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. W

    Is Substituting q(t) the Correct Method to Verify a Differential Equation?

    Homework Statement How does one show that q(t) is indeed a solution? Homework EquationsThe Attempt at a Solution My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS. Reason being that if q is indeed a solution, the result of the...
  2. Avatrin

    I Calculus of variations question

    Hey, I have a theorem I cannot prove. We have a function x^* that maximizes or minimizes the integral: \int^{t_1}_{t_0} F(t,x(t),\dot{x}(t))dt Our end point conditions are: x(t_0) = x_0, x(t_1) \geq x_1 I am told that x^* has to satisfy the Euler equation. That I can fully understand since...
  3. T

    Pre-university physics study group (calculus based)

    So here is a problem (more of a dilemma) I encountered in my mandatory physics class [(high school level) i say mandatory since I take an other optional calculus based one], many students are often mislead for example in kinematics the equations are very clunkily derived and when you finish...
  4. Another

    I Question about vector calculus

    particles in plane polar coordinates r = rcosθ i + rsinθ k F = Fer + Feθ ∂r/∂r =|∂r/∂r|er = (cos2θ + sin2θ)½er = er why ∂r/∂θ =|∂r/∂θ|eθ = (r2cos2θ + r2sin2θ)½eθ = reθ I understand that ∂r/∂θ = -rsinθ + rcosθ but why ∂r/∂θ = (r2cos2θ + r2sin2θ)½eθ
  5. E

    Programs Vector calculus and E&M physics as a engineering major?

    I am an engineering major at Los Angeles Pierce community college. I have been for the last years working 40 hours a week in order to sustain and put myself through community college. After I transfer, I don't plan on working. Now, each semester due to my work schedule and life happening, I can...
  6. A

    Classical Recommendation needed for integral calculus books

    I have learned an adequate amount of calculus including implicit, parametric differentiation as well as Upton second order differential equations in high school math course. It was really abstract and we were taught only how to solve mathematical problems. Now, I need to model those problems in...
  7. A

    Calculating the area of equilateral triangle using calculus

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  8. P

    Textbook for advanced calculus

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  9. J

    MHB Testing Convergence, Calculus II

  10. P

    Expressing series in terms of a Power Series

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  11. V

    Calculus in the velocity and acceleration of satellites.

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  12. F

    Preparing for Vector Calculus: What Topics Should You Focus On?

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  13. havonasun

    Courses Deciding Between Calculus I & Calculus w/ Analytic GeometryI

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  14. victorhugo

    Studying Starting 2 calculus classes without background in calculus

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  15. F

    Calculus List of GOOD multivariable calculus book

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  16. Felix Gonzales

    I don't understand the derivation process here, help?

    Homework Statement I understand the derivation it showed that included the sin (15.7 in the image) I just don't understand the following (15.8 in the image). Does "t" get pulled out of the equation? If so what do we derive for then? Does it become 0? If so, it would remain 0 and sin(0) is just...
  17. Rodrigo Schmidt

    Mathematically rigorous Calculus 2 book

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  18. C

    I Why div [grad(1/r)]=-4πδ(r vec)

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  19. K

    B Calculus of variation. Minimum surface

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  20. T

    What is the interval for a so that the given line is normal to the hyperbola?

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  21. GiantSheeps

    Courses Taking Multivariable calc after AP calc AB

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  22. Kiwigami

    Find time given a force with respect to displacement

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  23. M

    Calculus Rigorous calculus textbooks from intro to advanced

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  24. V

    Studying Am I Taking Too Long to Learn Calculus? Should I Give Up?

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  25. B

    I Calculus in the derivation of Euler-Lagrange equation

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  26. Vance Grey

    What is the meaning of tensor calculus?

    https://www.physicsforums.com/attachments/205736
  27. S

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  28. L

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  29. harpazo

    MHB Have you tried Demystified Calculus for a more student-friendly approach?

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  30. Blockade

    B Is dy/dx of x2+y2 = 50 the same as dy/dx of y = sqrt(50 - x2)?

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  31. O

    Courses Is undergrad analysis more difficult than calculus 1&2?

    I am considering taking it down the line.
  32. A

    Is it possible to self study calculus 2?

    I am a high school student entering 11th grade soon. I will be taking calculus ab (calc 1) in school, but I am wondering if it is possible to self study calculus 2 and take it over the summer at a cc?(and do well!) If everything goes right I want to do calculus 2 the summer after junior year and...
  33. R

    Line integral of vector field from Apostol calculus

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  34. Wrichik Basu

    Calculus Book Recommendations in Multivariable Calculus

    I am almost on the verge of completing single-variable Calculus, and I've got a book on the same by I. A. Maron. So, after getting a good grip on single-variable Calculus, I want to start with multivariable. Can anyone recommend me good books on multivariable Calculus with which I could begin...
  35. C

    Improper integral with spherical coordinates

    Homework Statement I have a question. I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent. x^2y^2z^2/r^(17/2) * f(x,y,z)dV. Homework Equations...
  36. manjuvenamma

    I Is it possible to find the limit of (1+1/x)^x as x approaches -infinity?

    Is it possible to find the limit of (1+1/x)^x as x approaches minus infinity using only the fact that it is e if x approaches infinity?
  37. R

    Line integral problems in Apostol calculus

    Homework Statement A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0 Homework Equations Find a value of a(in...
  38. danielhep

    Optimization w/ Constraint Question (Multivariable Calculus)

    Homework Statement Find any maxima/minima on f(x,y) = x2+2y2 on the unit circle, centered at the origin. Homework Equations grad f = λgrad g constraint: 1=x2+y2 The Attempt at a Solution grad f = 2xi+4yj grad g = 2xi+2yj 2x=λ2x 2y=λ4y How do I solve this? I don't see any way to get numbers...
  39. F

    Using dx/dy to find a y-parallel tangent

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  40. R

    Is there any way to calculate this integral?

    I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue. The theory is: where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2
  41. S

    Calculus Need help preparing for Courant's Calculus

    Hello, Just to give you some background info, I'm a software developer looking to make a career switch to Mechanical Engineering (At least I think so). My interests lie in design, complex systems, maglev, rockets and new forms of space propulsion. Before I really commit myself to University, I...
  42. S

    B Some help understanding integrals and calculus in general

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  43. R

    Calculus Reviewing Multivariable calculus to skip in college

    Hi, I'm asking for a friend who will be majoring in chemical engineering. We have already taken Calculus I, II, and III under a course offered by a local community college. Admittedly, it was taught from stewart's series of calculus books, and we did exactly zero proofs in the class, and all...
  44. Bunny-chan

    Epsilon distance between two terms

    Homework Statement How close is x to x_0 (x_0 \neq 0) so that 2. Homework Equations The Attempt at a Solution I tried to use absolute value properties:- \epsilon \lt \frac{\sqrt{x_0^2+1}}{x_0^3} - \frac{\sqrt{x^2+1}}{x^3} \lt \epsilonBy adding in the three sides, we...
  45. harpazo

    MHB How can self-study lead to long-term learning compared to classroom instruction?

    As I study calculus 3, I often revisit calculus 1 and 2. The following application is from single variable calculus, partucularly calculus 1, called RELATED RATES. I have not seen a related rates problem since the 2015. I am a bit rusty with the set up. Can someone help me set it up? I can take...
  46. harpazo

    MHB How do you solve this calculus 2 integration problem?

    I am currently studying calculus 3. However, once in a while, I like to review single variable calculus. I decided to tackle the following calculus 2 integration problem: ∫ tan^3 x dx Solution: Separate into two tangent functions. (tan^2 x)(tan x) Rewrite tan^2 x as sec^2 x - 1 using its...
  47. C

    Integral with transformations and bounded by x + y + z = 1

    Homework Statement I have a question. I need to know the integral dxdydz/(y+z) where x>=0, y>=0, z>=0.Homework Equations It is bounded by x + y + z = 1. The transformations I need to use are x=u(1-v), y=uv(1-w), z=uvw. The Attempt at a Solution y+z = uv. J = uv(v-v^2+uv) So I get the integral...
  48. EEristavi

    Basic Calculus: Differentiation usage

    Problem: How fast is the area of a rectangle changing if one side is I0 cm long and is increasing at a rate of 2 cm/s and the other side is 8 cm long and is decreasing at a rate of 3 cm/s?I have 2 approach and I want to know which is correct, why and what am I missing
  49. phasacs

    Why isn't Kinetic Energy always equal to Potential Energy?

    K= ∫mvdv = ∫m dx/dt dv = ∫m dx/dv dv/dt dv = ∫m dv/dt dx = ∫Fdx = U => K=U, why isn't this true? If it is, wouldn't that mean that Kinetic Energy is always equal to Potential Energy?
  50. Avatrin

    I Motivating definitions in calculus on manifolds

    Hi I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...
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