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. M

    A Calculus of Variations and Natural BCs

    Hi PF! Given a functional ##J[y]##, if the first variation is $$\delta J[y] = \int_D(ay+y'')y \, dV + \int_{\partial D} (y'+by)y\,dS$$ am I correct to think that when finding stationary points of ##J[y]##, I would solve ##ay+y''=0## on ##D## subject to boundary conditions, which would either...
  2. C

    A Computing de Rham Cohomology of Connected Sums of Objects

    Hi, I'd like to compute the de Rham cohomology of the 3 following objects : -A connected sum of ##g \in \mathbb{N}## reals projectives plans ##P_{2}(\mathbb{R})##. -A connected sum of ##g \in \mathbb{N}## torus without ##n## points ##\mathbb{T}^{2} - \{x_{1}, x_{2}, ..., x_{n}\}##. - A...
  3. Orodruin

    Insights The 10 Commandments of Index Expressions and Tensor Calculus - Comments

    Greg Bernhardt submitted a new PF Insights post The 10 Commandments of Index Expressions and Tensor Calculus Continue reading the Original PF Insights Post.
  4. shahbaznihal

    I Solving Tensor Calculus Question from Schutz Intro to GR

    I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...
  5. P

    MHB Jason's calculus questions

    Since the two functions touch at $\displaystyle \begin{align*} A\left( 0, \frac{15}{2} \right) \end{align*}$ that means that this point lies on the cubic function. Thus $\displaystyle \begin{align*} \frac{15}{2} &= a\left( 0 \right) ^3 + b\left( 0 \right) ^2 + c\left( 0 \right) + d \\...
  6. S

    Studying Physics students and proof based calculus

    Hey, I have been told to study calculus following Spivak's book. I was in an Engineering program and I have moved to a Physics one, and I want to retake calculus to really get good at it. The problem is, Spivak's seems to me like it's very proof based, and I'm having a hard time even with the...
  7. Cantor080

    Calculus Which self-contained calculus book explains the math in order....

    Which calculus book self contains experiences in order, and is stable to the max, for all the problems known in the subject? --------------------------------------------------------- I have got confused now. Math history seems to allow knowing all the data on how math formed, for any of a...
  8. Q

    I What are the insights into the Total Derivative formula?

    I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$ Any insight would be greatly...
  9. V

    Calculus Best textbook to truly understand single variable calculus?

    I am currently self learning high school level AP Calculus BC over the summer. From a mathematical perspective, I have heard that high school level calculus is just pure and shallow computational work. For this reason, I seek a deeper understanding in calculus. So far, the BC curriculum has been...
  10. M

    Find the height of each right cylinder

    Homework Statement I have to construct 8 concrete right cylinders for hurricane protection on windows, but I need to know the height to fit the given criteria. Each has a 4 inch radius and each weighs 1200 pounds. Concrete weighs 150 pounds per cubic foot. V= volume of right cylinder r=radius...
  11. IonizingJai

    Implicit differentiation problem

    Homework Statement If ##x\sqrt{1+y} + y\sqrt{1+x } = 0##, then prove that ##\frac {dy} {dx} = \frac {-1}{(x-1)^2}##. 2.Relevant Equations: $$ \frac {dy} {dx} = - \frac {\left (\frac {\partial f}{\partial x} \right)} {\left( \frac {\partial f} {\partial y} \right)}.$$ 3...
  12. prakhargupta3301

    Having a problem in steps while solving integrals

    Homework Statement My problem is in integral calculus (I'm new to it). I know what it is and how it works (basically. I'm not too advanced right now). The problem is as following: (I will be posting comments/reasons along with what I've done and with what logic/understanding I've done it...
  13. José Ricardo

    Calculus Exploring the Best Books for Studying Calculus: A Comprehensive Guide

    What are the best book to study Calculus? And there is a academic book with History of Calculus? I would like recommendations. I appreciate all.
  14. prakhargupta3301

    Having a problem in steps while solving integrals

    Okay. So I'm new to calculus. And this is the first time I'm solving a physics problem using integration. I understood that ∫dt (or ∫1dt) will be equal to just 't+C.' (Just like f ' (t+C) = 1). Though, that's not the problem. The problem is when I apply it this way: Question: The expression for...
  15. I

    B Can you deduce ##\tan(\theta) = \frac {df} {dx}## from this graph?

    Could someone explain to me how from this graph you can deduce that ##\tan(\theta) = \frac {df} {dx}##. Thanks
  16. Safder Aree

    Path Integral Setup for Given Initial and Final Points

    Homework Statement The path integral from (0,0,0) to (1,1,1) of $$<x^2,2yz,y^2>$$. I am a little confused about the setup.Homework Equations $$\int_{a}^{b} v.dl$$The Attempt at a Solution Here is how I set it up. $$\int_{0}^{1}x^2 dx + \int_{0}^{1}2yz dy + \int_{0}^{1}y^2 dz$$ Since the...
  17. M

    MHB Calculus of Measures: Mapping Natural Numbers to Rationals

    i have one question concerning measure theory and this...could we map the natural numbers to the positive rationals...then observe the measure between the rationals being mapped ,as a value giving us a sense of how many irrationals are between them...then generate a function where we take the...
  18. Adgorn

    Spivak's "Calculus": AM-GM inequality problem.

    Homework Statement The problem is stated as follows: "The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}## and "geometric mean" ##G_n=\sqrt[n] {a_1...a_n}## Satisfy ##G_n≤A_n## Suppose that ##a_1\lt A_n##...
  19. G

    How Is the Area Under a Gaussian Curve Computed?

    I have been teaching undergrad students informally, and one of the math problems that I have always enjoyed introducing them to is how to compute the area under a gaussian curve, or to keep it simple, the area under the curve ##z=e^{-x^2}## One of my students asked me a question that has...
  20. R

    What is the Taylor expansion of x/sin(ax)?

    Hey everyone 1. Homework Statement I want to compute the Taylor expansion (the first four terms) of $$f(x) =x/sin(ax)$$ around $$x_0 = 0$$. I am working in the space of complex numbers here. Homework Equations function: $$f(x) = \frac{x}{\sin (ax)}$$ Taylor expansion: $$ f(x) = \sum...
  21. CollinsArg

    I Surface area of a revolution, why is this wrong?

    Why is this way of thinking wrong?. can't I assume that when Δx tends to zero is a sufficient approximation of what I want to get? It confuses me with the basic idea of integrating a function to get the area beneath a curve of a function (which isn't also as perfect) . PD: I put Δx tends to...
  22. T

    B Find Perpendicular Forces Given 20 N at 60 Degrees

    I have a question. If you're given a force of 20 N and its 60 degrees to the horizontal, how could you find two perpendicular forces?
  23. gibberingmouther

    I A rigorous definition of a limit and advanced calculus

    i'm trying to review calculus and look a little deeper into proofs/derivations/etc. I'm doing this both for fun and to review before i go back to school. am i the only one who has difficulty understanding the "rigorous" definition of the limit? i found this web page...
  24. R

    Calculus derivatives word problem

    Homework Statement Is it possible to accurately approximate the speed of a passing car while standing in the protected front hall of the school? Task: Determine how fast cars are passing the front of the school. You may only go outside to measure the distance from where you are standing to the...
  25. A

    Calc 2 Integration Area Problem

    Homework Statement Please help me solve the calc problem pictured! Homework Equations y=3-x^2 and y=x+1 The Attempt at a Solution My attempt is in one of the photos!
  26. Connor

    Courses Should I wait to take calculus at another school?

    The current community college I'm at offers courses titled analytical geometry and calculus and only transfer to the universities in my state as calculus courses. I'm looking to move outside of South Carolina and going to a school like WSU. Should I wait to take calculus courses when I transfer...
  27. A

    Finding the limits of integration and quadratic formula

    Homework Statement Please look at the photo! Homework Equations -x^2+4x=x^2-6x+5 The Attempt at a Solution I got 2x^2-10x+5 but it says it's wrong
  28. Sune Irl

    Calculus 2 - Center of Mass and Pappus Centroid Theorem

    Homework Statement determine the center of mass of a thin plate of density 12 and whose shape is the triangle of vertices (1,0), (0,0), (1,1). Then, using the appropriate pappus theorem, calculate the volume of the solid obtained by rotating this region around the line x = -2. Homework...
  29. WeiShan Ng

    [Calc] Sign Convention in damping spring system

    Homework Statement A carriage is mounted on a spring, as shown in the diagram. The bottom of the spring is fixed to the ground. The carriage (loaded with its passenger) has a mass of 150kg. The carriage can only move vertically. The natural length of the spring is 10m and its spring...
  30. S

    MHB What is the Implication of a Formula in Propositional Calculus?

    Given : p\wedge\neg (q\vee r) then prove whether this formula implies:!) (p\wedge\neg q)\vee (p\wedge\neg r) OR 2) (p\wedge\neg q)\wedge (p\wedge\neg r)
  31. WhiteWolf98

    Vectors/ Calculus with i and j components

    Homework Statement I don't understand how to form an equation using the knowledge that, 'When ##t=4##, ##P## is moving parallel to the vector ##\mathbf {j}##'. I've seen the solution, and not a single part of it makes sense. I haven't attempted any question like this before, so I have no idea...
  32. Clara Chung

    Multivariable calculus problem

    Homework Statement Homework EquationsThe Attempt at a Solution I have attached the problem and solution. I don’t know how to do part b even I have looked at the solution. How to transform the original cartesian equation to the semi cylindral coordinate equation? Is there is systematical...
  33. bhobba

    Teaching Economics Without Calculus

    Just a quick question. Can you really teach economics without calculus? Reviewing it now just to refresh my memory since we have a budget being delivered tomorrow here in Aus. I have reached the point where it is proven that maximizing overall profit is different from profit per unit of...
  34. KFSKSS

    I need some help with implicit differentiaiton.

    Hello. My problem is as follows: Suppose x^4+y^2+y-3=0. a) Compute dy/dx by implicit differentiation. b) What is dy/dx when x=1 and y=1? c) Solve for y in terms of x (by the quadratic formula) and compute dy/dx directly. Compare with your answer in part a). I solved a) and b). a)=-4x^3/2y+1, and...
  35. A

    Calculus Best Books to Learn Calculus in 2021

    I have just started to learn calculus.Can you suggest the best book to grasp the concepts and to teach myself the true calculus.
  36. M

    Calculus Which books for Calculus AND Linear Algebra

    I wanted to go through Calculus and then Linear Algebra following either of two paths: a) Keisler's Infinitesmal approach>>>Nitecki Deconstructing Calculus>>>Nitecki Calculus in 3D>>>Freidberg's Linear Algebra OR b) Simmons Calculus with analytic geometry>>>Apostol Vol 1>>>>Apostol Vol...
  37. E

    Need help finding angular momentum of a particle

    1. At the instant of the figure, a 6.70 kg particle P has a position vector of magnitude 4.30 m and angle θ1 = 43.0° and a velocity vector of magnitude 3.40 m/s and angle θ2 = 32.0°. Force , of magnitude 7.40 N and angle θ3 = 32.0° acts on P. All three vectors lie in the xy plane. About the...
  38. E

    Physics Homework Help of converting to years (astronomy q)

    A star is 3.7 x 104 ly (light-years) from the center of its galaxy and is moving in a circle around that center at a speed of 170 km/s. (a) How long does it take the star to make one revolution about the galactic center? (b) How many revolutions has the star completed since it was formed about...
  39. H

    Other Books Famous Scientists Studied From

    Hey, I haven't seen a thread on this topic, so I figured I should start it. Hopefully others can contribute! Srinivasa Ramanujan - SL Loney, Trigonometry & GS Carr, A Synopsis of Elementary Results in Pure and Applied Mathematics SS Chern - Hall & Knight, Higher Algebra Bernard Riemann -...
  40. M

    Intro Physics Is Learning Calculus First Necessary for Self-Studying Physics?

    Hello. I’d like to self-learn physics. basically starting from a zero knowledge background to keep things simple. I have asked before and some individuals said that it is using your time to learn it once you know calculus? Is this true or should I start learning the basics now so that when I...
  41. H

    Calculus Who's familiar with all: Piskunov, Fichtenhols, and Smirnov?

    I am very interested in the "russian" type of math approach, a mix of rigor with lots of examples from physics and engineering mixed in with the calculus/analysis pedagogy. It also fascinates me that the same texts were studied by both physicists and engineers of the time period; so lauded...
  42. R

    Fourier transform of integral e^-a|x|

    Homework Statement I am supposed to compute the Fourier transform of f(x) = integral (e-a|x|) Homework Equations Fourier transformation: F(p) = 1/(2π) n/2 integral(f(x) e-ipx dx) from -infinity to +infinity The Attempt at a Solution My problem is, that I do not know how to handle that there...
  43. M

    Shoot a basketball with a minimum speed at some angle

    Homework Statement You should shoot a basketball at the angle ##\theta## requiring minimum speed. Avoid line drives and rainbows. Shooting from (0, 0) with the basket at (a, b), minimize ##f(\theta)= 1/(a \sin (\theta) \cos (\theta) -b \cos^2 (\theta))##. (a) If b =0 you are level with the...
  44. M

    Pick a,b,c,d for y=ax^3+bx^2+cx+d that models path of plane.

    Homework Statement A plane starts its descent from height ##y =h## at ##x = -L## to land at ##(0,0)##. Choose ##a, b, c, d## so its landing path ##y =ax^3 + bx^2 + cx + d## is "smooth". With ##\frac{\mathrm {d}x}{\mathrm {d}t} = V =##constant, find ##\frac{\mathrm {d}y}{\mathrm {d}t}## and...
  45. Guy Fieri

    Issue With Optimization Problem

    Homework Statement Homework Equations I have yet to figure out any relevant equations, but I do believe that the constraint equation for the optimization problem is the y=64-x^6 listed above. The Attempt at a Solution I am currently trying to figure out methods to begin my optimization...
  46. M

    B Area of a circle without calculus

    π is defined by the ratio of the circumference (R) of a circle to its diameter. The area of the circle is πR². Can this be derived without calculus (or Archimedes method)?
  47. M

    Interpret success-rate/time * $

    Homework Statement You are applying for a ##\$1000## scholarship and your time is worth ##\$10## an hour. If the chance of success is ##1 -(1/x)## from ##x## hours of writing, when should you stop? Homework Equations Let ##p(x)=1 -(1/x)## be the rate of success as a function of time, ##x##...
  48. Zack K

    Finding the Value of an Integral with U-Substitution

    Homework Statement Suppose that: 0∫2f(x)dx = 2 1∫2f(x)dx = -1 and 2∫4 = 7, find 0∫1f(x+1)dx Homework Equations a∫bf(x) = F(b) - F(a) The Attempt at a Solution So in these types of integration, we are needed to use u-substitution, the problem is, using u-substitution requires you to have...
  49. EEristavi

    Continuity of Function - f(x)=|cos(x)|

    Homework Statement [/B] We have a function f(x) = |cos(x)|. It's written that it is piecewise continuous in its domain. I see that it's not "smooth" function, but why it is not continuous function - from the definition is should be..Homework Equations [/B] We say that a function f is...
  50. S

    C/C++ How to use C++ in studying calculus

    How to use C++ in studying calculus I'm having a hard time.
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