Integrability Definition and 65 Threads

In Philippe G. Ciarlet's book 'An introduction to differential geometry', He gives the integrability conditions of the differential equations like this: $$ \partial_{i} F_{lj}=L^p_{ij} F_{lp},\,\,\,F_{ij}(x_0)=F^0_{ij}. $$ The integrability conditions for the existence of a global solution...

Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability...

Hi, starting from this thread, I'd like to clarify some mathematical aspects related to the notion of hypersurface orthogonality condition for a congruence. Let's start from a congruence filling the entire manifold (e.g. spacetime). The condition to be hypersurface orthogonal basically means...

Was reading the Reimann integrals chapter of Understanding Analysis by Stephen Abbott and got stuck on exercise 7.2.5. In the solutions they went from having |f-f_n|<epsilon/3(a-b) to having |M_k-N_k|<epsilon/3(a-b), but I’m confused how did they do this. We know that fn uniformly converges to...

Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1-form ##\omega##, namely that ##d\omega = \omega \wedge \alpha## is actually equivalent to the existence of smooth maps ##f## and ##g## such that ##\omega = fdg##. I found this About...

Let ##f## be a measurable function supported on some ball ##B = B(x,\rho)\subset \mathbb{R}^n##. Show that if ##f \cdot \log(2 + |f|) ## is integrable over ##B##, then the same is true for the Hardy-Littlewood maximal function ##Mf : y \mapsto \sup_{0 < r < \infty}|B(y,r)|^{-1} \int_{B(y,r)}...

Apologies for potentially being imprecise and clunky, but I'm trying understand integrability of the following Hamiltonian $$H(x,p)=\langle p,f(x) \rangle$$ on a 2n dimensional vector space $$T^{\ast}\mathcal{M} =\mathbb{R}^{2n}.$$ Clearly this is just the 1-form $$\theta_{(x,p)} =...

Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$ S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...

We show that there is a partition s.t. the upper sum and the lower sum of ##f## w.r.t. this partition converge onto one another. Let ##\epsilon>0##. Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a...

Suppose we have an infinite dimensional Hilbert-like space but that is incomplete, such as if a subspace isomorphic to ##\mathbb{R}## had countably many discontinuities and we extended it to an isomorphism of ##\mathbb{R}^{\infty}##. Is there a measure of integrating along any closed subset of...

I am trying to evaluate an integral with unknown variables ##a, b, c## in Mathematica, but I am not sure why it takes so long for it to give an output, so I just decided to cancel the running. The integral is given by, ##\int_0^1 dy \frac{ y^2 (1 - b^3 y^3)^{1/2} }{ (1 - a^4 c^2 y^4)^{1/2} }##

We have a function ##f: [a,b] \mapsto \mathbb R## (correct me if I'm wrong but the range ##\mathbb R## implies that ##f## is bounded). We have a partition ##P= \{x_0, x_1 , x_2 \cdots x_n \}## such that for any open interval ##(x_{i-1}, x_i)## we have $$ f(x) =g(x) $$ (##g:[a,b] \mapsto \mathbb...

Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if $$ sup \{L (f,P) : \text{P belongs to the set of...

So, I know that a function is integrable on an interval [a,b] if ##U(f,P_n)-L(f,P_n)<\epsilon ## So I find ##U(f,P_n## and ##L(f,P_n## ##L(f,P_n)=5(3-\frac{1}{n}-0)+5(3+\frac{1}{n}-(3-\frac{1}{n}))+7(4-(3+\frac{1}{n}))=22-\frac{2}{n} ##...

For a closed interval ##[a,b]## I have learned that ##U(f,P)-L(f,P)=\frac{(f(b)-f(a))\cdot(b-a)}{N}## where ##N## is the number of subintervals of ##[a,b]## (if ##f## is monotonically decreasing, change the numerator of the fraction to ##f(a)-f(b)##). However, if the interval is half-open, then...

Homework Statement Find a such that f is Riemann integrable on [0,1], where: ##f = x^acos(1/x)##, x>0 and f(0) = 0 Homework EquationsThe Attempt at a Solution I found at previous points a such that f is continuous, bounded and derivable, but I am not sure how to use that (as all these...

Prove that the function $$f(x) = 1+x, 0 \le x \le 1$$, x rational $$f(x) = 1-x, 0 \le x \le 1$$, x irrational (they are one function, I just don't know how to use the LATEX code properly) is not integrable on $$[0,1]$$ I don't know where to start, I tried to evalute the lower and upper Riemann...

Hello, I was wondering if two models show the same S matrix by a direct relation between their parameters, does that necessarily mean that both models are exactly equivalent? My idea is that this is true, but would like to know about a solid argument about it if possible, thank you!

Hi everybody! I have another question about integrability, especially about the limit comparison test. The script my teacher wrote states: (roughly translated from German) Limit test: Let -∞ < a < b ≤ ∞ and the functions f: [a,b) → [0,∞) and f: [a,b) → (0,∞) be proper integrable for any c ∈...

Homework Statement Here is a link to the proof I am reading: https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf Homework EquationsThe Attempt at a Solution The proof to which I am referring can be found on pages 8-9. At the top of page 9, the author makes an assertion which I endeavored to...

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of Riemann's Criterion for Integrability - Stoll: Theorem 6.17 Stoll's statement of this theorem and its proof reads as follows: https://www.physicsforums.com/attachments/3941In the above proof we...

Homework Statement Prove or give a counter example of the following statement: If f: [a,b] \to [c,d] is linear and g:[c,d] \to \mathbb{R} is Riemann integrable then g \circ f is Riemann integrable Homework EquationsThe Attempt at a Solution I'm going to attempt to prove the statement is...

I'm reading "The variational principles of mechanics", written by C. Lanczos and he said that, if one have the condition dq_3 = B_1 dq_1 + B_2 dq_2 and one want to know if there is a finite relation between the q_i, on account the given condition, one must have the condition \frac{\partial...

Homework Statement 1. Suppose that ##f = 0## at all points of a rectangle ##R## except on a set ##D## of outer content zero, where ##f \geq 0##. If ##f## is bounded, prove that ##f## is integrable on ##R## and ##\int \int f dA = 0##. 2. Now suppose ##f## is an integrable function on a...

Hi Let a classical particle with unit mass subjected to a radial potential V and moving in a plane. The Hamiltonian is written using polar coordinates (r,\phi) H(r,\phi) = \frac{1}{2}(\dot{r}^2+r^2\dot{\phi}^2) - V(r) I consider the angular momentum modulus C=r^2\dot{\phi}, and I...

Homework Statement Let h(x) = 0 for all x in [a,b] except for on a set of measure zero. Show that if \int_a^b h(x) \, dx exists it equals 0. We are given the hint that a subset of a set of measure zero also has measure 0. Homework Equations We've discussed the Lebesgue integrability...

Homework Statement Let ψ(x) = x sin 1/x for 0 < x ≤ 1 and ψ(0) = 0. (a) If f : [-1,1] → ℝ is Riemann integrable, prove that f \circ ψ is Riemann integrable. (b) What happens for ψ*(x) = √x sin 1/x? Homework Equations I've proven that if ψ : [c,d] → [a,b] is continuous and for every set...

Homework Statement Let f(x) be defined on [0,1] by f(x) = 1 if x is rational f(x) = 0 if x is irrational. Is f integrable on [0,1]? You may use the fact that between any two rational numbers there exists an irrational number, and between any two irrational numbers there exists a...

The title is self-explanatory. What is it meant in the physics and maths community by the words integrability and integrable system?

Hi, Homework Statement I am now asked to prove that f: [0,1]->[0,1] defined thus f(0)=0 and f(x)=1/10n for every 1/2n+1<x<1/2n for natural n, is integrable. Homework Equations The Attempt at a Solution Would it suffice to show that f is continuous? I.e. that lim x->0 f(x) =...

Let X be a uniform integrable function, and g be a continuous function. Is is true that g(X) is UI? I don't think g(X) is UI, but I have trouble finding counter examples. Thanks.

Homework Statement If f is integrable on [a,b], prove that there exists an infinite number of points in [a,b] such that f is continuous at those points. Homework Equations I'm using Spivak's Calculus. There are two criteria for integrability that could be used in this proof (obviously...

The problem states: Decide if the following function is integrable on [-1, 1] f(x)=\left\{{sin(\frac1{x^2})\;\text{if}\;x\in[-1,\;0)\cup(0,\;1]\atop a\;\text{if}\;x=0} where a is the grade, from 1 to 10, you want to give the lecturer in this course What I don't understand is how to find L(f...

Homework Statement Show the Thomae's function f : [0,1] → ℝ which is defined by f(x) = \begin{cases} \frac{1}{n}, & \text{if $x = \frac{m}{n}$, where $m, n \in \mathbb{N}$ and are relatively prime} \\ 0, & \text{otherwise} \end{cases} is Riemann integrable. Homework Equations Thm: If fn...

The book is saying if f is monotonic on a closed interval, then f is integrable on the closed interval. Or basically if it is increasing or decreasing on the interval it is integrable on that interval This makes sense, however this theorem seems to obvious because obviously if a function...

This isn't a homework question. My adviser has me studying basic analysis and has lately pushed me towards the following question: "Let f be any continuous function. Can we prove that there exists a SEQUENCE of step functions that converges UNIFORMLY to f?" I have noticed this idea is...

If a function f is continuous at a point p, must there be some closed interval [a,b] including p such that f is integrable on the [a,b]? As a definition of integrable I'm using the one provided by Spivak: f is integrable on [a,b] if and only if for every e>0 there is a partition P of [a,b]...

The canonical example of a function that is not Riemann integrable is the function f: [0,1] to R, such that f(x)=1 if x is rational and f(x)=0 if x is irrational ( i know some texts put this the other way around, but bear with me because i can reference at least one text that does not). Hence...

(I've been lighting this board up recently; sorry about that. I've been thinking about a lot of things, and my professors all generally have better things to do or are out of town.) Is there an easy way to show that if f is Lipschitz (on all of \mathbb R), then \int_{-\infty}^\infty f^2(x)...

Hello all, can someone please direct me towards an argument proving the Lebesgue integral from 0 to infinity of sin x / x does not exist? Many thanks

Homework Statement If f,g are Riemann integrable on [a,b], then for c,d real numbers, (let I denote the integral from a to b) I (cf + dg) = c I (f) + d I (g) Homework Equations The Attempt at a Solution I have the proofs for c I(f) = I (cf) and I (f+g) = I (f)...

a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous. b) show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0. i am a little stuck on part b). i am trying to use the dominated convergence theorem but i am a bit confused on what...

Homework Statement Let A={1/n, n =natural number} f: [0,1] -> Reals f(x) = {1, x in E, 0 otherwise Prove f is riemann integrable on [0,1] Homework Equations The Attempt at a Solution Not quite sure, but I think supf = 1 and inf f= 0 no matter what partition you take, then...

Homework Statement Hi guys. I'm really struggling with this problem. Any help is welcomed. Suppose I have a function f(y) = \intg(x)/(x^2) on the set [(y/2)^(1/2), \infty]. g(x) is known to be integrable over all of R. I want to show that f is integrable over [0,\infty], and that the...

Compositions of functions and integrability (is this right?) Homework Statement We know that if f is integrable and g is continuous then g\circf is integrable. Show to this is not necessarily true for piecewise continuity. We are given the hint to use a ruler function and characteristic...

Hi All, When computing the commutator \left[x,p_{y}\right], I eventually arrived (as expected) at \hbar^{2}\left(\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) - \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)\right) and I realized that, as correct...

Hi, I'm stuck on this problem here about composite function, help is appreciated: Let g : [a,b] -> [c,d] be Riemann integrable on [c,d] and f : [c,d] -> R is Riemann integrable on [c,d]. Prove that f o g is Riemann integrable on [a,b] if either f or g is a step function I was able to solve...

Hi there, My objective is to study Hamiltonian systems, integrable and non integrable systems, where there will be chaos, etc. I have a general idea of everything.. the destroyed tori, the symplectic structure of hamilton's equations, etc. But nothing is very clear to me! And the most...

Homework Statement Let Q=I\times I (I=[0,1]) be a rectangle in R^2. Find a real function f:Q\to R such that the iterated integrals \int_{x\in I} \int_{y\in I} f(x,y) \; and \int_{y\in I} \int_{x\in I} f(x,y) exists, but f is not integrable over Q. Edit: f is bounded Homework...

Hi I am a mathematics junior and I am doing a research project on hamiltonian systems and liouville integrability (don't ask why...). I am using the book by Vilasi, a graduate level book, but I am finding it quite difficult and badly written; for instance he uses functional analysis and...