Infinitesimal Definition and 143 Threads

While it's pretty easy to derive the infinitesimal version of the special conformal transformation of the coordinates: x'^{\mu}=x^{\mu}+c_{\nu}(x^{\mu} x^{\nu}-g^{\mu \nu} x^2) with c infinitesimal, how does one integrate it to obtain the finite version transformation...

I've already post this, but I've done it in the wrong section! So here I go again.. I've a doubt on the way the infinitesimal volume element transfoms when performing a coordinate transformation from x^j to x^{j'} It should change according to dx^1dx^2...dx^n=\frac{\partial...

When i develop integrals, changing the coordinates (cartesian-> polar for example), i always forget how to write infinitesimal surface or volume. Is there a sort of rule to derivate it? I mean, an intuitive way to remember it, not the mathematical derivation. (another thing: I've the same...

if I'm not mistaken, an infinitesimal is .000000...1 is this true?

Homework Statement Prove that the infinitesimal transformation generated by any dynamical variable g(q,p) is canonical. Homework Equations q' = q + e{q,g} p' = p + e{p,g} where e is some small number. The Attempt at a Solution Demanding that {q',p'} = 1 and that {q,q'} = {p,p'}...

What is dx? Wikipedia defines it as an infinitesimal change in x. Is Wikipedia correct? Assuming it is, what is not to say that if we halve a piece of string an infinite number of times, we shouldn't end up with zero? In that sense, d has to be zero!? More importantly, why is pdx =...

I have the folowing continuum mechanics problem which I can't solve: The unit elongations at a certain point on the surface of a body are measured experimentally by means of strain gages that are arranged at 60° in the direction of 0°, 60° and 120°. Coordinate system is rectangular Cartesian...

I have the folowing continuum mechanics problem which I can't solve: The unit elongations at a certain point on the surface of a body are measured experimentally by means of strain gages that are arranged at 60° in the direction of 0°, 60° and 120°. Coordinate system is rectangular Cartesian...

What does infinitesimal change in V mean? Can someone please illustrate with simple example. Lecture notes say infinitesimal change in V = dV And large change in V is delta V.. I don't understand what it means though

Hey guys, can anyone recommend me a book on infinitesimal calculus. What i mean by infinitesimal calculus is the derivation of differential equations by looking at infinitesimal changes (dx\,dt, etc...). Examples of this would be the Black-Scholes PDE, Euler-Bernoulli DE, etc.. I'm an EE...

We pretty much do derivations maybe 80% of the time in my Intermediate Mechanics class. I'm having a bit of trouble seeing the various infinitesimal areas or volumes when incorporating that into an infinitesimal mass and density equation in our gravitational chapter we're in right now. Is there...

Hello, I've just recently stumbled upon this forum, in search for an answer for my little dilemma, so I hope someone can help me. This is the question: Given that dx is an infinitesimal interval on set R, does it mean that it has infinitly many points in itself as well? If I understood...

I managed to show that the flux through an infinitesimally small cube equals the divergence of the vector field at that point. I also managed to show that the circulation around an infinitesimally small square equals the component of the curl perpendicular to that square at that point. Should...

Hi I have a question regarding unitary operators: If an infinitesimal operation (such as a rotation) is unitary does this guarantee that a finite transformation will also be unitary? thanks M

How can we proved that the infinitesimal angular displacement is a vector mathematically ? Or~how can we prove that a non-infinitesimal angular displacement is not a vector ?

Hello I am trying to understand the "infinitesimal arc-length square." So (ds)^2=(dx)^2+(dy)^2+(dz)^2. What does this means? And then what does (ds)^2=(dx)^2+(1+x^2)(dy)^2 -2x(dy) +(dz)^2 mean? And how does this apply to a space?

Is there a word that relates to infinitesimal in the way that zero relates to infinity?

Hi all. I have here a reference with a representation of the Lie algebra of my symmetry group in terms the fields in my Lagrangian. In order to calculate Noether currents, I would like to use this representation to derive formulae for the infinitesimal forms of the symmetry transformations...

Hello, in order to numerically solve a physics problem I think I need to add 2 (infinitesimal) rotations of one and the same segment each around a different point in 2D space in one iteration of numeric approximization. How does this addition work out? Is it the sum of the vectors connecting...

Is there an actual infinitesimal in the way that there is an actual infinity. Or would zero fill that role.

I read that http://img196.imageshack.us/img196/1705/71301190.png I am not so sure about the last term. Shouldn't it be http://img10.imageshack.us/img10/3962/88484785.png instead?

I have questions about the infinitesimal Lorentz transformation. but specifically about index manipulations. \Lambda^{\mu}_{}_{\nu}=\delta^{\mu}_{\nu}+\delta\omega^{\mu}_{}_{\nu} where \delta\omega^{\mu}_{}_{\nu} << 1 as found in many textbooks, we substitute this into...

I'm currently taking several physics courses (mechanics, thermodynamics etc) and common to them all is their frequent use of infinitesimals. I'll just give a short recap of how I was taught calculus, and this is how my math teacher would word it: [calculus training] \frac{dy}{dx} is not a...

When I was in calculus, the notion of the infinitesimal, the smallest possible unit, was really emphasized. When I switched into a more theoretical section of calculus (analysis in R^n), nothing about infinitesimals is said, instead all the proofs are in the style of epsilons and deltas...

Homework Statement The scale transformation is a continuous transformation which acts on a function f(x) according to D_{s}f(x) = f(sx) where s is a real number. There is a continuous family of such transformations, including the identity transformation corresponding to s = 1...

Hi all, I have been working on this problem for the past 2 days and can't seem to get it. I have gotten parts of the answer but not the answer. Could someone explain this to me? The answer is sqrt(2) - 1 or 1 / ( 1 + sqrt(2) ) Problem: H is infinity sqrt(H+1) / ( sqrt(2H) + sqrt(H-1) )...

Homework Statement In spherical polar coordinates, the infinitesimal displacement ds is given by: ds^2 = dr^2 + r^2 d\theta ^2 + r^2 \sin \left( \theta \right)^2 d\phi ^2 Can I find the volume of a sphere using ds? The Attempt at a Solution I know the spherical volume-element is given...

Hi folks, I have a question concerning the infinitesimal generator of a stochastic ;process, more specificaly of Brownian motion. Let X_t be a stochastic process, then the infinitesimal generator A acting on nice (e.g. bounded, twice differentiable) functions f is defined by...

When we compute scattering amplitude \mathcal{M}, using a coupling constant \lambda, and a cut-off energy \Lambda, it turns out that if \lambda is constant, then \mathcal{M}\to\infty when \Lambda\to\infty. The idea of renormalization seems to be, that we relate some physical coupling constant...

Homework Statement The book I am using (Zwiebach on page 66) uses the expression dF_v = T_0 \frac{ \partial{y}}{\partial{x}} |_{x+dx} -T_0 \frac{ \partial{y}}{\partial{x}} |_{x} for the force on an infinitesimal length of string. We assume dy/dx is much less than 1. I am not sure how the...

Homework Statement The book I am using (Zwiebach on page 66) uses the expression hello dF_v = T_0 \frac{ \partial{y}{\partial{x}} |_{x+dx} -T_0 \frac{ \partial{y}{\partial{x}} |_{x} Homework Equations The Attempt at a Solution Homework Statement Homework Equations The Attempt at a...

Homework Statement Can anyone help me to proceed with this? If we execute rotations of 90* about x-axis and 90* about y axis-what is the resulatant rotation matrix?Will the result commute if we rotate by changing the order?Will they commute if infinitesimal rotations are considered...

I know this isn't technically special or general relativity, but I'm posting this here since, hopefully, people in this forum will be familiar with the question! Suppose the metric tensor is form-invariant under the transformation x\rightarrow\tilde{x}, so we require...

Hey! Can it be concluded generally that: \sum_r dx_r = 0 ...because we are summing an infinitesimaly small variable a finite number of times, in contrast to an integral which is an infinite sum of infinitesimaly small variables? In one of my books a probability is given by: p_r...

Hi everybody, I have one question about integrals. I know the definition of an indefinite or definite integral but I am not sure I understand the notation. The indefinite integral of a function f:R->R (assuming that it exists) is noted like this \int f(x)dx Is the notation f(x)dx a...

PLEASE help asap with quantum physics! Hi there, i need help in a couple of questions that I'm just stumped one of them : A) use induction to show that [ x (hat)^n, p(hat) sub "x" ] = i (hbar)n x(hat)^(n-1) - so far I've figured out this equation is in relation to solve the above eq...

I thought this may be of some interest. http://arxiv.org/abs/hep-th/0505124 Authors: D. V. Ahluwalia-Khalilova Comments: 17 pages [This essay received an "honorable mention" in the 2005 Essay Competition of the Gravity Research Foundation.] Report-no: ASGBG/CIU Preprint: 29.03.2005A...

Hi everybody, I am trying to get addtionnal data on "infinitesimal numbers" dx. I am not sure about the terminology, I have heard it a long ... long time ago during a lecture (my memory may be wrong, so may be I was sleepind and it was during a dream? :rolleyes: ). I think (memory) that...

If you have two particles that are even billions of light years away from each other, is there any gravitational pull between then? (Considering the possibility that there is nothing else in the universe)

What IS Infinity? Reference: http://www.pbs.org/wgbh/nova/archimedes/contemplating.html http://www.pbs.org/wgbh/nova/archimedes/infinity.html

If each spacetime point p_i can be associated with a contant force f_i then the interaction \sum_{i=1}^\infty f_i between points can be described with the use of orthogonal forces.

LIM stands for Local Infinitesimal Motion. LIM is motion of two exclusive space points at the local infinitesimal region of space. The metric can be theorized to be smaller than the Planck length of 10^{-33} cm . It is known that all fermions possesses a magnetic moment. The existence of...

Any smooth connected 1 dimensional manifold is diffeomorphic either to the circle, or to some interval of real numbers. Take a line segment of length 1. It is one dimensional. A-------B Find the midpoint of the line segment and rotate it into 2 dimensions A | | |------B Each...