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. S

    A Differential forms and vector calculus

    Let ##0##-form ##f =## function ##f## ##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}## Then consider the following dictionary of symbolic identifications of expressions expressed in the language of differential forms on a manifold and expressions expressed in the...
  2. J

    I Extremal condition in calculus of variations, geometric

    Hi folks, I am a bit confused with the extreme condition used in the calculus of variations: δ = 0 I don't understand this rule to find extreme solutions (maximum or minimum) If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the...
  3. A

    Calculus of Variations; Maximum enclosed area problem.

    The problem reads: "You are given a string of fixed length l with one end fastened at the origin O, and you are to place the string in the (x, y) plane with its other end on the x-axis in such a way as to maximise the area between the string and the x axis. Show that the required shape is a...
  4. INAM KHAN

    Calculus Where Can I Find Calculus by James Stewart 8th Edition?

    I need this book Calculus by James Stewart 8th edition.
  5. E

    Calculus Updating my Electricity and Magnetism --> Vector Calculus?

    Dear all, I'n an EE that finished his degree more than 10 years ago. I wanted to refresh my Electricity and Magnetism knowledge. I bough Purcells book some weeks ago (https://www.amazon.com/dp/1107014026/?tag=pfamazon01-20) and I'm kind of struggling through the maths (Vector calculus). I've...
  6. Z

    B Help with understanding Nature of Roots for Quadratic and Cu

    Hi I am writing my final Mathematics exams for Grade 12 in South Africa in 5 days. I am well prepared with an aim of getting 100%, but one concept in functions might prevent that - the concept of how the nature of roots are affected by vertical/horizontal shifts in a function, and how to...
  7. caters

    Best Tunnel Shape for X,Y,Z Coordinates

    Homework Statement X,Y,Z(coordinates) What function corresponds to the best tunnel shape? g = 9.8 m/s^2(earth gravity) Homework Equations F(x)=Y G(x)=X^2 in the xy plane G(z)= sin(X) in the xz plane H(x)= parabolic sinusoid(X^2 and sin(X) both in the xy plane) The Attempt at a Solution I have...
  8. MiLara

    I Why do some but not all derivatives have physical meaning?

    I know that taking the derivative of certain functions that explain physical phenomena can lead to another statement describing the physical system, the most famous being the derivatives of position. That is, position-->velocity-->acceleration-->jerk-->jounce...and taking any other further...
  9. P

    I Numerical Calculus of Variations

    I attempt to solve the brachistochrone problem numerically. I am using a direct method which considers the curve ##y(x)## as a Lagrange polynomial evaluated at fixed nodes ##x_i##, and the time functional as a multivariate function of the ##y_i##. The classical statement of the problem requires...
  10. K

    Why is the curl of the electric dipole moment equal to zero in the far field?

    Hello.Looking at Jackson's ch 9 on radiation, I am trying to calculate the fields E and B from the potentials in the far field but it is very confusing. Given now the approximation for he vector potential \textbf{A}_{\omega}(x) = -ik \frac{e^{ikr}}{r} \textbf{P}_{\omega} with...
  11. T

    Implicit Differentiation Question

    << Mentor Note -- thread moved from the technical math forums at OP request, so no Homework Help Template is shown >> x2y + xy2 = 6 I know we use the chain rule from here, so wouldn't that be: (d/dx)(x2y + xy2) = (d/dx)(6) so using the chain rule of g'(x)f'(g(x) and the d/dx canceling out on...
  12. D

    [Multivariable Calculus] Implicit Function Theorem

    I am having trouble doing this problem from my textbook... and have no idea how to doit. 1. Homework Statement I am having trouble doing this problem from my textbook... Show that the equation x + y - z + cos(xyz) = 0 can be solved for z = g(x,y) near the origin. Find dg/dx and dg/dy (dg/dx...
  13. J

    Using integral calculus to find the equation of the quartic

    Homework Statement The question states Use integral calculus to find the euation of the quartic that has (1,23) and (3, 15) and a y-intercept of 24. Homework Equations The previous part of the question was A quartic has stationary points of inflection at x=1 and x=3. Explain why...
  14. M

    B Natural exponential function, calculus

    So I'm trying out various practice problems and for some reason I can't get the same answer when it comes to problems involving natural exponentials. Here's the problem A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of...
  15. L

    Variational calculus or fluid dynamics for fluid rotating in a cup

    my first post having just joined! Problem statement - what curve describes the surface of a rotating liquid? Stirring my cup of coffee years ago sparked this thought. Question - is the way to solve this problem to use variational calculus, or fluid dynamics? I have always thought the former but...
  16. i_hate_math

    Area of Region Vector Calculus

    I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre...
  17. I

    Weber-Fermat Problem, degenerate cases

    Homework Statement I have to prove some things on the Weber-Ferma problem. Here is the assignment : We want to find a point $$x$$ in the plane whose sum of weighted distances from a given set of fixed points $$y_1, ...,y_m$$ is minimized. 1-Show that there exist a global mimimum to the...
  18. ChloeYip

    Studying Are there any resources for studying calculus and physics?

    I am in year one, hope to get more exercises to work on practice... especially in calculus and physics (introductory level) My school is already provide webwork and masteringphysics as homework, but I don't think they are enough...(limited number of questions only...) Are there any more...
  19. weezy

    Proof of independence of position and velocity

    A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$ In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...
  20. karush

    MHB 10) AP Calculus linear functions

    $\textbf{10)} \\ f(x)\text{ is continuous at all } \textit{x} \\ \displaystyle f(0)=2, \, f'(0)=-3,\, f''(0)=0 $ $\text{let} \textbf{ g } \text{be a function whose derivative is given by}\\ \displaystyle g'(x)=e^{-2 x} (3f(x))+2f'(x) \text{ for all x}\\$ $\text{a) write an equation of the...
  21. A

    Calculus of Variations: Functional is product of 2 integrals

    Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...
  22. Brian T

    Computational Looking for resources on Discrete Exterior Calculus and FEEC

    Does anyone have recommendations for reading/resources on Discrete Exterior Calculus and/or Finite Element Exterior Calculus? In particular, I want to learn the topics to use in a project for a course and so would like to learn how to implement these methods (specifically geared toward...
  23. weezy

    Verifying the Correctness of My Proof

    1. I have to show: 2. Given: 3. My attempt : I just want to verify if what I've done is correct or not. Thanks!
  24. S

    Prove r(t) moves in a line, if a and v are parallel

    Homework Statement A point moves on a curve \vec { r } with constant acceleration \vec { A } , initial velocity \vec { { V }_{ 0 } } , and initial position { \vec { { P }_{ 0 } } } b. if \vec { A } and \vec { { V }_{ 0 } } are parallel, prove \vec { r } moves in a line c. Assuming \vec {...
  25. M

    I A-level differentiation/derivative dilemma

    Hello, and thank you for your time. I just started my A-levels derivatives/differentiation , and I would be more than happy if you could help me clarify it. For example I know that y is a function in terms of x right? y=f(x) The derivative of it is f'(x)=dy/dx . This means it is the rate of...
  26. Tspirit

    A problem on calculus in Griffiths' book

    I can't understand the solution to Problem 1.4(a). The solution is the following: What puzzles me is that ρ(θ)dθ=ρ(x)dx ? Why are they equal?
  27. A

    Calculus Calculus 1 text book - Need review of precalculus

    Hi, so I'm taking calculus 1 this year however I haven't taken precalculus in several years. I don't remember any of it, and the textbook of the course doesn't review it at all(they just sample you questions) and I'm having issues solving the precalculus review questions(how necessary is it that...
  28. WeiShan Ng

    Find Antiderivative of y: y^2=x^2+1

    Homework Statement x=sec(t), y=tan(t), -π/2 ≤ t ≤ π/2 Try to find y in terms of x Homework EquationsThe Attempt at a Solution 1.[/B] ∂y/∂x = sec(t)/tan(t) y=∫sec(t)/tan(t)∂x =∫x/y∂x =(1/y)*∫x∂x =x2/2y + C 2y2=x2 + C When t=π/4, x=√2, y=1 2(1)2 = (√2)2 + C C=0 So y2 = x2/2 2. y/x = sin(t)...
  29. S

    I Properties of Direct Product of Half Open and Open Intervals

    The 2-D plane is usually constructed as "ℝxℝ" and ℝ is both open and closed. My question is, what is the direct product of a half open and an open interval? Is it also open or half open?
  30. TheDemx27

    How to Calculate Heat Current in a Spherical Shell?

    Homework Statement A spherical shell has inner and outer radii r_a and r_b, respectively, and the temperatures at the inner and outer surfaces are T_a and T_b. The thermal conductivity of he shell material is k. Derive an equation for the total heat current thought the shell in the steady...
  31. B

    I Understanding "Terrible" Math Notation: A Calculus Guide

    Is there some standardized math text with "proper universal notation" I could read for calculus? In one of my courses, $$\int\frac{dx}{x}$$ had a red mark through it, with a note that said "impossible" or something. I earned a zero on the question due to the above. In another instance...
  32. S

    MHB Proving Russell's Paradox in Predicate Calculus

    Can we prove in the predicate calculus,that there does not exist someone who can shave all those that do not shave themselfs?? (Russell's Paradox)
  33. S

    I What is the purpose of Arc-Length Parameterization?

    My teacher just briefly introduced arc length parameterization and went on to frenet serret frames, without any explanation or motivation. What is the purpose of arc length parameterization? What role does it play in TNB? What is the purpose of TNB frames anyways?
  34. casualluchador

    Courses Should I take calculus 2,3 and diff. eqs at CC while in HS?

    Hi I was wondering if taking calculus 2,3, and differential equations by the end of my senior year at the local community college would be a wise choice. Would taking these math classes before i use them in physics hinder my learning? (I want to be a physicist). Would I gain an advantage in...
  35. Elvis 123456789

    Integration by parts and approximation by power series

    Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...
  36. P

    Courses Is Skipping Calc I and II Worth It? Considerations for Freshman Physics Majors

    Hi! I'm a freshman and I plan to get my bachelor's in Physics. I couldn't get any Physics or Math classes this semester as most were either full or clashing with my mandatory History and English classes. I gave the Math Placement Test at my university and was able to skip Precalculus classes...
  37. S

    I Proving Theorem 1 in Spivak's Calculus: Tips & Tricks

    Hello I am struggling with proving theorem 1, pages 98-99, in Spivak's Calculus book: "A function f cannot approach two different limits near a." I understand the fact that this theorem is correct. I can easily convince myself by drawing a function in a coordinate system and trying to find two...
  38. K

    Physics with calculus. Prior to Calculus

    This is my first year in college, and I am currently taking calculus 1, and physics w/ calculus. My academic adviser told me that I would be okay taking the two together since I'd be learning the calculus as i went. This doesn't seem to be the case because for a lot of my problems we are already...
  39. D

    I Connection Between Calculus and Physics

    I am at the stage of my education where I am seeing calculus (mainly differential equations) popping up in my engineering courses. However, I have just started multi-variable calculus, so I have not taken differential equations yet. My professors basically show us how certain equations are...
  40. S

    Prove this is a right triangle in a sphere

    Homework Statement Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle Homework Equations |AB|^2 = |AP|^2 + |PB|^2 |AB}^2 = 4r^2 The Attempt at a Solution Not sure if showing the above equations are true is the...
  41. C

    B How do you find the limit of this?

    Hi! First time poster, I'm about to enter first year calc and thought that I could get ahead of the curve by checking out some questions beforehand. This showed up on one of the university calculus exams but I couldn't figure out how to do it. I tried to finding a common denominator but then was...
  42. Z90E532

    I Notation in Spivak's Calculus on Manifolds

    I have a question regarding the usage of notation on problem 2-11. Find ##f'(x, y)## where ## f(x,y) = \int ^{x + y} _{a} g = [h \circ (\pi _1 + \pi _2 )] (x, y)## where ##h = \int ^t _a g## and ##g : R \rightarrow R## Since no differential is given, what exactly are we integrating with...
  43. M

    I Vector Calculus: Divergence of Dyadic AB

    So I have a quick question that will hopefully yield some clarification. So the divergence of a dyadic ##\bf{AB}## can be written as, $$\nabla \cdot (\textbf{AB}) = (\nabla \cdot \textbf{A}) \textbf{B} + \textbf{A} \cdot (\nabla \textbf{B})$$ where ##\textbf{A} = [a_1, a_2, a_3]## and...
  44. Nader AbdlGhani

    B Difference between these functions .

    What's the difference between f(x)=3 and f(x)=3x^0 ? and why Limit of the second function when x\rightarrow0 exists ? and is the second function continuous at x=0 ?
  45. N

    I Re-derive the surface area of a sphere

    Hey everyone, I've been stuck on this one piece of HW for days and was hoping someone could help me. It reads: The surface area, A, of a sphere with radius R is given by A=4πR^2 Re-derive this formula and write down the 3 essential steps. This formula is usually derived from a double...
  46. W

    Redesigning the math/physics curriculum, take 2

    I saw the topic and have given it a lot of thought over the past few years, and my take is so different from the other thread I'm starting another. Problems with the beginning math curriculum: 1. Too abstract and difficult 2. Totally unmotivated These problems were unavoidable 50 years ago...
  47. J

    Applied Books like J. Callahan's Advanced Calculus: A geometric view

    Hello, do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)? This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology). Thanks in advance!
  48. A

    B Calculus Project - Requesting Advice

    Good afternoon, I'm looking to complete a Mathematics Calculus Exploration for my course, before starting to write up my project I was hoping to find someone that could offer some advice or comments to make my project more successful. First of all my project seems very simple at first glance...
  49. ibkev

    B Tensor Calculus vs Tensor Analysis?

    I've seen the terms tensor calculus and tensor analysis both being used - what is the difference?
  50. D

    I The two equivalent parallel velocity vectors

    This is an exercise from the textbook Apostol Vol 1 (page 525, second edition), and I do not know how to prove it: Suppose a curve C is described by two equivalent functions X and Y, where Y(t) = X[u(t)]. Prove that at each point of C the velocity vectors associated with X and Y are parallel...
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