What is Calculus: Definition and 1000 Discussions

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. doktorwho

    What is this expression equal to?

    Basically i need to evaluate the limit of this expression ##\lim \sqrt[3]{n^6-n^4+5}-n^2=?## I want to know if this is correct and why: ##\lim \sqrt[3]{n^6-n^4+5}-n^2=\lim...
  2. cathal84

    Finding length of vector with unknown variable

    Finding length of vector with unknown variable. Purely for study purposes. Find the smallest possible length of the vector →v . Let vector V = (-2/3,b,16/7) Equation for finding length of vector : Sqrt(a^2+b^2+c^2)Question would be quite straight forward had there been no unknown variable but...
  3. T

    Finding Volume and Surface Area of a Banana Using Calculus

    Homework Statement We are given a Banana, and asked to find the volume and surface area of the function, using calculus. So far, we have learned elementary calculus (derivatives, limits, and integrals) as well as volumes of revolutions. We traced the banana on graph paper, plotted points on the...
  4. H

    Studying How can I teach myself physics?

    So, I am majoring in mechanical engineering with a minor in physics at San Jose State University. I want to learn as much as possible with physics by reading books and taking my future courses at SJSU, but I don't know what books to read. Any recommendations? I want to get pretty close to...
  5. H

    Studying What books can I learn from the most?

    Okay, so I'm currently a mechanical engineering major at San Jose State University, but I just want to become much more engaged with physics and mathematics. I do pretty good with calculus, math comes easy to me. I'm a first year student taking calculus 2. I was just wondering what books I can...
  6. C

    I Convergence of Taylor series in a point implies analyticity

    Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
  7. ParabolaDog

    Struggling immensely with tensors in multivariable calculus

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  8. M

    Vector Calculus - Tensor Identity Problem

    Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...
  9. dfklajsdfald

    Solve Antiderivative of xe^x | Step-by-Step Guide

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  10. W

    Derivative in spherical coordinates

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  11. B

    Absolutely Convergent, Conditionally Convergent, or Divergent?

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  12. PhotonSSBM

    B Finding Maxima/Minima of Polynomials without calculus?

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  13. S

    I Vector Calculus: What do these terms mean?

    In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...
  14. N

    Schools Need Help: Calc, GPA, and Grad School.

    Hello everyone. I'm currently a sophomore working toward a BS in physics (and a minor in astronomy) at a top private engineering school. With the semester finishing up, I'm a little worried about where I am now and where I will be after graduation and I have a few questions. A little...
  15. G

    Fluid Pressure & Force Calculus 2

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  16. K

    I What is the boundary surface of a collimator?

    Hi everybody, I’m trying to calculate the shape of a boundary line f(x) between two mediums that collimates rays from a point light source. This requires the rays to hit the boundary line under a certain angle, so I calculated the slope m(φ) of the boundary line for a ray with polar angle φ (φ...
  17. G

    Finding maximum height of a string before it goes slack

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  18. S

    I Spivak's Definition of Integrals vs Stewart's

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  19. O

    A Frechet v Gateaux Derivative and the calculus of variations

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  20. M

    Electric field from a non-uniformly charged disk

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  21. Cjosh

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  22. C

    I Find potential integrating on segments parallel to axes

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  23. T

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  24. Kaura

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  25. HRubss

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  26. Kaura

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  27. Ketav

    Calculus: Verify Thick Walled Cylinder Equations

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  28. victorhugo

    Studying What is the best way to learn calculus?

    I didn't take calculus at school so I'm going to learn it during summer holidays before doing it at uni. The thing is, the only reason I got so good at General Mathematics is because I didn't follow the step by step to doing a problem and just remembering the rules, I took the time to ask lots...
  29. Y

    MHB Calculus - definite integral

    Consider the definite integral ∫202x(4−x2)1/5 dx. What is the substitution to use? u= 4-x^2 Preview Change entry mode (There can be more than one valid substitution; give the one that is the most efficient.) For this correct choice, du/dx= -2x Preview Change entry mode If we make this...
  30. IntegralBeing

    Calculus I've just finished Stewart's calculus, now what?

    I hated that book so much; I had the opportunity to change to Spivak or Apostol in holidays but I didn't do it. I feel like I will have to read a good rigorous calculus text from the beggining since Stewart's textbook is sheer rubbish in many senses. Which book should I read to continue my...
  31. T

    I Derivative of A Def. Integral Equals Another Def. Integral?

    I'm going through the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman. On Page 138 (below) ) The authors take the derivative of a definite integral and end up with a definite integral plus another term. How did...
  32. merricksdad

    Find the center of mass of a cone with variable density....

    Homework Statement Find the center of mass of an inverted cone of height 1.5 m, if the cone's density at the point (x, y) is ρ(y)=y2 kg/m. Homework Equations The formula given for this problem is rcm=1/M * ∫rdm, where M is total mass, r is position, and m is mass. The Attempt at a Solution...
  33. dykuma

    How Does Viscosity Affect Motion in a Fluid?

    Homework Statement A mass M falls under gravity (force mg) through a liquid with decreasing viscosity so that the retarding force is -2mv/(1+t). If it starts from rest, what is the speed, acceleration, and distance fallen at time t=1. Homework Equations F=ma The Attempt at a Solution F =...
  34. G

    Finding the time period using the potential

    Homework Statement A particle of unit mass moves in one dimension with potential V(x) = ½μ2x2 + εx4 (ε>0). Discuss the motion of the particle. If the particle released from rest at x=a (a>0) express the time period T for the particle to return to a in the form of an integral and show that when...
  35. C

    Maximum Torque / Evenly Spread across a lever.

    Homework Statement The maximum torque on a lever is 1.5 x 10^6 Newtons. How many people of weight 750N can stand evenly spaced on this lever, which has a length of 20 meters? Homework Equations T=FR Weight=mg W=Fd X = Number of people The Attempt at a Solution I have set 1.5x10^6 N =...
  36. A

    Unsure if this is a trig identity or calculus

    Homework Statement $$sinx - cosx = 1/3$$ solve for $$sin(2x)$$ Homework Equations $$sin^2x + cos^2x = 1$$ $$sin2x = 2cosxsinx$$ The Attempt at a Solution I think you can square both sides and get: $$sin^2x - cos^2x = 1/9$$ But how can I use this information to solve for sin2x? Is there a...
  37. 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...
  38. 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...
  39. 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...
  40. INAM KHAN

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

    I need this book Calculus by James Stewart 8th edition.
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. J

    Using integral calculus to find the equation of the quartic

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  50. 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...
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