Mathematics Definition and 994 Threads

do you know any books, videos, or notes that can help me to understand these topics: 1-primitive roots for primes 2-the existence of primitive roots I am using right now elementary number theory and its application by Kenneth H. Rosen to understand these topics, do you know another source that...

Hi. What exactly is happening mathematically when you integrate ##\frac{1}{x}## $$\int_a ^b \frac{1}{x} dx=\ln{b}-\ln{a}=\ln{\frac{b}{a}}$$ if there's units? Sure, they cancel if you write the result as ##\ln{\frac{b}{a}}##, but the intermediate step is not well-defined, so why should log rules...

I would appreciate if someone could help me to understand what is happening in section 12.3 from the Howard George's book. First of all, the propose of the section is to show how $SU(3)$ decomposes into $SU(2) \times U(1)$. But i can't understand what is happening. First of all, i can't get the...

I only have my AA so far (working towards physics degree), but I would like to start changing directions career-wise. I'm completely burnt out on my current field (went through several careers over the years, but lately, it's been sales, and I HATE sales, just good at it). I would love to do...

I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book) There exists a set...

TL;DR Summary: Got my AA with a focus in physics nearly a decade ago. Looking to go back and finish what I started, but need to brush up and looking for good resources to do so? Looking for resources that are thorough that can help me brush up on calculus 1-3, physics 1-2, and possibly...

From the Introduction to https://arxiv.org/abs/2106.11285 "Since the dawn of time, human beings have asked some fundamental questions: who are we? why are we here? is there life after death? Unable to answer any of these, in this paper we will consider cohomology classes on a compact projective...

I want to calculate eccentric anomaly of all points of ellipse-circle intersection. Ellipse is not rotated and its center is in origin. Circle can be translated to (Cx, Cy) coordinates. I am using python for calculations. Only solution I found, is this...

In fact, it WAS a homework couple of years ago, and I've solved it, kind of (below). I still would like to find a cleaner solution. Here is what I did. Let's say, the apples are labeled, and their weights are ##x_1, x_2, ...##. He takes out the apple #1 and finds that, e.g., ##x_2+x_5+x_9+... =...

"Former math teacher explains why some students are good at math and others lag behind" The title of a news article shown on todays Yahoo site, https://www.yahoo.com/news/former-math-teacher-explains-why-122744193.html Looking in the section called " Why are some students ‘good’ at math and...

I know that we have to assume certain things for the math to be achievable (at my level). for instance, I assumed that the rocket goes in a straight line instead of orbiting around the earth at an angle. but I can't develop any further assumptions as the task is so generalised and open-ended...

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Firstly, the exercise itself is not difficult: On one hand, $$|(a + ib)(c + id)|^2 = |a + ib|^2|c + id|^2 = (a^2 + b^2) (c^2 + d^2) = MN.$$ On the other hand, ##(a + ib)(c + id) = p+ iq## for some integers p and q, and so $$|(a + ib)(c + id)|^2 = |p + iq|^2 = p^2 + q^2.$$ Thus, ##MN = p^2 +...

How did you find PF?: I was randomly searching the net for info on calculus books for self study, found a math reddit that brought me here. I'm 65. Not working since 2000. In HS, 9th Algebra 1 = A, 10th Geometry/Trig =A, 11th, Algebra 2= D, (long story)....so no Calculus in Sr yr. Through...

I want YouTube channels that systematically raise my mathematics and physics level

I understand the mathematics that 1 divided by infinity is virtually zero and so equals zero. I look on the internet and that is the answer that I get. Is this a simplification for early mathematics learning and, if I continue, will I find a more complex answer? The reason that I ask is that I...

Epsilontic – Limits and Continuity I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I think I’m not the only one who struggled at that point. We mainly learned algorithms at school, i.e. how things are...

Going through this, am still checking but will post all the same; which method did they apply to find the roots of the attachment below. My thinking; Let ##p+qi## be the cube root of ##x^3-6x+2=0## then, ##\sqrt{x(x^2-6)}=i\sqrt{2}## ##(p^2-q^2+2pqi)(p+qi)= x^3-6x+2## We know that...

I am just starting on mathematics and physics so I might need your help. I look forward to learn from all of you.

Hi, PF There are two ways to write domain and range of a function: through set notation, or showing intervals. I've chosen the set notation, and, for ##y=\csc x##, this is the attempt: $$\text{D}:\{x\,|\,x\not\in{n\pi},\,n\not\in{\mathbb{Z}}\}$$ $$\text{R}:\{f(x)\,|\,x=\mathbb{R}\(-1,1)\}$$...

I realize of course that this will probably not apply to all physicists, but at least every physicist in my university's math department is very unrigorous when it comes to mathematics. This is frustrating because some of the physics material seems genuinely interesting, but the lack of an...

What is the difference between these to. -3² and (-3)² ? I know - x - = + I am told -3²= -9 not +9 -3² means -3 x -3 = 9 I am told (-3)² = 9

Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants. I will provide introduction in very condensed form to get quicker to the essense. Conservative part First of all, let us...

##\frac {1} {x^2 -c^2}## with ##c \neq {0}## So the first thing I do is split the ##x^2 -c^2## into the difference of squares so ##x +c## and ##x - c## I then do ##\frac {A} {x + c}## ##+## ##\frac {B} {x-c}##, and then let ##x=c## to zero out the expression. And that is where I am getting...

Hi everyone, I'm fibrebundle. I actually joined this forum because I'm really interested in abstract maths. I'm particularly intereseted in alegebraic topology and geometry at the moment. But I'm also really interested in spectral graph and graph theory. I'm starting grad school in engineering...

I'm in my last 2 years of high school, and I have to pick a speciality to study before becoming an undergraduate and studying in college. In the future, I'm hoping to become an experimental physicist. My high school offers 3 specialities that are relevant to physics to pick from, all of them...

TL;DR Summary: new book with interesting problems There's a new book out by Routledge called Mathematical Conundrums with many interesting problems in algebra, arithmetic, route-drawing, and logic. Good for schools as algebra is no higher than high school. Challenging though.

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I am a curious physics student who wants to learn how to use its knowledge to create things, to understand phenomenons and so on. I am looking for detailed explanations that use physics and maths. (books, websites, videos, etc.)

I remember there was a method of learning/teaching mathematics where all they do in class is to force students to prove the theorems themselves. What was this method again? It was named after someone.... @fresh_42 ?

IAM from Argentina. Aficionated physics and mathematics

Doing some self study here; my understanding of order of an element in a group is as follows: Order of ##3## in ##\mathbb{z_4}## can be arrived by having, ##3+3+3+3=12≡0## likewise, the order of ##12## in ##\mathbb{z_{20}}## can be arrived by ##12+12=24 ≡4≠0## ##12+12+12=36≡16≠0##...

Hello all, I've taken math through differential equations and linear algebra, am in my senior year of physics curricula while conducting McNair research regarding General Relativity. I found a NASA document outlining Einstein's field equations, which suggests only preparative familiarity with...

In short, I'm interested in working on a web-app to make landmark papers in theoretical physics and mathematics more broadly accessible, especially to undergraduate and graduate students who are looking to catch up to modern topics (without sacrificing rigor or exactness of understanding), and...

What is physical applied mathematics?

...Out of interest am trying to go through the attached notes, My interest is on the highlighted, i know that in ##\mathbb{z}/\mathbb{6z}## under multiplication we shall have: ##1*1=1## ##5*5=1## am assuming that how they have the ##(\mathbb{z}/\mathbb{6z})^{*}={1,5}## is that correct...

((x+1)^2 -1)/(x+1)^2 dx 1-1/(x+1)^2 dx Let u=x+1 1-1/u^2 du u+1/u +c (u^2+1)/u +c Not as answer given in the book.

Is there any particular reason as to why certain texts use ##j## and others ##i## when looking at complex numbers? Maths is a relatively easy subject but at times made difficult with all this mix-up... i tend to use a lot of my time in trying to understand author's language and this is also...

My wife was in Las Vegas over the weekend to visit her sons and grandson, and saw this new structure there, the MSG sphere. In this image the sphere appears to be a very large basketball. Other images I've seen are of a huge eye, animated fireworks displays, an image of the earth, along with...

In mathematics, there is the Ramanujan summation: $$1+2+3+4+...=-\frac{1}{12}$$ https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯ This sum is used in physics for predicting the Casimir effect: https://en.wikipedia.org/wiki/Casimir_effect I have also heard that this sum was used in the string...

I'm trying to figure out how the mathematics of conservation of mass in compressible flow works for a simple setup. I posted this problem in hw physics section, and the conversation turned to the physics model and appears to have gone kaput. This was supposed to be a mathematics question (but...

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HI, I am an International Student studying in the US (at a "liberal arts" small college ranked in the Top 20 as per US News Rankings). I will graduate in 2024 and intend to apply for PhD programs in Mathematical Physics starting October 2023 for admission in 2024. GPA: I am a double major in...

I stumbled across this article from decades past written by the best EE instructor I ever had. I thought it might be of some passing interest to others in highlighting the difference between memorizing a mathematical result vs. truly understanding it. The essence of engineering, in effect. We...

Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more? I feel...

I found one ages ago about the hyperbolic functions, but it hadn’t been translated to English from German yet. Anyone know of a good book on hyperbolic functions and other transcendental functions besides the circular functions (trigonometric)?

Hello I am from Egypt my interests fall into the sciences mainly mathematics and chemistry

TL;DR Summary: Suggestions for the publication of an article Hello everyone, I was reviewing and I can't find much content on truncated octahedron formulas, can it be useful to publish an article in a magazine on the subject?. Thank you.

##f(x) = \begin{cases} \frac {x^3-x^2}{x-1} & \text{if } x\neq1 \\ 1 & \text{if } x=1 \end{cases}## Find ##f'(1)##. Using derivative definition: ##f'(1)=\lim_{x \rightarrow 1} {\frac {\frac {x^3-x^2}{x-1}-f(1)} {x-1}}## ##f'(1)=\lim_{x \rightarrow 1} {\frac {\frac {x^3-x^2}{x-1}-1} {x-1}}##...

TL;DR Summary: Here I am asking for some opinions and recommendations for mathematically rigorous books that should be taken as an interested physics student. I know the question is quite subjective but any insightful answer is appreciated. I am willing to join undergraduate physics classes...