What is Calculas: Definition and 44 Discussions

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.
Lambda calculus consists of constructing lambda terms and performing reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules:

producing expressions such as: (λx.λy.(λz.(λx.z x) (λy.z y)) (x y)). Parentheses can be dropped if the expression is unambiguous. For some applications, terms for logical and mathematical constants and operations may be included.
The reduction operations include:

If De Bruijn indexing is used, then α-conversion is no longer required as there will be no name collisions. If repeated application of the reduction steps eventually terminates, then by the Church–Rosser theorem it will produce a β-normal form.
Variable names are not needed if using a universal lambda function, such as Iota and Jot, which can create any function behavior by calling it on itself in various combinations.

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  1. Safinaz

    Simpliying this partial differential equation

    Does this equation mean that: ## x+y =z ##, and ## x = y##? I mean ## \delta_{ij} ## terms in the LHS of the eqaution equal those at the RHS ? with knowing that ## \partial_i \partial_j x ## term dose not vanish for ## \delta_{ij}=1 ## Any help is appreciated!

    I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?

    Here is a proof of mean value theorem: Consider a line passing through the points (a, f(a)) and (b, f(b)). The equation of the line is y-f(a) = {(f(b)-f(a))/(b-a)} (x-a) or y = f(a)+ {(f(b)-f(a))/(b-a)} (x-a) Let h be a function that defines the difference between any function f and the...
  3. Mr X

    Derivation or proof of derivative sin (x)

    How do I do this from here without using the derivatives of sin or cos ?
  4. Saracen Rue

    I Why is the integral of ##\arcsin(\sin(x))## so divisive?

    Initially, I was attempting to find the function which expresses the area under enclosed between the function ##\arcsin(\sin(x))## and the ##x##-axis (so technically I am looking for ##\int_{0}^{x} \arcsin(\sin(t)) dt## specifically, but got caught up on finding the general antiderivative)...
  5. S

    MHB Derivative of f(x), g(x) and h(x) from Calculus Problem A.1: Solve Urgent"

    Please also solve this : Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ0(x) of the following function with respect to x: λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)Note: here problem A.1 is the upper problem.Please solve both the problem.
  6. Trey Chandler

    Need help solving this trapezoidal problem

    This is my attempt at the problem I'm pretty sure the answer is supposed to be in gallons but I don't know what I'm doing clearly.
  7. S

    I Calculus- Area between two curves (minimize it)

    Hi, This is my first question here, so please apologise me if something is amiss. I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can...
  8. R

    B A proof of the fundamental theorem of calculus

    is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it? i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it. i know...
  9. Z

    Differential Equation with an Initial condition

    Homework Statement x(dy/dx) = 3y +x4cos(x), y(2pi)=0 Homework Equations N/A The Attempt at a Solution I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential...
  10. navneet9431

    Yes, your answer is correct.

    <Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x\rightarrow 0} (x-tanx)/x^3$$ I solve it like this, $$\lim_{x\rightarrow 0}1/x^2 - tanx/x^3=\lim_{x\rightarrow 0}1/x^2 - tanx/x*1/x^2$$ Now using the property $$\lim_{x\rightarrow 0}tanx/x=1$$,we have ...
  11. Z

    Kinematics in Cylindrical Coordinates

    Homework Statement A small bead of mass m slides on a frictionless cylinder of radius R which lies with its cylindrical axis horizontal. At t = 0 , when the bead is at (R,0), vz = 0 and the bead has an initial angular momentum Lo < mR sqrt(Rg) about the axis of the cylinder where g is the...
  12. L

    Pressure and Volume -- are the growth/decay rates exponential?

    Homework Statement forumlate if its growth/decay is exponential I have equation that i intergrated and found Pressure over volume = work done of pressure P = 3.2c^-1.4 f(v) = -8v^-0.4 i set limits of 10x10^-6 --> 100x10^-6 and 10x10^-6 --> 100x10^-6 but i increase both values by 20x10^-6 every...
  13. M

    B Understanding the basics of integration

    I tried learning calculus using the book by Spivak.In this text, while introducing integrals the author explained a lot about partitioning the area under the curve and defined the integral.The way I understood this is, as we increase the number of divisions in the partition the lower sum and the...
  14. A different perspective to derivatives rather than slopes-3b1b

    A different perspective to derivatives rather than slopes-3b1b

    A different perspective to derivatives.
  15. T

    Is this derivative in terms of tensors correct?

    Homework Statement Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$ where q is a constant vector. Homework EquationsThe Attempt at a Solution $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...
  16. YoungPhysicist

    B Is This a Valid Proof for the Number of Roots in a nth Degree Polynomial?

    Recently I came up with a proof of “ for a nth degree polynomial, there will be n roots” Since the derivative of a point will only be 0 on the vertex of that function,and a nth degree function, suppose ##f(x)##has n-1 vertexes, ##f’(x)## must have n-1 roots. Is the proof valid?
  17. T

    What does “Media Upgrade” mean in Thomas' Calculus textbook?

    What is the difference between: THOMAS' CALCULUS, media UPGRADE (11TH EDITION) [ISBN-13 9780321489876] and THOMAS' CALCULUS (11TH EDITION) [ISBN-13 9780321185587]? could you provide detailed answer please? what does the title-addition "Media Upgrade" mean and what changes does it make on the...
  18. Scrope

    Solve Multi Var Limit: Help Appreciated!

    Homework Statement https://gyazo.com/268bef206850bfbf30fb0cca3f783599 <----- The question Homework EquationsThe Attempt at a Solution Had this on a test today, honestly not sure how to evaluate. I know you can pass the limit to the inside of arctan but I can't see how the inside goes to...
  19. bornofflame

    [MultiVarCalc] Find the volume of the solid

    Homework Statement Find the volume of the solid between the cone ##z = \sqrt{x^2 + y^2}## and the paraboloid ##z = 12 - x^2 - y^2##. Homework Equations ##x^2 + y^2 = r^2## The Attempt at a Solution I drew a simple diagram to start off with to visualize the solid formed by the intersection of...
  20. Aniket13

    Area of common surface of two bubbles

    What will be the area of common surface of two identical bubbles of radius R , i know there common surface will be flat as the radius of curvature of comman surface will tends to Infinity , but how do i relate with area of flat surface I tried to use Energy = Surface tension * area And then...
  21. H

    I Equality of two particular solutions of 2nd order linear ODE

    I got the following two integral for the a particular solution of a 2nd order linear ODE $$(D-a)(D-b)y = g(x)$$ by using inverse operators ##\frac{1}{D-a}## and ##\frac{1}{D-b}##. The two different integrals are obtained by operating these operators in different order on y to get a particular...
  22. EastWindBreaks

    Error propagation, is this correct?

    Homework Statement given E is constant, find the uncertainty in the angular frequency, ω. can someone please check my work? Homework EquationsThe Attempt at a Solution
  23. J

    Analytic Integration of Function Containing the Exponential of an Exponential

    Homework Statement Can this function be integrated analytically? ##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32 \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),## where ##a##, ##b## and ##L## are some real positive...
  24. J

    A Analytical Integration of a Difficult Function

    Is it possible to integrate the following function analytically? ##\int_{0}^{\infty} \frac{\exp{-(\frac{A}{\tau}+B\tau+\frac{A}{\beta-\tau})}}{\sqrt{\tau(\beta-\tau)}}d\tau,## where ##A##, ##B## and ##\beta## are real numbers. What sort of coordinate transformation makes the integral bounded...
  25. Alex Salazar

    Schools Can I take College Algebra and Trig at the same time?

    Hello everyone, After about 6 years I'm finally getting out of the military in May and start going to school full time starting in the fall. I want to do engineering, but I know I'm behind in math and am worried about starting calculus. I want to improve my math by taking both College Algebra...
  26. J

    I Is the Pole at u=2 Integrable?

    Hi All, I am wondering if the function below is Integrable: $$\frac{\exp{(-\frac{1}{2}(u-2)^2-2u^2)}}{u-2}$$ When I work it out on computer, the integral is finite from -Inf to Inf. But clearly it has a pole at u=2. Is this pole integrable? If yes, what kind of coordinate transform is...
  27. Delta31415

    Newton's method and complex roots

    1) the problem I understand Newton's method and I was able to find all the real roots of the function.However, I don't understand how to find the complex roots. I know that z=x+yi, and that I can plug in z for the formula. However I, don't know how to change the function (...
  28. Summer95

    I Can anyone evaluate this integral?

    Hello All! I am trying to solve the simple pendulum without using a small angle approximation. But I end up with this integral: $$\int_{\frac{\pi}{4}}^{\theta}\frac{d\theta}{\sqrt{cos(\theta)-\frac{\sqrt{2}}{2}}}$$ Is this possible to evaluate? If so, could I get a hint about what methods to...
  29. J

    Integral Convergence: Examining 1/(e^x sqrt(x))

    http://‪C:\Users\johny\Downloads\q4.jpg 1. Homework Statement Hi, so the question is I have to tell if this integral diverges or converges.(without solving it) integral(1/(e^x sqrt(x)))dx from 1 to +inf Homework Equations integration techniques. The Attempt at a Solution my answer: let 1/e^x...
  30. L

    Couple Simple Grade 12 Calculas Homework Help NEEDED

    TO EVERYBODY, THIS IS NOT MY HOMEWORK. I HAVE A TEST TOM and these are some questions that I just don't know how to do, I gave attempts to some and some I am just missing something. Plus, these questions are from all over the unit so its not my homework. I need it done asap. So then I know how...
  31. S

    Calculas of variation /euler,s equation

    i m little confusd in (finding the shortest distance b/w two points is a straight line in three dimensions)i have solved it but nothing found any accurate result [ds][/2]=[dx][/2]+[dy][/2]+[dz][/2] is the distance in 3-dimension b/w 2 points then how we can start and how we can take...
  32. V

    Tell me about these basic powefull rules of calculas.

    There are several rules in calculus which have far reaching applications in field of mathematics, physics etc. unfortunately there is no derivation or visualisation of those rules in my maths book. SO SHAME!. one of these is that definite integral of function from x1 to x2 gives geometrical...
  33. S

    Solved: Find "k" for Distinct Real Roots of f(x) = x3 + 3x2 + k

    Question 1: Homework Statement Find "b" so that the difference between the x co-ordinates of the two inflection points of y=x4 + 4x3+bx2+5x+7 is 3 The Attempt at a Solution I.P. at y' = 0 two x co-ordinates are x and x+3 y' = 4x3 + 12x2 + 2bx +5 0 = 4x3 + 12x2 + 2bx +5 0 =4(x+3)3 +...
  34. H

    How to Find the Volume of a Bounded Region with Sphere and Cone Equations

    Homework Statement Find the volume of a region bounded above by the unit sphere x^2+y^2+z^2=1 and below by the cone z=sqrt(x^2+y^2). I am really confuse here.. >< Homework Equations Sphere: x^2+y^2+z^2=1 Cone: z=sqrt(x^2+y^2) The Attempt at a Solution I had plot the graph of the...
  35. J

    Partial Derivatives in Differential Calculus: Need Help

    im trying to get a function R^2 --> R whose partial derivatives of all orders exist in a neighborhood of (0,0) but it, itself, is not continuous at the origin. thing is, i think I am thinking of the question wrong because, I am getting examples and checking if partial der. exist but how can i...
  36. G

    Vector Calculas: Simplifying expressions

    Homework Statement (a -b) \cdot (b - c) \times (c - a) Homework Equations The Attempt at a Solution Honestly I have no idea where to being. I believe this expression is a scalar triple product but I do not know how to use the properties to simplify this expression. Sorry, kind...
  37. M

    Vector Calculus Homework: Evaluate \int F dr

    Homework Statement Evaluate \int F dr if F(x,y) =(6x2 +4y) i + (4x-2y) j and the curve C is a smooth curve from (1,1) to (2,3). Homework Equations The Attempt at a Solution I took partial derivatives with respect of y for the first term and with respect of x for the second...
  38. K

    Finding the Normal Equation for a Curve at a Given Point

    Find the equation of the line normal to the curve at (0,0). (Normal lines are perpendicular to the tangent lines) 2y + sinx = xcoxy I found the derivative to be (cosy - cosx)/ (2+ xsiny) then the slope tangent is 0...then the slope perpendicular to it is undefined...so then is the...
  39. S

    Home school Calculas( high school level)

    Hi.. New at this site, hope I'm posting these stuff on the write place... well i had these question below for review for an exam, hope someone can help me solve them, i did my solution in Paint program and on paper and am linking where i uploaded them, ( image shack) -Find the second...
  40. J

    Finding the Indefinite Integral: A Calculus Homework Question

    Hi everyone. I have this problem for my calculas class. So far i can only get a word which adds up to 94. any help is greatly appreciated. For the alphabet, let a=1, b=2, ... z=26 then, you can start adding up letters of a word. For example, "love" = l+o+v+e = 12+15+22+5 = 54. So, love is...
  41. L

    In Calculas I've learned that 'F means the integral of a function.

    In Calculas I've learned that 'F means the integral of a function. SO what does ''F mean, th integral or an integral of a function?
  42. J

    Calculas Related Rates and Angle Problem

    So i have receive a problem in my calculas class that i have been working on for about 2 weeks and have come up empty handed in my attempt to find the answer, the question is quite lengthly, but is as follows: a cruiser is steaming on a straight course at 20 knots. An airplane, flying so low...
  43. L

    Is Calculus Just an Extension of Pre-Calculus?

    I've studied Pre-Calculas with derivitaves and all but not Calculas. Is it that much different from Precal or is it about the same as Precal?
  44. T

    Proving dy/dx = (a+4b)x^(a+4b-1) for y = x^(2a+3b)/x^(a-b), a and b Integers

    Show that dy/dx = (a+4b)x^(a+4b-1) if y = x^(2a+3b) / x^(a-b) and a and b are integers thx