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. jlmccart03

    Courses Calculus 3 -- looking for ways to help me understand

    So I am in calculus 3 this year and have passed both calc 1 and 2 with a B and C+ respectively. I could have gotten a better grade but was lazy. I was lazy by using calculators and not actually learning the arithmetic and algebra. Now one serious issue I have is Trig. I can never remember trig...
  2. L

    Linear and Non-linear Equations (QM)

    1. Problem Recall that we defined linear equations as those whose solutions can be superposed to find more solutions. Which of the following differential/integral equations are linear equations for the function u(x,t)? Below, a and b are constants, c is the speed of light, and f(x,t) is an...
  3. D

    Vector Geometry: Quadrangular Pyramid with Inner and Cross Products

    Homework Statement A quadrangular pyramid OABCD with square ABCD as the bottom. OA = 1, AB = 2, BC = 2 Also, OA perpendicular to AB, OA perpendicular to AD. Question 1 : Find the inner product \overrightarrow {OA}.\overrightarrow {OB} and the size of the cross product |\overrightarrow...
  4. G

    Help with this differential calculus

    <Moderator's note: Moved from a technical forum and therefore no template.> Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order...
  5. L

    If a constant number h of fish are harvested from a fishery

    Hi! Can anyone help me? If a constant number h of fish are harvested from a fishery per unit time, then a model for the population P(t) of the fishery at time t is given by: dP/dt = P(5-P) - h, P(0) = P0. a. Solve for the IVP if h = 4. b. Determine the value of P0 such that the fish...
  6. S

    Finding the Derivative of y=sqrt(x+sqrt(x+sqrt(x)))

    Homework Statement This is a chain rule problem that I can't seem to get right no matter what I do. It wants me to find the derivative of y=sqrt(x+sqrt(x+sqrt(x))) Homework Equations dy/dx=(dy/du)*(du/dx) d/dx sqrtx=1/(2sqrtx) d/dx x=1 (f(x)+g(x))'=f'(x)+g'(x) The Attempt at a Solution My...
  7. D

    Find the limit using Riemann sum

    Homework Statement i want to find limit value using riemann sum \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br> question : <br> \lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br> Homework EquationsThe Attempt at a...
  8. M

    Engineering Fluid Mechanics book that's pure integration/vector calculus

    Hello. The textbook that was assigned to my class is not a good fit for my professor. The textbook simplifies the basic force equations so that there are no general equations with integrals or vector calculus calculations. Can anybody recommend a Fluid Mechanics book or books that deal only...
  9. X

    I How much would time pass between watching the sun set ....?

    How much would time pass between watching the sun set from ground level and then watching it set again from the top of a sky scraper? I heard once that this could be done using one of the towers of the World Trade Center. So I assume one could also do this using the Sears / Willis Tower in...
  10. D

    Trying to find this double integral using polar coordinates

    Homework Statement question : find the value of \iint_D \frac{x}{(x^2 + y^2)}dxdy domain : 0≤x≤1,x2≤y≤x Homework Equations The Attempt at a Solution so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x and i decided to convert...
  11. A

    B Quick question about calculus (derivatives)

    I thought Differentiation is all about understanding it in a graph. Every time I solve a question on differentiation I visualise it as a graph so it's more logical. After all, that IS what the whole topic is about, right? Or am I just wrong? But when you look at these questions...
  12. B

    Calculus What are some good resources for studying Spivak's Calculus?

    I'm currently struggling a lot in self-studying Spivak's Calculus; I've tried numerous online resources including math stackexchange, I'm currently about half-way through How To Prove It as many people have suggested. I'm only in high school, so the calculus class I'm taking barely coincides...
  13. V

    Force due to two thin charged rods acting on each other

    Homework Statement I have 2 thin rods of length L in the axial plane(x axis), they're of the same uniform linear charge distribution and are separated by a distance a . Homework Equations $$E = \frac{kq} {R^2}$$ $$F = qE$$ The Attempt at a Solution [/B] Let 0 be one at one end of the two rod...
  14. A

    Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

    Homework Statement Solve the following heat Eq. using FFCT: A metal bar of length L is at constant temperature of Uo, at t=0 the end x=L is suddenly given the constant temperature U1, and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any...
  15. A

    Solving partial differential equation with Laplace

    Homework Statement am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it? appreciated, Homework EquationsThe Attempt at a Solution my attempt is the same as in the attached picture...
  16. A

    Schools Taking Calculus 1 over the summer -- question?

    I was talking to my friend and she told me how she was taking Calculus 1 over the summer in 5 weeks. Everyone else averaging a 30 percent as their overall grade. The teacher was planning to raise the grade. So if you get a 30%, you'll get a C. However, my question is that is it okay to pass...
  17. S

    I Can Cycloid Paths Determine Least Time Courses in 3D?

    when inducing that the cycloid is the least time-taking course between the two points in the two dimension, we have to use calculus of variations. Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?
  18. 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...
  19. Avatrin

    I Why Must the Transversality Condition Be Less Than or Equal to Zero?

    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...
  20. 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...
  21. 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
  22. 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...
  23. 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...
  24. A

    Calculating the area of equilateral triangle using calculus

    Homework Statement Calculating the area of equilateral triangle using calculus. Homework EquationsThe Attempt at a Solution The area of the triangle is the area of the circle minus 3 times the area of the sector shown in (light blue). So, the target is to calculate the pink area first...
  25. P

    Textbook for advanced calculus

    Oh wait, I am not supposed to ask questions here, sorry. I can't find where i should post questions though
  26. J

    MHB Testing Convergence, Calculus II

  27. P

    Expressing series in terms of a Power Series

    Hello and thank you for trying to help. In spite of the fact that this seems a very simple problem, I do not find myself able to get a solution. Here it goes: Let $$f(x)=\displaystyle \sum_{k=3}^\infty a_k \frac{x^k}{k(k-1)(k-2)}$$ and $$g(x)=\displaystyle \sum_{k=0}^\infty a_k x^k$$. Express...
  28. V

    Calculus in the velocity and acceleration of satellites.

    Homework Statement I am working on a project dealing with the velocity and acceleration of satellites based on their distance from Earth. I was recommended to include some calculus in this project. Originally I thought I could just take the derivative of the orbital speed equation to find...
  29. F

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

    Hi, In next semester, I am going to take vector calculus. Here is the course description: Vector fields, line and surface integrals, Green's Theorem, Stokes' Theorem, Divergence Theorem and advanced topics such as differential forms or applications to mechanics, fluid mechanics, or...
  30. havonasun

    Courses Deciding Between Calculus I & Calculus w/ Analytic GeometryI

    I work full-time, so evening or online classes are all I can do. I need to take calculus from the community college I've been attending, but it is not offered in the evening. In fact, none of the local colleges or universities offer it in the evening, so I must do it online. So I go to register...
  31. victorhugo

    Studying Starting 2 calculus classes without background in calculus

    I started studying calculus last week and in 2 weeks I start my double degree in: Science (Physics) / Engineering (Honours) My 4 classes will be: FUNDAMENTALS OF PHYSICS B MATHEMATICS 2: SERIES AND INTEGRAL CALCULUS ELECTRICAL SYSTEMS ENGINEERING COMPUTING AND ANALYSIS So, what tips do you...
  32. F

    Calculus List of GOOD multivariable calculus book

    Books on multivariable calculus that I often see get good recommendations are, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard and Hubbard Vector Calculus by Colley What are other good books with some material on differential forms like Hubbard and Colley? Books by Edwards...
  33. 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...
  34. Rodrigo Schmidt

    Mathematically rigorous Calculus 2 book

    So, i am currently studying physics in a brazilian university. I am going to have a Calculus 2 course which, in Brazil, covers Ordinary Differential Equations and multi-variable differential calculus. So which challenging and rigourous books would you guys recommend for that? Thanks for the...
  35. C

    I Why Does div[grad(1/r)] Equal -4πδ(r vec)?

    I don't understand why grad(1/r)= (-1/r^2)ê_r. I know its derivate, but in cero the function is not definite and you can't find its derivative in cero.And, why div [grad(1/r)]=-4πδ(r vec) ?
  36. K

    B Calculus of variation. Minimum surface

    so df/dy' is yy'/ √(1+y'^2) then we are supposed to do y' . [ yy'/ √(1+y'^2) ] - y√(1+y'^2) how does this bring equation 2 in the image ?
  37. T

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

    Homework Statement Find the interval for a so that (3-a)x+ay+(a^2-1)=0 is normal to the curve xy=4Homework Equations If two curves intersect each other orthogonally (at right angles) m1×m2=-1 Slope of a curve y=f(x) at a point(x1,y1) is equal to value of (dy/dx) at x1,y1 The Attempt at a...
  38. GiantSheeps

    Courses Taking Multivariable calc after AP calc AB

    I am going to be a freshman in college and I intend to major in astrophysics. This past year I took AP calculus AB, and I got an A and I believe I did well on the AP test (results haven't come back yet though). This morning I took my school's math placement test, and I was placed into...
  39. Kiwigami

    Find time given a force with respect to displacement

    Homework Statement With a function that gives Force with respect to Displacement (F(x) = 2000-100x), is there a way to find a general formula for the time it takes from displacement 0 to 20? The force is acting on an object, pushing it horizontally. Assume no friction and no air resistance...
  40. M

    Calculus Rigorous calculus textbooks from intro to advanced

    My ultimate goal is to become a theoretical physicist, a great one at that. I have mastered the prerequisites and I am now looking for rigorous calculus textbooks that make some references to physics, or are more orientated for people who want to become physicist.Thanks for your help.
  41. V

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

    I studied calculus (equivalent to calc 2 perhaps) back in school last year but it was just procedure, patters and tricks. I wanted to do more so I picked up spivaks calculus and it's taking me far too long. It's been a week or two with some 5 hours almost everyday and I'm only in chapter 4...
  42. B

    I Calculus in the derivation of Euler-Lagrange equation

    In the derivation of Euler-Lagrange equation, when differentiating S with respect to α, there is a step: $$\frac{\partial f(Y,Y',x)}{\partial\alpha}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial\alpha}+\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial\alpha}$$ Where $$ Y =...
  43. Vance Grey

    What is the meaning of tensor calculus?

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

    Integral Equation (or I think so) Calculus I problem

    Homework Statement Find a continuous funciton ##f## such that $$ f(x) = 1+ \dfrac{1}{x} \int_{1}^{x} f(t)dt $$ I think I solved it but I would like to see if it's right. Well, first of all, by the fundamental theorem of calculus I know that $$ \left( \int_{1}^{x} f(t)dt \right) ' = f(x) $$...
  45. L

    Volume of a Solid using Cylindrical Shells

    Homework Statement Find the volume of the region bounded by the curves y=3x-2, y=6-x, and the x-axis when the region is rotated around the y-axis. Homework Equations Volume using cylindrical shells: 2π∫r(x)h(x)dx The Attempt at a Solution I graphed the curves and then found the x-intercept...
  46. harpazo

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

    I just purchased, through amazon.com, Demystified Calculus (covering the first-two semesters of Calculus) and Demystified Advanced Calculus (Calculus 3). I like the language used in both books, language that does not include math jargon. Each book is divided into several sections with a test...
  47. Blockade

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

    For implicit differentiation, is dy/dx of x2+y2 = 50 the same as y2 = 50 - x2 ? From what I can take it, it'd be a no since. For x2+y2 = 50, d/dx (x2+y2) = d/dx (50) --- will eventually be ---> dy/dx = -x/y Where, y2 = 50 - x2 y = sqrt(50 - x2) dy/dx = .5(-x2+50)-.5*(-2x)
  48. O

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

    I am considering taking it down the line.
  49. 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...
  50. R

    Line integral of vector field from Apostol calculus

    Homework Statement Here are the three problems that i couldn't solve from the book Calculus volume 2 by apostol 10.9 Exercise 2. Find the amount of work done by the force f(x,y)=(x^2-y^2)i+2xyj in moving a particle (in a counter clockwise direction) once around the square bounded by the...
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