# Calculas Definition and 24 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. ### 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.
2. ### 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...
3. ### 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...
4. ### 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...
5. ### Evaluating This limit

<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 ...
6. ### 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...
7. ### 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...
8. ### 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...
9. ### A different perspective to derivatives rather than slopes-3b1b

A different perspective to derivatives.

24. ### Comparison test

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...