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. D

    I Geometric Point of View of sets

    A set is nothing more than a collection. To determine whether or not an object belongs to the set , we test it against one or more conditions. If it satisfies these conditions then it belongs to the set, otherwise it doesn't. The geometric point of view of sets- a set can be viewed as being...
  2. Z

    Spivak, Ch 5 Limits, Problems 10c: Proving limit relationship

    c) Why is the assertion ##\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)## obvious? First of all I don't think it is obvious but here is an explanation of why the limits are the same. ##\lim\limits_{x\to0} f(x^3)=l_2## means we are looking at points with ##x## close to zero and...
  3. Z

    Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

    I believe the x-axis is vertical here. The graph is composed of i) an infinite number of intervals that start on the ##y=x## line and finish at some ##x## with a decimal expansion ending in ##7\bar{9}##. E.g., from ##0.67## to ##0.67\bar{9}## which is considered ##0.68##. Other examples of...
  4. IAmGroot48

    Calculus 3 Multi-Integration (volume)

    Pending
  5. Z

    Spivak, Ch. 5 Limits, Problem 3 viii: Prove a limit of a function

    Consider item ##vii##, which specifies the function ##f(x)=\sqrt{|x|}## with ##a=0## Case 1: ##\forall \epsilon: 0<\epsilon<1## $$\implies \epsilon^2<\epsilon<1$$ $$|x|<\epsilon^2\implies \sqrt{|x|}<\epsilon$$ Case 2: ##\forall \epsilon: 1\leq \epsilon < \infty## $$\epsilon\leq\epsilon^2...
  6. P

    A Need some clarifications on tensor calculus please

    I've started reading up on tensors. Since this lies well outside my usual area, I need some clarifications on some tensor calculus issues. Let ##A## be a tensor of order ##j > 1##. Suppose that the tensor is cubical, i.e., every mode is of the same size. So for example, if ##A## is of order 3...
  7. T

    A Euler Lagrange and the Calculus of Variations

    Good Morning all Yesterday, as I was trying to formulate my confusion properly, I had a series of posts as I circled around the issue. I can now state it clearly: something is wrong :-) and I am so confused :-( Here is the issue: I formulate the Lagrangian for a simple mechanical system...
  8. S

    Schools Calculus vs Calculus for Engineers (The community college only has the former)

    I'm doing 2 years at a community college before transferring to a university for the last 2 years to become a Civil engineer. I was looking at the courses at the university, granted these are for all 8 semesters not simply post transfer courses. One thing I saw was that the 3 calculus classes...
  9. Darsh_22

    How do I sketch a flow profile and solve for curl in vector calculus?

    Hello, Can someone explain how to sketch the flow profile in detail. Also, I solved for curl, but I'm getting a zero while the answer is the differentiation of the function f(y). Pls do help me out!
  10. S

    MHB Solve Calculus Problem: Answers to d, e, f in Math Paper

    Please Solve this, I am unable. This problem requires you to read the following scientific article: http://math.colgate.edu/~integers/u76/u76.pdf These answers should be related to this paper. Questions are attached in the image; please answer mainly d, e and f Those are the main problems I...
  11. M

    How to prove that the shortest distance between two points is a line?

    I tried using hamilton method but i don't think that's correct
  12. A

    Calculus Advanced Calculus with Applications in Statistics

    Is someone has already heard about this book wrote by Andre I. Khuri (Professor emeritus in science at university of Florida) ? By the table of contents the book seems to cover a lot of things in calculus/multivariable calculus and in a rigourous way according to the preface (they argue that...
  13. hackedagainanda

    Intro Math Introductory Probability without Calculus

    Are there any good introductory texts aimed at students with only a working knowledge of High School Algebra? I currently have: Probability: An Introduction by Samuel Goldberg I do plan on eventually learning Calculus, but I would like to start learning probability sooner.
  14. jorgeluisharo

    Vector calculus — Computing this Divergence

    I really don't know how to proceed if I'm not using an specific coordinate system, Is there a way of doing this using only indices, in general form?
  15. Mr.Husky

    Calculus Just got Courant's calculus text -- Need some tips to work through it

    Hi everyone! I just bought the courant John calculus volume-1 from uni bookstore. I need some tips like errors, topics which are not explained well and topics which are really important and need to focus more on etc.. My background:- have Spivak as a supplement. Knows how to prove and read...
  16. M

    MHB Relation Algebra - Relational Calculus

    Hey! :giggle: Give for the following expressions of relation algebra the equivalent expression in relational calculus. 1. $\sigma_{B=A}(R(A,B,C))$ 2. $\pi_{B,C}(R(A,B,C))$ 3. $R(A,B,C)\cup S(A,B,C)$ 4. $R(A,B,C)\cap S(A,B,C)$ 5. $R(A,B,C)\setminus S(A,B,C)$ 6. $R(A,B,C)\times S(D,C,E)$ 7...
  17. Haorong Wu

    Classical Looking for simple materials for calculus of variations

    Hi, there. I have not systematically learned the calculus of variations. I would like to learn it now. Are there simple materials for the purpose of learning how to do the calculation in physics? No need for deeper consideration in mathematics. Is Mathematical methods for physicists by Arfken...
  18. M

    I Calculus of Variations on Kullback-Liebler Divergence

    Hi, This isn't a homework question, but a side task given in a machine learning class I am taking. Question: Using variational calculus, prove that one can minimize the KL-divergence by choosing ##q## to be equal to ##p##, given a fixed ##p##. Attempt: Unfortunately, I have never seen...
  19. A

    Calculating Work, Energy, and Power: Integrating Force and Velocity

    I tried to solve it by integrating force from 0 to L dF=dm.g where dm=λdx And then I multiplied it with velocity to get power because velocity is constant ∫(vdF)=v ∫(dF) But the book used integration to find work done and divided it by time for power My answer was λlgv(Option B) Giving...
  20. S

    Intermediate Value Theorem Problem on a String

    Preceding Problem. Let ##y=f(x)## be a continuous function defined on a closed interval ##[0, b]## with the property that ##0 < f(x) < b## for all ##x## in ##[0, b]##. Show that there exist a point ##c## in ##[0, b]## with the property that ##f(c) = c##. This problem can be solved by letting...
  21. Poetria

    Chain rule (multivariable calculus)

    ##f_x=3*x^2+y## ##f_y=2*y+x## ##(3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1}## Well, I am not sure how to evaluate it. I got a wrong result by multiplying by 0.1, i.e. ##((3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1})*0.1## I guess it is trivial but I am lost. :(
  22. Parkour

    Self teaching Calculus for physics?

    Summary:: I want to teach myself physics and was wondering if Thomas' Calculus is a good book to learn the Calculus I need to learn introductory physics. Hi, I am following a guide by Susan Rigetti (https://www.susanrigetti.com/physics) which aims to help and give sort of structured learning...
  23. mcastillo356

    Calculus Want to read a paragraph at "Calculus", by R. A. Adams, 8th or 7th ed.

    Hi,PF I've got "Cálculo", Spanish 6th edition, by Robert A. Adams. At Chapter 4.2, would like to read an updated (guess it might be better) 1st. paragraph of the section I translate "Test of the first derivate". Regards
  24. R

    Cauchy Riemann complex function real and imaginary parts

    Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...
  25. Strange_Baryon

    Calculus Learning Calculus with Stewart: Exercise Sets Explained

    Hello PF, After thinking, googling, and reading many threads here, I decided that I'm going to start learning calculus using Stewart. The problem is that the book puts too many exercises after each section, in addition to those at the end of each chapter. What's the difference between these...
  26. D

    B Are there mathematicians that dislike integral calculus?

    Solving integrals by hand is difficult and prone to errors, and the techniques such as integration by parts, partial fraction decomposition, and trig substitutions only work for a small subset of integrals and I do not see the point of avoiding technology like Wolfram Mathematica for...
  27. Istiak

    Understanding the Chain Rule in Multivariable Calculus

    But, If I use chain rule than, I get that. ##\vec v_i = \frac{dr_i}{dt}=\sum_k \frac{\partial r_i}{\partial q_k} \cdot \frac{\partial q_k}{\partial t}## But, they found that?
  28. Istiak

    Book Suggestion for Tensor Calculus

    Homework Statement:: Book suggestion Relevant Equations:: Calculus Book suggestion for tensor calculus.
  29. A

    I Purpose of Tensors, Indices in Tensor Calculus Explained

    I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it. Tensors. As...
  30. Poetria

    Good approximation - multivariable calculus

    I tried to use a Taylor series expanded at 3 and set to 3.01: https://www.wolframalpha.com/input/?i=27+++9+(-3+++x)^2+++(-3+++x)^3+++3+y^2+++y^3+++(-3+++x)+(27+++y^2)=3.01 I got the vector ## (\Delta x, \Delta y)= (0.37887, -0.54038)## It does give a desired result but it is marked as wrong...
  31. A

    MHB Question about the differential in Calculus

    Question about the differential in Calculus. Assume a function y = f(x) , differentiable everywhere. Now we have for some Δx Δy = f(x + Δx) - f(x) The differential of x, is defined as “dx”, can be any real number, and dx = Δx The differential of y, is defined by “dy” and dy = f’(x)...
  32. Poetria

    Partial differential (multivariable calculus)

    Intersecting the graph of the surface z=f(x,y) with the yz -plane. This is the option I have chosen, but it's wrong. I don't understand why. x is fixed so I thought the coordinates: y and z are left. I thought this source may be helpful...
  33. K

    I Basic question about variational calculus

    We've a functional ##J(\alpha)=\int_{x_{1}}^{x_{2}} f\left\{y(\alpha, x), y^{\prime}(\alpha, x) ; x\right\} d x## It's derivative with respect to the parameter ##\alpha## is given in textbook Thornton Marion as ##\frac{\partial J}{\partial \alpha}## Shouldn't it have been ##\frac{d J}{d...
  34. mcastillo356

    Calculus Calculus: TEOREMA 2? Investigating a Typo in Adams' 6th Edition

    Hi, PF I've got the spanish sixth edition of "Calculus", by Robert A. Adams. When it comes to introduce the exponential function properties, calls it "TEOREMA 2", but could it be a typo?. Just a page before, there is another TEOREMA 2: Natural logarithm properties. Greetings
  35. Asker321

    Courses Should I take Calculus 2 my freshman year of college?

    I self-studied Calculus 1 (AP Calc AB) in my sophomore year of high school and I got a 4 on the AP exam. I took the class Calculus 2 (AP Calc BC) in my junior year of high school. I did not take the exam at the end of the year. I didn't take any math in my senior year of high school. Because...
  36. B

    Multivariable Calculus proof for Optics

    Part A) For part A I forgo breaking down the identity into it's component x, y, and z parts, and just take the r derivative treating r' as a constant vector. This seems to give the right answer, but to be entirely honest I'm not sure how I'd go about doing this component by component. I figure...
  37. mcastillo356

    Calculus Sentence in English in textbook "Calculus", by Robert A. Adams

    Hi, PF I've got a translation into spanish of the sixth edition of "Calculus", by Robert A. Adams. At the second chapter, "Differentiation", eleventh section, I would like to know: the title of the section (Could it be "Velocity and Acceleration"?); and a sentence of the Example 2 ("A point P...
  38. LCSphysicist

    Calculus: Integral along a curve.

    Let $F = (P(x,y),Q(x,y))$ a field of vector class 1 in the ring $A={(x,y): 4<x²+y²<9}$ and $x,y$ reals. I am having trouble to understand why this alternative is wrong: If $ \partial P /\partial y = \partial Q /\partial x$ for every x,y inside A, so $\int_{C} Pdx + Qdy = 0$ for every...
  39. vibha_ganji

    Calculus Calculus Textbook: Courant vs Apostol

    Hi! I was looking for a Calculus textbook to buy and was debating between Courant and Apostol. I was planning on reading Spivak parallel to one of these. My question is does Apostol cover more material than Courant does or are they practically the same. Sorry if this has already been asked, I...
  40. Nick tringali

    I want to learn calculus on a budget

    Hello everyone, Last year I took pre calc and that was the highest math I was required to take for my biology degree. I’ve recently became interested in learning calculus. Does anyone have a good online course recommendation I could buy in order to learn all of calc 1? I was looking on udemy...
  41. Flamitique

    I Solving an Integral using Feyman's trick

    Hey guys ! I just need a little help on a integral I was trying to solve using feyman's technique. This is the integral from 0 to 1 of (sin(ln(x))/ln(x) dx, which has been solved in one of the videos of bprp, but I'm trying to solve it using a different technique, and I end up with a different...
  42. R

    I How can we define a limit approaching negative infinity?

    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 $$\lim_{x\to p^-}=-\infty? $$
  43. Mathman2013

    Find maximum of the function N(t) without Calculus

    How would you go about finding maximum value for this function without Calculus? You can draw in it a CAS tool like Geogebra, NSpire or Maple. And use the maximise ability. But is possible to do it by hand? Pre-Calculus?
  44. AdvaitDhingra

    Calculus What are some affordable textbooks for learning math concepts for physics?

    Hey guys, so I was on this thread on tips for self studding physics as a high schooler with the aim to become a theoretical (quantum) physicist in the future. I myself am a 15 year old who wants to become a theoretical physicist in the future. A lot of people in the thread were saying that...
  45. V

    Determine the limit of 2^x/x^2 as x approaches infinity....

    How would I determine the following limit without substitution of large values of x to see what value is approached by the complex function? ## \lim_{x \rightarrow +\infty} {\dfrac {2^{x}} {x^{2} } } ## where ## x\in \mathbb{R}##
  46. N

    Studying Ready to start Calculus (I think), but unsure where to start

    Hi all, Over the last few weeks I've been catching back up with High School Algebra, Trigonometry, along with some Geometry. I'm now looking for the next challenge and unsure where to start. Due to studying Astrophysics/Physics topics, I'm assuming studying applied mathematics topics is the...
  47. X

    Maximizing Profit with Calculus: Understanding the Profit Formula

    I know the formula profit = (price-cost)quantity. but when ı applied the numbers ı can not substitute them profit = ((30-2q-2-(40/q)) but don't know what ı need to do ? would appreciate any help given please.
  48. C

    Vector calculus - show that the integral takes the form of (0, a, 0)

    Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right. I also tried to compute the right-hand side of...
  49. T

    I Have you ever used "symbolic logic" to help you learn calculus?

    I failed to become a mechanical engineer because I could not learn how to do sequences and series in Calculus II. I could get Cs and Bs on all the concepts of Calculus II until I got to Sequences and Series. Then I would get F minuses on any tests involving series problems such as Infinite...
  50. A

    MHB Calc Solved! Get Quick Help With Your Calculus Q

    please I need solve of this question very quickly
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