics from more basic material that does not rely as much on calculus, and also read on the history. Personally, I first looked into...

Ok, I got around to reading #5, turns out I didn't need to brush up on thermodynamics so much for a simple reason: you are putting the cart before the horse. Let me explain. You are trying to learn material that uses partial derivatives and functions of two variables, before you even understand...

That is correct. Notice that as long as you have the initial and final velocity and the time difference, you can always compute this, irrespective of the body's motion. Let me give you some history on this. Both Newton and Leibniz are considered to have been the inventors of calculus. The...

I think you're confusing average and instantaneous acceleration. This webpage defines the terms: http://physics.info/acceleration/. As I said ##a=u/t## is not always right, but let us, for the sake of argument say it is (and I'm once again assuming ##a## and ##u## are functions of time. Then...