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

    Line Integral of circle in counterclockwise direction

    My attempt: Let ##x=a \cos \theta## and ##y=a \sin \theta## $$\int_{L} xy^2 dx-x^2ydy$$ $$=\int_{0}^{2\pi} \left( (a\cos \theta)(a\sin \theta)^2 (-a\sin \theta)-(a\cos \theta)^2 (a \sin \theta)(a\cos \theta)\right) d\theta$$ $$=-a^4 \int_{0}^{2\pi}\left( \sin^3 \theta \cos \theta+\cos^3 \theta...
  2. chwala

    Show that ##f(x,y)=u(x+cy)+v(x-cy)## is a solution of the given PDE

    Looks pretty straightforward, i approached it as follows, ##f_x = u(x+cy) + v(x-cy)## ##f_{xx}=u(x+cy) + v(x-cy)## ##f_y= cu(x+cy) -cv(x-cy)## ##f_{yy}=c^2u(x+cy)+c^2v(x-cy)## Therefore, ##f_{xx} -\dfrac{1}{c^2} f_{yy} = u(x+cy) + v(x-cy) - \dfrac{1}{c^2}⋅ c^2 \left[u(x+cy)+v(x-cy)...
  3. I

    Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates

    I want to prove that ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates where ##a,b,c \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch ) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1...
  4. I

    Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##

    I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book) There exists a set...
  5. L

    Proving limits for roots and exponents

    Hi I have to prove the following three tasks I now wanted to prove three tasks with a direct proof, e.g. for task a)$$\sqrt[n]{n} = n^{\frac{1}{n}}= e^{ln(n^{\frac{1}{n}})}=e^{\frac{1}{n}ln(n)}$$ $$\displaystyle{\lim_{n \to \infty}} \sqrt[n]{n}= \displaystyle{\lim_{n \to \infty}}...
  6. gleem

    I Wish to Give Away a Textbook Set on The Calculus

    I have a two volume set of Differential and Integral Calculus textbooks by a well-known mathematician, his second edition (1959 printing) the last of twenty printings. This set was recommended by one of my math professors. Although it was my intention to read them it was not to be. Below is a...
  7. Tobi9242

    Mathematical model for drag on tether

    I have a model for airs density as a function of height I would imagine the speed can be describes as the angular velocity times length The coefficient of drag can be found online, seems to be around 1.17 for a cylinder It seems to me that im going to need an integral somewhere, but can't quite...
  8. E

    I have troubles finding the limit of this piecewise function

    I have troubles finding the limits at the designated points , should i only find the limit at infinity where f(x) has belongs to an interval containing inifinity? (sorry for english) and for the a this is what i attempted. i am unsure. Our textbook never talks about piecewise functions and...
  9. mcastillo356

    B Some questions regarding the integral of cos(ax), "a" not zero

    Hi PF, $$\int \cos ax\,dx,\quad a\in{\mathbb R-\{0\}}\quad x\in{\mathbb{R}}$$ Let's make $$u=ax,\quad du=adx$$ and apply $$\int \cos u\,du=\sin u+C$$ $$\frac{1}{a}\int \cos ax\,adx=\frac{1}{a}\sin u+C$$ Substituting the definition of u $$=\frac{1}{a}\sin ax+C$$ Doubts: (i) Have I written well...
  10. Z

    Calculation of moment of inertia of cylindrical surface

    Here is the homogenous paper rectangle And if we roll it we get a cylinder with base radius ##a##. It is not clear to me what "an axis through a diameter of the circular base means". Let's imagine such as axis is ##\alpha## in the following figure Then we have...
  11. Memo

    Help understanding this integral solution using trig substitution please

    Here's the answer: Could you explain the highlighted part for me? Thank you very much!
  12. Memo

    Integral involving powers of trig functions

    Could you check if my answer is correct? Thank you very much! Is therea simpler way to solve the math?
  13. I

    A Integration of trigonometric functions

    Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).
  14. chwala

    Find rate at which the liquid level is rising in the problem

    I was able to solve it using, ##\dfrac{dV}{dt} = \dfrac{dV}{dh}⋅\dfrac{dh}{dt}## With, ##r = \dfrac{h\sqrt{3}}{3}##, we shall have ##\dfrac{dV}{dh} = \dfrac{πh^2}{3}## Then, ##\dfrac{dh}{dt}= \dfrac{2×3 ×10^{-5}}{π×0.05^2}= 0.00764##m/s My question is can one use the ##\dfrac{dV}{dt} =...
  15. chwala

    Show that acceleration varies as cube of the distance given

    In my approach i have distance as ##(x)## and velocity as ##(x^{'})##, then, ##(x^{'}) = kx^2## where ##k## is a constant, then acceleration is given by, ##(x^{''}) = 2k(x) (x^{'})## ##(x^{''}) = 2k(x)(kx^2) ## ##(x^{''}) = 2k^2x^3##. Correct?
  16. Infrared

    Challenge Math Challenge Thread (October 2023)

    The Math challenge threads have returned! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Do not solve problems that are way below your level. Some problems...
  17. S

    Existence of directional derivative

    My attempt: I have proved (i), it is continuous since ##\lim_{(x,y)\rightarrow (0,0)}=f(0,0)## I also have shown the partial derivative exists for (ii), where ##f_x=0## and ##f_y=0## I have a problem with the directional derivative. Taking u = <a, b> , I got: $$Du =\frac{\sqrt[3] y}{3 \sqrt[3]...
  18. S

    Find f(x,y) given partial derivative and initial condition

    My attempt: $$\frac{\partial f}{\partial x}=-\sin y + \frac{1}{1-xy}$$ $$\int \partial f=\int (-\sin y+\frac{1}{1-xy})\partial x$$ $$f=-x~\sin y-\frac{1}{y} \ln |1-xy|+c$$ Using ##f(0, y)=2 \sin y + y^3##: $$c=2 \sin y + y^3$$ So: $$f(x,y)=-x~\sin y-\frac{1}{y} \ln |1-xy|+2 \sin y + y^3$$ Is...
  19. M

    Determine whether ## S[y] ## has a maximum or a minimum

    a) The Euler-Lagrange equation is of the form ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ##. Let ## F(x, y, y')=(y'^2+w^2y^2+2y(a \sin(wx)+b \sinh(wx))) ##. Then ## \frac{\partial F}{\partial y'}=2y' ## and ## \frac{\partial F}{\partial...
  20. F

    Insights Epsilontic – Limits and Continuity

    Epsilontic – Limits and Continuity I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I think I’m not the only one who struggled at that point. We mainly learned algorithms at school, i.e. how things are...
  21. mcastillo356

    B Integral of cosecant function: understanding different approaches

    Hi, PF Trigonometric Integrals "The method of substitution is often useful for evaluating trigonometric integrals" (Calculus, R. Adams and Christopher Essex, 7th ed) Integral of cosecant...
  22. P

    How to get general solution via Green's function?

    I'll start with a characterization of the Green's function as a fundamental solution to a differential operator. This theorem is given in Ordinary Differential Equations by Andersson and Böiers. ##E(t,\tau)## is known as the fundamental solution to the differential operator ##L(t,D)##, also...
  23. S

    Multivariable calculus problem involving partial derivatives along a surface

    I just wanted to know if my solution to part (b) is correct. Here's what I did: I took the partial derivative with respect to x and y, which gave me respectively. Then I computed the partial derivatives at (-3,4) which gave me 3/125 for partial derivative wrt x and -4/125 for partial derivative...
  24. J

    Introduction to Modern Astrophysics -- What are the prerequisites?

    I was recently recommended this book and told it was a standard textbook at an upper undergraduate level or lower graduate level. Well that's certainly above my level, but specifically what would be the prerequisites? I've no formal math training but self taught calculus at a level somewhere...
  25. S

    Unit tangent vector and curvature with arc length parameterization

    (a) $$\frac{ds}{dt}=|r'(t)|$$ $$=\sqrt{(x(t))^2+(y(t))^2+(z(t))^2}$$ $$=\frac{2}{9}+\frac{7}{6}t^4$$ $$s=\int_0^t |r'(a)|da=\frac{2}{9}t+\frac{7}{30}t^5$$ Then I think I need to rearrange the equation so ##t## is the subject, but how? Thanks Edit: wait, I realize my mistake. Let me redo
  26. M

    Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

    a) Observe that ## \frac{\partial}{\partial z}F(y, z)=y^{n-1}\cdot \frac{2z}{2\sqrt{y^2+z^2}}=\frac{zy^{n-1}}{\sqrt{y^2+z^2}} ##. This means ## G(y, z)=\frac{z^2\cdot y^{n-1}}{\sqrt{y^2+z^2}}-y^{n-1}\cdot \sqrt{y^2+z^2}=\frac{z^2\cdot...
  27. M

    Proofs about the second-order linear differential equation?

    Proof: (i) Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##. Then ## u''+\frac{f}{2}u=0\implies r^2+\frac{f}{2}=0 ##, so ## r=\pm \sqrt{\frac{f}{2}}i ##. This implies ## u_{1}=c_{1}cos(\sqrt{\frac{f}{2}}x) ## and ##...
  28. A

    A Summing over continuum and uncountable numerocities

    Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants. I will provide introduction in very condensed form to get quicker to the essense. Conservative part First of all, let us...
  29. mcastillo356

    B Trigonometric substitution, a case I'd like to share

    Hi, PF First I will quote it; next the doubts and my attempt: "In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expresions. In calculus, trigonometric substitution is a technique for evaluating integrals. (...) Case I: Integrands containing...
  30. P

    I Lipschitz continuity of vector-valued function

    I'm reading Ordinary Differential Equations by Andersson and Böiers, although this is more related to multivariable calculus. There is a Lemma regarding Lipschitz continuity which I have a question about. Below ##\pmb{f}:\mathbf{R}^{n+1}\to \mathbf{R}^n ## is a vector-valued function defined by...
  31. KungPeng Zhou

    A question about definite integrals and series limits

    In my opinion , if it can be shown that this is a monotonically bounded sequence, one can confirm that there is a limit. First，we know $$ \frac{1-x^{4n}}{1+x^{2}}dx=(1-x^{2}) (1+x^{2}) ^{n-1}=(1-x^{4}) ^{n-1}(1+x^{2}).$$ According to the integral median theorem，we can get $$a_n=(2- \sqrt{3} )...
  32. KungPeng Zhou

    How Do You Solve Calculus Homework Problem Six Using LaTeX?

    How to solve the sixth problem?
  33. bhobba

    Insights Beginner's Guide to Precalculus, Calculus and Infinitesimals

    [url="https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/"]Continue reading...
  34. bhobba

    Insights Precalculus, Calculus and Infinitesimals

    See my insights article for those interested in an unconventional approach to doing Precalculus at an accelerated pace and beginning Calculus. It is different from the usual way that a precalculus is done text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and...
  35. chwala

    Find the derivative of the given function

    Let's see how messy it gets... ##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...
  36. Anonymous001

    I Difference between d/dt and d(theta)/dt? Why is it dr or ds/dt?

    So, first of all, why and how are we taking the derivative of the vector r or s as d/dt if t is not a parameter of the equations? Second question is what is the difference between d/dt(r) and d(theta)/dt(r) and also between d/dt(s) and d(theta)/dt(s)? Like, both of these appear at the bottom of...
  37. L

    I Taylor series of 1/ln(t+1) at t=0

    I have tried to use the natural Taylor expansion of ln(t+1) and working with long division but I get the result 1/t+1/2+t/12−t^2/24+⋯ instead of 1/t+1/2-t/12+t^2/24+... that is the right result I have tried to do this several time but still don't works. Is there a miscalculation in my long...
  38. I

    Show that the given function is decreasing

    As a follow up for : https://www.physicsforums.com/threads/let-k-n-show-that-there-is-i-n-s-t-1-1-k-i-1-2-k-i-1-4.1054669/ show that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for ##...
  39. M

    Can anyone please verify/confirm these derivatives?

    Note that ## \frac{\partial F}{\partial x}=\frac{2x}{2\sqrt{x^2+y'^2}}=\frac{x}{\sqrt{x^2+y'^2}}, \frac{\partial F}{\partial y}=0, \frac{\partial F}{\partial y'}=\frac{2y'}{2\sqrt{x^2+y'^2}}=\frac{y'}{\sqrt{x^2+y'^2}} ##. Now we have ## \frac{dF}{dx}=\frac{\partial F}{\partial x}+\frac{\partial...
  40. chwala

    Solve the given differential equation

    My interest is only on the highlighted part, i can clearly see that they made use of chain rule i.e by letting ##u=1+x^2## we shall have ##du=2x dx## from there the integration bit and working to solution is straightforward. I always look at such questions as being 'convenient' questions. Now...
  41. chwala

    Solve the given differential equation

    I am on differential equations today...refreshing. Ok, this is a pretty easier area to me...just wanted to clarify that the constant may be manipulated i.e dependant on approach. Consider, Ok I have, ##\dfrac{dy}{6y^2}= x dx## on integration, ##-\dfrac{1}{6y} + k = \dfrac{x^2}{2}##...
  42. I

    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    let ##k \in\mathbb{N},## Show that there is ##i\in\mathbb{N} ##s.t ##\ \left(1-\frac{1}{k}\right)^{i}-\left(1-\frac{2}{k}\right)^{i}\geq \frac{1}{4} ## I tried to use Bernoulli's inequality and related inequality for the left and right expression but i the expression smaller than 1/4 for any i...
  43. G

    Show that the Taylor series for this Lagrangian is the following...

    We have ##L(v^2 + 2v\epsilon + \epsilon^2)##. Then, the book proceeds to mention that we need to expand this in powers of ##\epsilon## and then neglect the terms above first order, we obtain: ##L(v^2) + \frac{\partial L}{\partial v^2}2v\epsilon## (This is what I don't get). We know taylor is...
  44. mcastillo356

    I Substitution in a definite integral

    Hi, PF I am going to reproduce the introduction of the textbook; then the Theorem: The method of substitution cannot be forced to work. There is no substitution that will do much good with the integral ##\int{x(2+x^7)^{1/5}}\,dx##, for instance. However, the integral...
  45. A

    Calculus Can I skip the introduction in Apostol's calculus?

    I'll be starting Apostol's calculus book in a little less than two weeks, as I'm finishing up an easier textbook. I looked ahead and I see that there is a four part introduction, and I was wondering if I could just skip that. I briefly skimmed it, and it just looks like a review of summation...
  46. akrill

    Finding the Centre of Mass of a Hemisphere

    Place hemisphere in xyz coordinates so that the centre of the corresponding sphere is at the origin. Then notice that the centre of mass must be at some point on the z axis ( because the 4 sphere segments when cutting along the the xz and xy planes are of equal volume) y2 + x2 = r2 We want two...
  47. L

    I Taylor expansion of f(x)=arctan(x) at infinity

    I have to write taylor expansion of f(x)=arctan(x) around at x=+∞. My first idea was to set z=1/x and in this case z→0 Thus I can expand f(z)= arctan(1/z) near 0 so I obtain 1/z-1/3(z^3) Then I try to reverse the substitution but this is incorrect .I discovered after that...
  48. mcastillo356

    B I don't recognize this limit of Riemann sum

    Hi, PF, I hope the doubts are going to be vanished in a short while: This is the limit of Riemann Sum ##\displaystyle\lim_{n\rightarrow{\infty}}\displaystyle\frac{1}{n}\displaystyle\sum_{j=1}^{n}\cos\Big(\displaystyle\frac{j\pi}{2n}\Big)## And this is the definition of the limit of the General...
  49. chwala

    I Integration of ##e^{-x^2}## with respect to ##x##

    My first point of reference is: https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2 I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial...
  50. mcastillo356

    I Solve Int. Eq.: Exponential Growth Diff. Eq.

    Hi, PF There goes the solved example, the doubt, and the attempt: Example 8 Solve the integral equation ##f(x)=2+\displaystyle\int_4^x\,f(t)dt##. Solution Differentiate the integral equation ##f'(x)=3f(x)##, the DE for exponential growth, having solution ##f(x)=Ce^{3x}##. Now put ##x=4## into...
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