Calculus Definition and 1000 Threads

My background is calculus, but I feel that, in calculus (my textbook is Calculus, by Robert Adams), I study limits, continuity, differentiation, integrals, etc, but, I realize that if we want cool things to happen in calculus, we need some basic topological concepts: I've taken a look at metric...

I have made an attempt, which I give below. There are two issues. The first is that I am missing a factor of 2 in the MOI of the sphere. The second issue is that the integral I am getting for the spherical shell is very complex. I have tried symbolab and it chucks out some horrible things. This...

I have found the following proof of remainder term for midpoint rule of integration: and I'm trying to understand the part of it where author is applying MVTI to composition of functions ##f''(\xi_i(x))## and move it out of integral sign as ##f''(\xi_i)##. If we solve Taylor's series for this...

The closest I can get is ##k=\frac{\sqrt{2}a}{\sqrt{3}}## as follows: ##I=\int_0^\pi y^2dA## ##\therefore I = \int_0^\pi a^2 \sin^2 x dA## ##dA = y.dx = a \sin x dx## ##\therefore I = \int_0^\pi a^3 \sin^3 x dx = a^3 \int_0^\pi \sin^2 x.\sin x dx## Let ##u=\sin^2x##, ##\therefore...

Maybe it's obvious why and I'm just dumb, but I hope you can help me understand this -----------------------------+∞ I sorta get why the Ʃ(1/n) diverges to +∞ -------------------------------n=1 ------------------+∞ But why the Ʃ(1/n)^p (p>1), converges? ------------------n=1

I had posed this problem as the OP here . I am trying to do the same problem here, without numbers. Drawing : I put the image of the problem to the right. O is the center of the circle distant ##a## from the circumference like points P and R. At a height ##h## from the base Q, a segment is cut...

Is the fact, that the sequence of arguments and functions matter, for lambda calculus, the reason it's universal. I thought about it because expressions like true and false depend on the idea that the order in which the arguments get substituted matters. Keep in mind that I have zero background...

Hello, I am studying arc lengths and areas for parametric curves from the Adams & Essex Calculus book and I am a bit baffled by example 2 in the image attached. I understand the solution in the book where they integrate from t=0 to t=pi/2 (first quadrant) and multiply by two to get the full...

Anybodys whos read the book Second year of Calculus, from celestial mechanics to Special relativity, what are your thoughts on it? Would you recommend it? What faults does it have? Any information about the book would be helpful.

This is for a math report that I'm supposed to write, which means I'm not supposed to use conservation of energy. This makes life much harder... so please bear with me. I am interested to see how you'd solve this purely kinematically though (if it can be solved that way). Please tell me if this...

1. Does it make sense to learn math in a foreign language(English) from book because I will take some engineering courses on English or better stay at mother language? 2. What book do you recommend for self learning calculus, maybe better engineering oriented?

What level of math is needed for IPhO? I know that basic knowledge of calculus (basic differentiation and integration) and vectors (addition, dot product, cross product, etc) is definitely required, even just to derive basic results. However I've heard that you need other stuff as well such as...

TL;DR Summary: Does the book linked in the post contain answers to odd numbered questions at the back of the book? I'm considering buying the following book by James Stewart: Calculus: Early Transcendentals, Metric Edition. I asked ChatGPT and it said that most of Stewart's books have answers...

Part (a) is fine. I have attached my work for this here anyway because I use the first two lines of (a) in my proof of (b). In part (b), I get the extra factor of ##\frac{1}{r} \frac{\partial z}{\partial r}## and I don't know why. Can anyone help? I am unable to figure out the flaw in my steps :(

I have an expression of a vector field ##\vec J = J_r \hat r + J_\theta \hat \theta## in polar coordinates, in a region that's a quarter of an annulus, where theta ranges between 0 and pi/2 and r goes from ri to ro. I need to compute ##-\int _\Omega T(S_{xx}-S_{yy}) \frac{\partial J_x}{\partial...

Source: Paul's Notes (https://tutorial.math.lamar.edu/Solutions/CalcIII/ChainRule/Prob6.aspx) In his solutions: where he computes the stuff in red. My question is, why is $$\frac{\partial y}{\partial p}$$not $$1+3\frac{\partial t}{\partial p}=1+3\left(\frac{-1}{2}\right)=\frac{-1}{2}$$ Thanks

Cobbling together a definition of the infinitesimal from bits and pieces of info gathered from books and the internet: The infinitesimal ##d## is the positive real number greater than ##0## but less than any other positive real number. My problem is how to express the above in logical...

This really cracked me up! Unless there is something i am not seeing! part (a) is straightforward, using quotient rule: ##\dfrac{dy}{dx} = \dfrac{x⋅\dfrac{1}{x}- \ln x}{x^2}=\dfrac{1-\ln x}{x^2}## From here i was able to see that, ##\int \dfrac{\ln x}{x^2} dx= \int \dfrac{1}{x^2}- \dfrac{\ln...

I could understand the problem perfectly; however, I do not know how to construct the problem. The problem states that two forces are acting simultaneously on the object, but how could I represent that fact mathematically? I really want to solve it, but I am facing this roadblock, so please...

Hello, I've been doing Calculus for a few days now and I'm beginning to wonder whether or not I should be memorizing definitions. By memorizing definitions I mean memorizing them by heart (word for word). This type of memorization hasn't worked very well for me in the past. At this very moment...

This is part of the working from f(3x^2-1)^2xdx; I don't understand from when 6x becomes 1/6

I've got this integral I'm trying to find: $$ \int \frac{d \theta}{ \sqrt{1 - \cos \theta}} $$ To me it smells like trig sub, so I investigate the right triangle: Such that: $$ \cos u = \sqrt{1-cos \theta} $$ we also have from the same triangle: $$ \sin u = \sqrt{\cos \theta} $$ Square...

Hello, I would like to start off my saying how much Calculus I have done so far. I am familiar with the idea of limits, derivatives and integrals (though I do have some holes in my knowledge). So far, I have only done Calculus I. I was introduced to some ideas of Calculus II, but those were...

I was watching an explanation of why the spacetime interval is invariant in all inertial frames (even when it's not lightlike) and the author made the assertion that if we have the relationship ds'=f(ds), we can expand the function as A+B*ds+C*ds^2+... (where C is not the speed of light). That's...

TL;DR Summary: Continuity of a function, Calculus newbie, delta, epsilon, Greetings! I have just started studying Calculus for my engineering course, and I am already facing some problems to understand the fundamental ideas regarding the continuity of a function. I'd be very much grateful if...

Hi, The task is as follows In order for it to be a norm, the three properties must be fulfilled. 1. Positive definiteness 2. Absolute homogeneity 3. Triangle inequality ##\textbf{Positive definiteness}## Since all three elements are given in absolute value, the result of ##\max{}## will...

I have been a long proponent of beginning calculus being introduced early in math education at about grade 7 or so, and in the US (with or just after Algebra and Geometry), Calculus BC is taken in grade 10 or even earlier. It's not well known, but believe it or not, a few hundred students in...

Hi, Im completly lost regarding the following exercise: Unfortunately, I don't understand how to prove the statement using the chain rule. The chain rule is always used if there is a composition, i.e. ##f\circ g=f(g(x))## then I first have to calculate ##g(x)## and insert this result into...

Is this correct? Ignore my bad drawings

Hi, I am having problems with the following task: My lecturer gave me the tip that I should apply the multidimensional chain rule to obtain the following transformation ##\sum\limits_{i=1}^{n} \frac{\partial f}{\partial x_i}(x) \cdot x_i= \frac{d f(tx)}{dt} \big|_{t=1}## Unfortunately, I...

I intuitively understand that it has 2 degrees of freedom (rolling without slipping - RWS), but I struggle to formalize this according to the rules of the art: I obtain: 10 - 3 (ground) - (rolling without slipping constraint) = 2 how to precisely calculate the RWS constraint? what formula and...

Which exactly year Newton first time write about calculus, 1664, 1665 or 1666? I find three years circles in many sources, 1664, 1665 and 1666 , do yo maybe know some thrusted source where we can find correct information? Here is 1665...

I'm looking for good books on Tensors. I have "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" from Pavel Grinfeld. But i look for others. [Mentor Note: Thread moved from the Relativity forum]

Hi everyone, I'm a new here, this is my presentation https://www.physicsforums.com/threads/new-self-study-member.1061374/ I want to study physics for my own intrest and understanding of the universe. A few years ago, after high-school, I studied Halliday and Resnick quite thoroughly, both...

STARTING IPHO. Hi guys, I am going to grade 10 this year and wanted to start physics olympiad preparation from basics. I dont know calc 1 too. Can you suggest me a good calculus book to get my hands on? Moreover, is it late to start my preparation? Thanks in advance.

My hypothesis: The number of red giants is equal to the number of stars times the given fraction f. The number of stars in a solid angle omega, is given by the density distribution of stars in the Galaxy times the volume of the observed solid angle: #RG = f*n(r)*V where V = (d^3*omega)/3. I...

I'm trying to pace myself to get up to speed with my calc and physics before going back to college. It's been so long that I don't know how much we got up to in calc I. Can someone tell me what chapter in Thomas Calculus does calc I go up to, or do they cover the whole thing?

This is my attempt at the solution. x from the equation dF = GmdM/x^2 represents the distance between the circular plate’s center of mass and mass m.

I would like to take the derivative of the following function with respect to Gt: $$\mathrm{G}_{t+1}=\mathrm{g}_{0}\mathrm{e}^{-qHt}$$ I think that the answer is either -1 or ##\mathrm{e}^{-qHt}-1## If you could show the calculations that would be a great help. Thanks very much.

My lines are as follows; ##y=\sqrt x## and ##y=x^2## intersect at ##(0,0## and ##(1,1)##. Along ##y=\sqrt x##, from ##(0,0)## to ##(1,1)## the line integral equals, $$\int_0^1 [3x^2-8x] dx + \dfrac{4\sqrt x-6x\sqrt x}{2\sqrt x} dx $$ $$=\int_0^1[3x^2-8x+2-3x]dx=\int_0^1[3x^2-11x+2]dx =...

I'm trying to brush up on my algebra, trig, and calculus, and one thing I know I was always weak on before was proofs. I was never sure what equations would suffice as "proof," and which equations did not. Maybe this is an inane question, and maybe there is a really simple answer to this. I...

In free time I start to solve differentials and integrals, I am doing fine, I just follow rules and solve the tasks. I start solve some applied calculus tasks, but I dont really understand why for exmple second derivative represent acceleration, why first is speed, why I need to derivate...

Like the title and the summary suggest, I can derive the volume ##V=2\,\pi^{2}\,r^{2}\,R## for a ring torus - a doughnut-style toroid (one such that the major radius ##R## > the minor radius ##r##, and it therefore has a hole at the center) that is of circular cross-section. But I want to be...

Hi all currently got a lagrangian function which i've found to be : \begin{equation}\mathcal{L}=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+4x^2\dot{x}^2+4y^2\dot{y}^2+8xy\dot{x}\dot{y})- mg(x^2+y^2) \end{equation} Let us first calculate $$(\frac{\partial L}{\partial \dot{x}})$$ which leads us to...

To use the formula above, I have to prove that $$\lim_{n\rightarrow \infty}f(x)=\lim_{n\rightarrow \infty}\left(\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+........\frac{1}{n}}{n^2}\right)=1$$ To prove so, I tried using L'Hopital's Rule: $$\lim_{n\rightarrow \infty}f(x)=\lim_{n\rightarrow...

Hi; please see below I am trying to understand how to get to the 2 final functions. They should be the same but 6 for the first one and 2 for the second? (I hope my writing is more clear than previously) There is an additional question below. thanks martyn I can't find a standard derivative...

(Disclaimer: I don't know whether this type of post encouraging discussion on a function is allowed, if not please close this) Hello PF, If you're a fan of integrating, you'll hit a ton of special functions on the way. Things like the Harmonic Numbers, Digamma function, Exponential Integral...

##(f(c) - f(a))((f)(b) - f(c)) <0## tells us that there are two cases: ##f(c) >f(a), f(b) ## ##f(c) <f(a), f(b) ##. I guess we need to define a new function here that let us use the Rolle's theorem.. But it is not clear enough how to do so.

Hello PF! This is my very first post here. Just yesterday my paper was accepted by ArXiV, called "A Simple Continuation for Partial Sums". If you have time (it's 14 pages) you can take a look at it here. I was just interested in ways I could improve my paper or if it was completely useless in...