Mathematics

How did mathematicians come up with Area of circle = ( pi * r^2 ) ? Why did they choose r^2 as one of the products? Was pi found by brute forcing numbers until it worked?

In a chemistry book I saw the following calculation with regard to entropy:- S = 1.38 x 10^-23J/K x ln2^10^22 (or if it helps 2superscript10superscript22 or 2 raised to 10 to the power of 22) The answer is apparently 0.096 J/K So how would you calculate ln2^10^22 using a calculator? Thanks G56

I thought it might be fun to show you some elementary aspects of superanalysis and in particular how we might generalise two well known differential equations on the line to superlines. First what is a superline, [math]\mathbb{R}^{1|m}[/math]? We will define the superline as the space with local coordinates [math]\{t, \theta^{\alpha}\}[/math] where [math]\alpha[/math] runs from [math]0[/math] to [math]m[/math] and [math]\theta^{\alpha}\theta^{\beta} = - \theta^{\beta} \theta^{\alpha}[/math]. In particular, [math]\theta^{\alpha}\theta^{\alpha}=0[/math]. As we will be doing some elementary analysis, we won't worry about coordinate transformations or anything like that. …

I wanted to share with you some useful results concerning differential forms on a manifold. For those that know, the coordinate free definitions of differential forms and their derivations (de Rham, interior product and Lie derivative) are I feel quite complicated. I have very little feeling for them. If we turn to a "super" description in terms of local coordinates then the definitions are much more workable. What I am about to say is not new. Definition(ish) A supermanifold is a "manifold" with both even and odd coordinates. Locally, we mean that a supermanifold can be endowed with coordinates [math]z^{A}=\{x^{a}, \theta^{\alpha}\}[/math] such that [math]x^…

There is this problem from my textbook but I can't seem to figure it out. A rectangular swimming pool is to be built with the area of the pool being 2500 square feet. In addition, the owner wants 6 foot wide decks along the sides of the pool and 8 foot wide decks at the two ends. Determine the dimensions (to the nearest tenth of a foot) of the pool that allows the pool and decks to be built on the smallest possible piece of property. Thank you

I'm taking part in a course with the Open University and I have been given an assignment, which includes a question asking me to calculate the speed of someone diving off a diving board. The distance given is 4.6m and the acceleration due to gravity is 9.8 m s-2. I'm totally lost and have no idea where to start in trying to calculate this. The context of the question is in relation to solving equations by changing an equation round. If anyone can shine some light on this for me I would be forever grateful.

[ATTACH]1664[/ATTACH] As a practising geometist I was intrigued by the attached diagram which I drew a long while ago. I endevoured to draw eight circles around a larger circle so that all the circumferences touched. By rough measurement it appeared that the ratio of the diameters of the smaller circles to the diameter of the larger centre circle was very close to the Golden Section. A good friend and mathematician advised me that it was close but no coconuts were to be won. Well I have since wondered about the fact that it was so close. I have expanded my wondering to go 3 dimensional and to consider spheres instead of circles. So the question is gentlemen wou…

Just how small is a nanometer? Liked this guys analogy with the aircraft carrier... anyone got a better one?

Discrete Mathematics or Number Theory? I have never done discrete mathematics before but I'm interested in it. Number theory sounds fun, too.

I have over a 100 pages of attempts to find a pattern in Prime numbers. However you only have to read one of them. Go to the Homepage at www.constructorscorner.com and under the date of the updates click the link of the new work. A lot of work still needs done, but I believe I have found a way to find “x” on the Parabola. I am looking for feedback on my work. Let me know what you think. Can it work?

there are 40 balls in a bag, each numbered idividualy 1 through to 40. what are the odds of picking say 1,2,3, and 4? I know it`s 1 in 10 of picking at least ONE ball correctly, I`m stuck what happens after that though any help?

Simplify applyin' properties: [math](p\implies q)\implies[p\implies(\sim q\wedge p)].[/math]

Hi. Would polar coordinates be preferred for calculations on a heliostat mirror aiming? Get about a metre long piece of string, anchor one end to anything, call that T 'target' and also anchor about the middle of the string to anything else, higher or lower, being that point M -mirror- Grabbing the end of the string point S -sun- , a solar ray path can be visualized by moving the string along a plane. Now, what are convenient reference axis to use for calculating the azimuth and elevation the mirror M should aim so S always illuminates T ? Or, any other suggestions on how is it done ?

I never studied the curve algorithms; I just use 'em. So I really have no idea how to answer this question. Once again working in SVG. When using the cubic Bézier curve commands, I want to find out the coordinates of the actual peaks. For instance, c 3.75,20.0 5.00,-20.00 8.75,0.00 approximates (!) a one-cycle sine curve. I want to be able to figure out the curve's minimum and maximum (y-axis) values. How can I do this? Thanks!

...are computer using the Newton-Raphson method to evaluate square roots ?

Hey guys, I came accross this proof while I was studying this material but I am have a hard time with it. Does anybody know how to go about showing those too equivalencies?? Prove that log(n!) = Theta(n log n) by proving the following two claims: • log(n!) = O(n log n). • log(n!) = *omega(n log n). Thanks!!

Anybody here know how to compute pi factorial? Pi isn't an integer, or a rational number, so I'm at a loss at how to do it....

A coworker of mine phoned with this problem. You have been asked to create at least 1000g of a mixture of sunflower seeds and raisins. The mixture must contain at least 75g of dietary fibre, but no more than 40mg of iron. Use the chart below to determine how many units of sunflower seeds (1 unit is 100g) and how many units of raisins (1 unit is 100g) should be used to minimize the cost of producing this mixture. Dietary fibre (g) 10 g / unit sunflower seeds Dietary fibre (g) 5 g / unit of raisins Iron (mg) 4 mg / unit Sunflower seeds Iron 2mg / unit raisins I’ve worked out a system of e equations, and have tried to graph the problem. But I don’t know…

I'm learning how to use MatLab & Maple. But I wonder if there is a really good, free alternative to these programs. What do you think of SciLab & GNU Octave (vs. MatLab) ? ...or Maxima (vs. Maple or Mathematica) ?

I have been doing a little fooling around with math(yes I realize how uber nerdy that is) and I may have found a Fermat’s last theorem type problem that is: If x is an integer grater then 1 there is no solution for y^2 = x! or possibly(I doubt it) A^b = x! if y, a, and b are integers. Probably one of the larger factorials proves me wrong but I am just putting it out there.

Hello, Is it possible to combine the following functions into one? x := 0.5m ft(z) := if(z<=x, f(z), f(x)+f2(z-x)) f(z) := 850*g*z f2(z) := 1024*g*z ft(z) calculates the static pressure at depth z in a 0.5m layer of oil on top of water. So if z is in the oil the result is just f(z), but if in the water layer, f(0.5)+f2(z-x). My fluids lecturer hinted that this kind of problem can be solved without the use of ifs, but didnt seem eager to expand! I'm wondering if it can be solved through knowledge of the pressure-depth gradients of each fluid... but it's mirky!

Hi all. Looking how to calculate the power developed by an undershot water wheel spun by a laminar flowing river in function of -Flat paddles submerged area, -Speed of river ignoring friction losses. http://commons.wikimedia.org/wiki/Image:Undershot_water_wheel_schematic.svg Thanks.

Hey All, At first watch I was at a loss of how they developed such creative mathematics and architecture. After no sleep I came with this conclusion, if you would be so kind as to scrutinize: In the development of mathematics the progression could be described as having evolutionary traits. As much as we love to learn just for the sake of sating a ferocious curiosity we unfortunately would not get as many grants as the people claiming that their math/physics can better mankind. So our math tends to follow the needs of the time. In the case of western civilization you see an abundance of grid/simple geometric systems from street layout to pipeline. In the case o…

I saw a very interesting thread in the archive on infinity. Why was it archived, which basically killed the thread? I would have replied, but I can't.