Calculus Definition and 1000 Threads

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus (plural calculi) is a Latin word, meaning originally "small pebble" (this meaning is kept in medicine – see Calculus (medicine)). Because such pebbles were used for counting (or measuring) a distance travelled by transportation devices in use in ancient Rome, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.

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  1. Agent Smith

    B The infinitesimal in Calculus

    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...
  2. chwala

    Solve the given problem involving integration

    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...
  3. BurpHa

    Two Forces Acting Simultaneously

    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...
  4. Heisenberg7

    Studying Should One Memorize Definitions?

    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...
  5. M

    I Question about this Integration by Substitution

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

    I Integration Using Trigonometric Substitution

    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...
  7. Heisenberg7

    Calculus What Is the Best Calculus Book for Self-Learners?

    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...
  8. Sciencemaster

    I Series Expansion of the Infinitesimal Spacetime Interval

    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...
  9. T

    f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε

    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...
  10. L

    Is the Max Function a Norm in R^3?

    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...
  11. bhobba

    Kumon Math and Similar Programs

    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...
  12. L

    How Do You Prove the Multivariable Chain Rule?

    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...
  13. L

    Question involving the Divergence Theorem and Surface Integrals

    Is this correct? Ignore my bad drawings
  14. L

    Why Does the Multidimensional Chain Rule Lead to Euler's Theorem Proof?

    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...
  15. D

    How to calculate the degrees of freedom of a simple multi-body system?

    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...
  16. U

    Which year did Newton first work with calculus: 1664,1665 or 1666?

    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...
  17. Alaindevos

    What are Good Books on Tensors for Understanding Einstein's Field Equation?

    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]
  18. Giovanni04

    Is this a good self-study program?

    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...
  19. itsdavid

    Classical Starting IPhO preparation in grade 10 -- Calculus book suggestions

    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.
  20. Rageuke

    Need to derive a formula for the number of red giants

    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...
  21. A

    Calculus What Chapters in Thomas' Calculus Would Calc I be?

    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?
  22. Sam Jelly

    Proving the gravitational force of a solid sphere using integration

    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.
  23. mathgenie

    I Taking the derivative of a function

    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.
  24. chwala

    Verify Green's Theorem in the given problem

    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 =...
  25. A

    I Practice With Proofs? (Algebra, Trig, and Calc)

    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...
  26. U

    B Some questions while I self-learn Applied Calculus

    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...
  27. S

    B Having trouble deriving the volume of an elliptical ring toroid

    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...
  28. Hennessy

    I Calculus Question within Lagrangian mechanics

    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...
  29. A

    Evaluate the limit of this harmonic series as n tends to infinity

    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...
  30. M

    Partial Differentiation of this Equation in x and y

    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...
  31. mathhabibi

    I Alternating Harmonic Numbers are cool, spread the word!

    (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...
  32. A

    A question regarding continuous function on a closed interval

    ##(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.
  33. mathhabibi

    I Requesting constructive criticism for my paper

    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...
  34. A

    Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

    I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ Which leads...
  35. MAXIM LI

    Limit of probabilities of a large sample

    My first thought as well but I think the problem is deeper than that. I think that as the n tends towards infinity the probability of the the sample mean converging to the population mean is 1. Looking at proving this. By the Central Limit Theorem the sample mean distribution can be approximated...
  36. A

    How Does Basic Calculus Open Doors in Physics?

    I am quite interested in math and physics and keep on trying to discover new stuff. I recently learnt badic calculus and i find math cool
  37. Brix12

    I How do I format equations correctly? (Curl, etc.)

    A question in advance: How do I format equations correctly? Let's say $$\mathbf{k}\cdot\nabla\times(a\cdot\mathbf{w}\frac{\partial\,\mathbf{v}}{\partial\,z})$$ - a is a scalar Can I rewrite the expression such that...
  38. Safinaz

    How Does the Dirac Delta Function Identity Apply in Equation (27) Derivation?

    I need help to understand how equation (27) in this paper has been derived. The definition of P(k) (I discarded in the question ##\eta## or the integration with respect for it) is given by (26) and the definition of h(k) and G(k) are given by Eq. (25) and Eq. (24) respectively. In my...
  39. L

    Dirac delta function approximation

    Hi, I'm not sure if I have calculated task b correctly, and unfortunately I don't know what to do with task c? I solved task b as follows ##\displaystyle{\lim_{\epsilon \to 0}} \int_{- \infty}^{\infty} g^{\epsilon}(x) \phi(x)dx=\displaystyle{\lim_{\epsilon \to 0}} \int_{\infty}^{\epsilon}...
  40. A

    Electromagnetism: Force on a parabolic wire in uniform magnetic field

    I know the easier method/trick to solve this which doesn't require integration. Since parabola is symmetric about x-axis and direction of current flow is opposite, vertical components of force are cancelled and a net effective length of AB may be considered then ##F=2(4)(L_{AB})=32\hat i## I...
  41. M

    How should I show that ## B ## is given by the solution of this?

    a) Consider the functional ## S[y]=\int_{0}^{v}(y'^2+y^2)dx, y(0)=1, y(v)=v, v>0 ##. By definition, the Euler-Lagrange equation is ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ## for the functional ## S[y]=\int_{a}^{b}F(x, y, y')dx, y(a)=A...
  42. M

    How should I calculate the stationary value of ## S[y] ##?

    Consider the functional ## S[y]=\int_{1}^{2}x^2y^2dx ## stationary subject to the two constraints ## \int_{1}^{2}xydx=1 ## and ## \int_{1}^{2}x^2ydx=2 ##. Then the auxiliary functional is ## \overline{S}[y]=\int_{1}^{2}(x^2y^2+\lambda_{1}xy+\lambda_{2}x^2y)dx, y(1)=y(2)=0 ## where ## \lambda_{1}...
  43. Frabjous

    I Heavyside’s operational calculus vs. transforms

    Are there features of operational calculus (or operator methods) that are advantageous over transforms for DE? I know that the techniques are closely related.
  44. G

    Help me prove integral answer over infinitesimal interval

    In the book, I see the following: ##\int_{x_1}^{x_1 + \epsilon X_1} F(x, \hat y , \hat y') dx = \epsilon X_1 F(x, y, y')\Bigr|_{x_1} + O(\epsilon^2)##. My goal is to show why they are equal. Note that ##\hat y(x) = y(x) + \epsilon \eta(x)## and ##\hat y'(x) = y'(x) + \epsilon \eta'(x)## and...
  45. G

    Why Does Taylor's Theorem Use +O(ε) Instead of -O(ε)?

    I am trying to grasp how the last equation is derived. I understand everything, but the only thing problematic is why in the end, it's ##+O(\epsilon)## and not ##-O(\epsilon)##. It will be easier to directly attach the image, so please, see image attached.
  46. M

    How should I use the Jacobi equation to determine the nature of this?

    Here's my work: Let ## n>1 ## be a positive integer. Consider the functional ## S[y]=\int_{0}^{1}(y')^{n}e^{y}dx, y(0)=1, y(1)=A>1 ##. By definition, the Jacobi equation is ## \frac{d}{dx}(P(x)\frac{du}{dx})-Q(x)u=0, u(a)=0, u'(a)=1 ##, where ## P(x)=\frac{\partial^2 F}{\partial y'^2} ## and...
  47. P

    Problem involving space elevators

    (a) The length ##h = L## for which the tension is minimum is the length that corresponds to the geostationary orbit, where the angular velocity of the cable matches the angular velocity of the Earth. This is because at this point, the centrifugal force balances the gravitational force, and the...
  48. chwala

    Find the local maxima and minima for##f(x,y) = x^3-xy-x+xy^3-y^4##

    Ok i have, ##f_x= 3x^2-y-1+y^3## ##f_y = -x+3xy^2-4y^3## ##f_{xx} = 6x## ##f_{yy} = 6xy - 12y^2## ##f_{xy} = -1+3y^2## looks like one needs software to solve this? I can see the solutions from wolframalpha: local maxima to two decimal places as; ##(x,y) = (-0.67, 0.43)## ...but i am...
  49. chwala

    Find the dimensions that will minimize the surface area of a Rectangle

    My interest is on number 11. In my approach; ##v= xyz## ##1000=xyz## ##z= \dfrac{1000}{xy}## Surface area: ##f(x,y)= 2( xy+yz+xz)## ##f(x,y)= 2\left( xy+\dfrac{1000}{x} + \dfrac{1000}{y}\right)## ##f_{x} = 2y -\dfrac{2000}{x^2} = 0##...
  50. G

    I Integrating 1/x with units (logarithm)

    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...
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