henceproved

if you allow me to be politically incorrect, is it fair to say that integration happens on measure space and differentiation happens on norm spaces and these two concept coincide/related only when the underlying spaces is both norm and measurable space.

A while ago I was reading Frobenius theorem (manifold) and although I didn't understand it completely, a key takeaway was the existence of integral manifolds which was used to prove the existence of solution on 1st order homogenous linear PDE. This got me thinking that to prove the higher order inhomogenous case, we (if possible) can extend this theorem for a tensor field. Is there any research/result in this direction.

Hi to everybody and nobody

start ChatJax

Is it something related to the det that occurs in a change of variable

I am reading Lee book on smooth manifold. In the chapter Differential form, he starts with saying that we cannot define multiple integral of a function in a coordinate independent way of a manifold and gives the following example Let $f: C \rightarrow \mathbb{R}$ where C is a unit cube and f is a const funtion = 1. Then $\int_C fdx = $ Vol(C) is not invariant under coordinate transf. My problem is I dont understand the above statement that why Vol(C) not invariant under coordinate tranf.

(X,M,u) be a measurable space. Suppose $A \in M$, then is it always true that $A$ has a measure (finite or infinite)

this time it is word to word exact

actually I asked the same question on the main site. feel free to look at it math.stackexchange.com/questions/3005915/…

Yes I am drawing from sequence, but I have to find words from which words like BAD or CAD are formed. hence words like ADB are legit

the answer is given to be 6/216, but I got 12/216 (which is 2/216*3! if you consider the permutation of letters)

216 it is

sorry typo

yes accepted

also you are forgetting to permute it among themselves, then you will get the same answer as me

sorry I didn't understand what you are trying to say

*chosen random with replacement

but the textbook answer is 6/256

my answer is 12/256

From letteres A,B,C,D,E,F three letters are choosen at random. What is the probability that words BAD or CAD will be formed from choosen letter

Hi basic prob question?

oh I get your point, so you mean saddle like situation occurs here

The step in the derivation looked if and only if

Existence of extremal function

for minimal

Euler Lagrange equation (calculus of variatio) is necessay but not sufficient why?

now I got it

yes for every step of induction we need one eigen value

that would be an intresting example to check for me

Scheme of proof: One show that V has one invariant subspace W (which will turn out to eigen space), and then we use induction on V/W

suppose T is a linear tranf. over a vector space V over F sulch that all eigen value of T lies in F, then T can be written as triangular matrix? I think I weaken the hypothesis of the above theorem by requiring atleast eigen value is in F. Am I right in saying so?

Is my proof right?

= v(m(y)) = 0 implies m(y) = 0

then m(T)v = 0v = 0

yeah i think earlier I was mixing newtonian wave with wave equation.

actually mass controls the acceleration due to external potential. And also shrondinger equation has mass term, hence the evolution of wave equation is determined by the mass of the particle.

let me think about it for a moment

actually that what i wrote that mass should be irrelevant in physics in the last part of my statement. But somehow this statement seems to me a bit radical and hence wanted to confirm

and wave and particles both have momentum (although defined by different formula in each case). Is this the right intrepretation.

Hi, everyone. I was thinking about quantum-mechanics when the following thought came to me. Consider a particle like electron. By de-borgli equation, this electron has a wave equation $\psi$. Now we know that electron has mass but waves doesn't, so how do I account for mass of electron in the $\psi$ description. One way I thought a way out of this conundrum is the mass is irrelevant and only momentum is relevant,

okay nice talking to u, see u soon,

so if i have to understand the quatum behavious of light, I have to study QFT?

quantum mechanics

from ur preivous comment i get the impresssion that study of photon doens't come under QM, is that what u were saying

noted

@vzn if you are not busy I would like to hear both the pov

Greeting everyone, I have a conceptual difficulty. Light, as we know, can be considered as a photon as well as EM wave. The photon has a QM description given by wavefunction. Now I want to know what is the relation between wavefunction and EM wave equation (Similar questions have been asked before in physics se but I didn't understand it well).

but the answer says it is a noun

WHILE according to me should be a adverb

Identify the part of speech: Sit down and rest a WHILE