Analysis and Calculus

so i broke the problem into 2 problem i think that this might be possible. so 1 https://docs.google.com/spreadsheets/d/11GqvUjDHVbl05w-LkLpZ6OWWWyX4v0uo/edit?usp=sharing&ouid=102842103828562427147&rtpof=true&sd=true proving that every even number is here note: their is an algoritme (in the colour )in every column for example column a is 4/4/4/ b is 3/2/3/2/3 and every column has one 2 https://docs.google.com/spreadsheets/d/1NJ4RALu6jXOD67SJ38OLLtuBDRYqZccI/edit?usp=sharing&ouid=102842103828562427147&rtpof=true&sd=true proving that when using the conjecture for ev…

To gain true understanding of a subject it can help to study its origins and how its theory and practice changed over the years – and the mathematical field of calculus is no exception. But calculus students who do read accounts of its history encounter something strange – the claim that the theory which underpinned the subject for long after its creation was wrong and that it was corrected several hundred years later, in spite of the fact that the original theory never produced erroneous results. I argue here that both this characterization of the original theory and this interpretation of the paradigm shift to its successor are false. Infinitesimals, used properly, were…

This problem is from the book Calculus Made Easy by Silvanuws P. Thompson, copyright 1946 (3rd Ed.) How on earth is answer not du/dt = 8u7? The author uses a dot for showing both multiplication and as a divider between the ones column and the tenths column (see pics) Thx

https://www.researchgate.net/publication/242501702_The_Mathematical_Description_of_Shape_and_Form On page 16, there appears to be a typo on formula (2.13). Should there be a dot above psi? So curvature should equal a derivative of the angle with respect to arc length. Thank you.

How do you parametrize an implicit surface? The surface in question is quadratic in terms of x, y, and z. Ie., 0 = ax squared + bx + cy squared + dy + ez + fz + gxy + hxz + uyz +j. I want general parametrization for all arbitrary constants from a-j. Answer should probably involve some combination of trigonometric functions of uv. I am asking this question for my son who does not have internet access. I will copy, paste, print, and send the answer to him. My mathematics background is far short of being able to understand this. Thank you.

Wikipedia article for laplacian says that it is how much the average value of the function deviates from the value of the function but this doesn't makes sense to me. Would it be the average value surrounding the point minus the value at the point divided by the area of the surface or the volume confined within? A while ago I realized that the laplacian equals the curl of the contour line of the function, but I forgot my logic. I know that the laplacian is the sum of second-order partial derivatives, but I would like to know what it means geometrically. Thank you.

Hi everyone, I'm trying to solve some Lebesgue problems from my exercise book and I got stuck in some of them: Prove that if a set A has zero measure, then its interior is empty. I've thinking on suppose the contrary and find an open subset of A with positive measure, but I'm not really sure if it's the right way. True or false: f is integrable if and only if |f| is integrable over Rn. I know that if f is measurable, then it's correct, but here there is no previous condition so I don't know if the statement is true or false. Could you give me some tips to solve it? Thanks in advance.

I just came across this paper https://figshare.com/articles/preprint/Primorial_numbers_and_the_Riemann_Hypothesis/13838111, claiming to prove the Riemann Hypothesis. I'm not an expert on this subject, but the proof seems to be valid. I have also attached the file below. Primorial numbers and the Riemann Hypothesis..pdf

I have attached a picture below, and the thought of this idea is confusing me too much. -> In the first graph, I have taken the area of a single rectangle (say, first rectangle) as M * Δx , where M represents the arithmetic mean of f(x) and f(x) + Δy ( Δy is f(x+Δx) - f(x) ) which i thought would givea better approximation of the area as opposed to directly taking area as f(x)Δx So the question,(please see the pic first) why dont we take the area under a curve as ∫ (f(x) + dy/2)dx? because when Δx is big, the expression of area under a curve as ∫ (f(x) + Δy/2)Δx would give a more precise result. will it give the same as Δx approaches 0? Please …

In this desmos graph: https://www.desmos.com/calculator/f607tsq5ux I need to find max for the elliptical paths of rab and rcd so that I can determine the foci... I have researched on the web finding max of a function, but just cannot seem to get my head around it. Would greatly appreciate some help.

I am making notes on real analysis by using propositions and truth tables. I decided to make it in this way in order to present a straightforward and convincing proofs even for skeptical students. The content is taken from link removed Lecture 1 talks about properties of real numbers. The lecture starts with a very short introduction to real numbers followed by the irrationality of √2. I decided to replace this very short introduction with two lengthy sections. Logic and Elementary Set Theory Real Numbers In the first section, I discuss statements, elementary set theory and some elementary aspects of logic 101. I would kindly …

I have the following definition: $$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$

There is a paper at vixra (dot) org Number theory, citation 2103.0181 Instantly Factorize Any Product Of Two Small Or Large Twin Prime Numbers. Simple Method - 72 is the constant integer used in the process to find repeated addition in the series. First Step – Repeated Addition Series. Following the steps ask your colleague to add 72 and 36 as show below. 72 * 1 = 72 + 36 = 108 72 * 2 = 144 + 36 = 180 72 * 3 = 216 + 36 = 252 72 * 4 = 288 + 36 = 324 ......... 'Last Sum Of Series' Counting can be done as many times like 72 *5 , 72 * 6, 72 * 7 ........ and one time adding 36 for each series. Series can go up to infinity. …

Hi, I've been working on some Lebesgue measure and Lebesgue integral exercises for a few days and I have some doubts. I need to say if the statements are true (I need to prove it) or false (I need to give a counterexample). Let $f:E\subset\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that $|f(x)-f(y)|\le|x-y|$ for $x,y\in\mathbb{R}^n$, then $f$ transforms null measure sets into null measure sets. If two integrable functions agree in a dense set, the value of the integrals is the same. If two functions agree in a dense set, one of then is measurable if and only if the other one is also measurable. For the first one, \textbf{I have no idea. I only now it's true …

I didn't like approximate definitions of trigonometric functions (it was about 34 years ago). Then I made speculation that side of angle (if angle is less or equal to Pi/2 rad) proportionally divides arc Pi/2 and its chord (21/2). Then my math teacher corrected me with her speculation that there is especial arc in which if to connect any two points of proportional division of this arc and its chord by straight line and to connect any two points of any another proportional division of this arc and its chord by another straight line , then the straight lines cross in one point of definition of trigonometric functions and angles(arcs). Is there any prize for exact definitio…

Dear all, I have a system of N non-linear equations in N variables and 1 parameter. The system has solutions until the parameter is less than a certain value. I am interested in finding the maximum value of the parameter for which the system has solutions. Could you help me to find a way to do that? I have tried to use the parameter as an extra variable of the system and to add one more equation, but I struggle to find the new equation... I try to give you an example in case I have not been clear enough: −x^2 + 10x − 1/2y^2 + 6y − K = 0 −2x^2 + 20x + 3/2y^2 − 18y = 0 How do I find the greatest K for which this system has a valid solution? Note that this…

What are numbers between 0 and 1??

Hi everyone. Tina here. This is my first ever post on such a forum and would apologise in advance for my lack of formal clarity and perhaps very amateurish presentation etc. But I have a burgeoning interest in the beautiful mystery and potential of mathematics and reason in general. And would love that anyone might direct me in my latest field of curiosity, which is....well...."things to do with primes." I may not be able to formalise my question very well....as intimated....and do understand in general that people often may not even be sure what they are asking.. But here goes.... Consider, if you would....…

Can anyone suggest a good lecture series on Complex Analysis on YouTube? I have already searched on YouTube myself, and there are a few. But I wanted to know if any of you would recommend some particular lecture series which you consider to be good.

The Taylor series has been analyzed in the enclosed paper in several ways to reveal discrepancies. The analysis is of course of a mathematical nature. Requesting the attention of the audience to the mathematics of the paper and the issues ensuing from it.... Taylor1000.pdf

How might one manage to find the surface area and volume of the following constructs? Rotini: A geometric figure I've found to be similar is the helicoid. However, the helicoid (having been formed from a 2D plane) has an edge width equal to epsilon, whereas the edges of a fully 3D rotini-like construct would be roughly catenarian in shape. Fusillibucati/Cavatappi: The only difference between the two is length. Cavalieri's principle could possibly be at play here, but the topography could potentially suggest otherwise. Campanelle: No hints for this one, good luck. Casarecce/Gemelli: The only difference between them is the topographic helical pitch. C…

I found this question in an Oxford admissions quiz (2016 question 2) and was wondering how you would go about this. I am not applying for Oxford just thought that I might need to know how to do a question like this. Thanks!

From my understanding, it is not only possible to have a rotating vector field with a point in its centre where the velocity vector is precisely zero, but that the reasons to expect one follow from the fact that if the surroundings are rotating about one point, they are not moving at that point. But what if you had a moving vector field? For instance, let's say you had the centre of some galaxy, about which all the stars around it were orbiting, moving through space. Obviously in the reference frame that is that moving galaxy, the velocity of that centre is zero. But in some external reference frame not moving parallel to that galaxy, that centre is not zero. …

There’s a reference in Bertrand Russell's book: Analysis of matter, pg 3, third paragraph - continuing onto Pg 4. He talks about R as relating a term to its successive term in a sequence - using the sum of first "n" odd numbers equaling n^2 as an example. I'm not sure how he defines "R" to start with in this example. Then introduces Rxn, which relates a sequence of numbers together?? I'm not sure if I'm reading his notation correctly. Any input on this passage would be helpful. There is link to the book below. http://strangebeautiful.com/other-texts/russell-anal-matter.pdf

Is it possible to find an analytical solution to the following differential equation Thank you in advance to those who will respond