Infinitesimal Definition and 135 Threads

Homework Statement [/B] A Lorentz transformation ##x^{\mu} \rightarrow x'^{\mu} = {\Lambda^{\mu}}_{\nu}x^{\nu}## is such that it preserves the Minkowski metric ##\eta_{\mu\nu}##, meaning that ##\eta_{\mu\nu}x^{\mu}x^{\nu}=\eta_{\mu\nu}x'^{\mu}x'^{\nu}## for all ##x##. Show that this implies...

So the question goes something like this: Now I've already found the solution to the problem, so I don't need any assistance there, and why I'm not posting this in homework help. What I'm having trouble with is visualizing the situation at some instant right before the cat catches the mouse...

I have light incident from plane with velocity v0 into plane with velocity v1. Obviously, according to Snell's law, v0/v1=sin(theta0)/sin(theta1), where theta0 and theta1 are the angles with regard to the vertical line. How to calculate d(theta0)/d(theta1)? There are probably arguments from...

This is my second term in my master's and one of the courses I've taken is QFT1 which is basically only QED. In the last class, the professor said the Klein-Gordon Lagrangian has a global symmetry under elements of U(1). Then he assumed the transformation parameter is infinitesimal and , under...

Homework Statement Say x is an infinitesimal number on the hyperreal line, is this expression finite, infinite or infinitesimal Homework Equations (sqrt(4+x)-2)/x The Attempt at a Solution [/B] My approach so far has been that sqrt(4+x) is (2+y) where y is another infinitesimal and y<x...

Hello, In the context of QFT, I do not understand the statement: ##\frac{\delta \phi(x)}{\delta \phi(y)}=\delta (x-y)## I understand the proof which arises from the definition of the functional derivative but I do not get its meaning. From what I see is generalizes ##\frac{\partial...

Hello, I do not understand how to compute the infinitesimal variation of the field at fixed coordinates; under lorentz transformation . I am doing something wrong regarding the transformation of the ##x## coordinate. I am looking for: ##\Delta_a=\phi_a'(x)-\phi_a(x)##, variation appearing in...

Homework Statement Find the infinitesimal dilation and conformal transformations and thereby show they are generated by ##D = ix^{\nu}\partial_{\nu}## and ##K_{\mu} = i(2x_{\mu}x^{\nu}\partial_{\nu} - x^2\partial_{\mu})## The conformal algebra is generated via commutation relations of elements...

Hi, Could you please explain me why, under the transformation of a complex valued field Φ→eiαΦ, for an infinitesimal transformation we have the following relation? δΦ=iαΦ Thanks a lot

I am wondering how can I find the infinitesimal displacement in any coordinate system. For example, in spherical coordinates we have the folow relations: x = \, \rho sin\theta cos\phi y = \, \rho sin\theta sin\phi z = \, \rho cos\theta And we have that d\vec l = dr\hat r +rd\theta\hat \theta...

Homework Statement I'm reading something in my quantum physics book that says given a wavefunction ψ that is even, if we evaluate its integral from -ε to ε, the integral is 0. How can this be? I thought this is the property of odd functions. Homework Equations ψ=Aekx if x<0 and ψ=Be-kx if x>0...

All throughout calculus texts, the authors have always put conditions on the manipulation of differentials. They say that for the chain rule, the cancellation of the differentials is simply a way to remember the formula. When doing separation of variable for ODEs, texts always say something...

Homework Statement A disk of radius R carries a uniform surface charge density σ. (a) Using the results for the uniformly charged ring given in the lectures, or otherwise, compute the infinitesimal electric field dE due to the circular slice the disk of charge dQ shown in the figure. Express...

Euler was the master in analysisng anything. This can be seen in his words in the preface of his book "Mmathematica" (translated by Ian Bruce), where he speaks on the text of Hermann "Phoronomiam": Euler has given many insightful words on analysisng things in his preface of many other books...

I don't understand what is meant by "derive the formula for finding the volume of a sphere that uses infinitesimals but not the standard formula for the integral"? Is this talking about Gauss or what? I'm completely self taught in calculus and I did three proofs already... the old cylinder /...

Homework Statement So we have infinitesimal transformations from ##q_i## to ##\bar{q_i}## and ##p_i## to ##\bar{p_i}## ( where ##p_i## represents the canonical momentum conjugate of ##q_i##) given by $$\bar{q_i} = q_i + \epsilon \frac{\partial g}{\partial p_i}$$ $$\bar{p_i} = p_i - \epsilon...

Hi I'm reading Elementary calculus - an infinitesimal approach and just wan't to make sure my understanding of what dy, f'(x) and dx means is correct. I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the...

Hello buddies, Could someone please help me to understand where the second and the third equalities came from? Thanks,

I'm trying to derive what ##ds^2## equals to in spherical coordinates. In Euclidean space, $$ds^2= dx^2+dy^2+dz^2$$ Where ##x=r \ cos\theta \ sin\phi## , ##y=r \ sin\theta \ sin\phi## , ##z=r \ cos\phi## (I'm using ##\phi## for the polar angle) For simplicity, let ##cos...

How does the wavefunction propagate for infinitesimal times? Please no Feynman path integral arguments!

Homework Statement Derive the infinitesimal rotation operator around the z-axis. Homework Equations My book gives this equation (which I follow) with epsilon the infinitesimal rotation angle: $$ \hat{R}(\epsilon) \psi(r,\theta, \phi) = \psi(r,\theta, \phi - \epsilon) $$ but I just don't get...

Hello, I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that is moving on a circle with a generic potential. (I used simple polar coordinates in two...

If we have: $$F_{\mu\nu} \rightarrow \cos\alpha F_{\mu\nu} +\sin\alpha \star G_{\mu\nu}$$ $$G_{\mu\nu} \rightarrow \cos\alpha G_{\mu\nu} +\sin\alpha \star F_{\mu\nu}$$ for rotation $\alpha$. If infinitesimal transformation for small alpha one gets $$\delta F_{\mu\nu} = \delta\alpha~\star...

Hi, I am studying path integral formulation from Ballentine. Till equation 4.50, I follow quiet well. G(x,t;x_0,t_0) = \lim_{N \to \infty}\int\ldots\int\left(\frac{m}{2\pi i\hbar\Delta t}\right)^{\frac{N+1}{2}}\exp{\sum_{j=0}^{N}\left(\frac{im(x_{j+1}-x_j)^2}{2\hbar\Delta...

I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between...

I want compute the following integral: $$\\ \int f(x,y) \sqrt{dx^2+dy^2}$$ Is correct this pass-by-pass: $$\\ \sqrt{\left( \int f(x,y) \right)^2} \sqrt{dx^2+dy^2} = \sqrt{\left( \int f(x,y) \right)^2 (dx^2+dy^2)} = \sqrt{\left( \int f(x,y) \right)^2 dx^2 + \left( \int f(x,y) \right)^2 dy^2}...

Helloo, I don't understand how one arrives at the conclusion that the hamiltonian is a generating function. When you have an infinitesimal canonical transformation like: Q_{i}=q_{i}+ \delta q_{i} P_{i}=p_{i}+\delta p_{i} Then the generating function is: F_{2}=q_{i}P_{i}+ \epsilon...

dx as change in distance vs dx as infinitesimal x? Why are they the same notation?Sent from my iPhone using Physics Forums

In K&K's Intro to Mechanics, they kick off the topic of rotation by trying to turn rotations into vector quantities in analogy with position vectors. It's quickly shown, however, that rotations do not commute, making them rather poor vectors. They then show, however, that infinitesimal rotations...

Hello, I've been looking at the derivation of the exponential function, here http://www.statlect.com/ucdexp1.htm amongst other places, but I don't get how, why or what the o(delta t) really does. How does it help? It's really confusing me, and all the literature I've looked at just...

I think you know definition of line infinitesimal: [ds]^2 = \begin{bmatrix} dx & dy & dz \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}^2 \begin{bmatrix} dx\\ dy\\ dz\\ \end{bmatrix} = \begin{bmatrix} dr & d\theta & dz \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0...

Hi, I don't understand why in some texts they put that infinitesimal volume dV = dx dy dz. If V = x y z, infinitesimal volume should be dV = y z dx + x z dy + x y dz, by partial differentiation. Thanks

I tried to verify that the SYM lagrangian is invariant under SUSY transformation, but it turned out there is a term that doesn't vanish. The SYM lagrangian is: \mathscr{L}_{SYM}=-\frac{1}{4}F^{a\mu\nu}F^a_{\mu\nu}+i\lambda^{\dagger a}\bar{\sigma}^\mu D_\mu \lambda^a+\frac{1}{2}D^a D^a the...

Everything I've encountered in physics so far allows infinitesimal numbers to be manipulated as real numbers. But there has been much criticism towards Leibniz's notation, and I assume it is for a reason. When in mathematics will the infinitesimal notation not work? Including treating...

Homework Statement The problem can be found in Jackson's book. An infinitesimal Lorentz transform and its inverse can be written under the form ##x^{'\alpha}=(\eta ^{\alpha \beta}+\epsilon ^{\alpha \beta})x_{\beta}## and ##x^\alpha = (\eta ^{\alpha \beta}+\epsilon ^{'\alpha \beta})...

Is there a treatment of "infinitesimal operators" that is rigorous from the epsilon-delta point of view? In looking for material on the infinitesimal transformations of Lie groups, I find many things online about infinitesimal operators. Most seem to be by people who take the idea of...

Hello everyone, I'm going through some lecture notes and there are some things I don't understand about the whole derivation of the angular momentum multiplet. It's said that the skew-symmetric 3x3 matrices J_i are the infinitesimal generators of the rotation group SO(3). Later, however...

I hope someone can put me on the right track here. I need to derive the infinitesimal generator for a bridged gamma process and have come a bit stuck (its for a curve following stochastic control problem - don't ask). Any tips, papers, books that could guide me out of my hole would be greatly...

We know that we calculate the volume of sphere by taking infinitesimally small cylinders. ∫ ∏x^2dh Limits are from R→0 x is the radius of any randomly chosen circle dh is the height of the cylindrical volume. x^2 + h^2 = R^2 So we will get 4/3∏R^3 Now the question is why cannot we...

My two questions are related to the title. The problematic is: "How can we connect the coordinates ... xα... for α = 0, 1, 2 and 3 to the coordinates of the same object in another frame, say ...yλ for λ = 0, 1, 2, 3 in preserving the quantity η(x)αβ. dxα.dxβ = η(y)λμ. dyλ. dyμ? Can someone...

Homework Statement Let \begin{equation} \delta _{1} \, and \, \delta _{2} \end{equation} be two infinitesimal sypersymmetry transformations on xμ compute \begin{equation} [\delta _{1}, \delta _{2} ]x^{μ}. \end{equation} Homework Equations The commutator is: \begin{equation}...

Write the expression for the infinitesimal radial inertal force dfr "look at pic" Consider the non-uniform rigid disc of Figure 1. The density of the material is variable and depends on the radial position: p = p (r). Its analytical expression is p=po+p(1) and where Po and P(1) are known and...

So Reynold's transport theorem states that \frac{\mathrm d}{\mathrm d t} \int_{V(t)} f \; \mathrm d V = \int_{V(t)} \partial_t f \; \mathrm d V + \int_{V(t)} \nabla \cdot \left( f \mathbf v \right) \; \mathrm d V. Now I would expect (on basis of conceptual reasoning) that if I were to apply...

Homework Statement In order to determine the infinitesimal generators of the conformal group we consider an infinitesimal coordinate transformation: x^{\mu} \to x^\mu+\epsilon^\mu We obtain \partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu=\frac{2}{d}(\partial\cdot\epsilon)\eta_{\mu\nu}...

Consider a point p in a manifold with coordinates x^\alpha and another point nearby with coordinates x^\alpha + dx^\alpha where dx are infinitesimal or arbitrarily small. Suppose we have a function f on this manifold. Then we can write df=f(x^\alpha + dx^\alpha)-f(x^\alpha)=\partial_\mu f...

Homework Statement Show that the heat transferred during an infinitesimal quasi static process of an ideal gas can be written as dQ = (Cv/nR)(VdP) + (Cp/nR)(PdV) where dQ = change in heat Cv= heat capacity while volume is constant n= number of moles of gas R= ideal gas constant Cp=...

I know how to find out if an infinitesimal delta is of the first order (if delta*y/delta*x=dy/dx+epsilon is approximately equal to (dy/dx)*delta*x, but how do you find out if an infinitesimal is of second order or higher and how do you find out what that order is? Note: I will update this...

let's say heat change between system and surrounding, so the process must occur in infinitesimal steps ie, infinitesimal temperatures here. my problem is, how can we causes that to happen this way? i found some sources saying that, this is due to the large size of the surrounding. How are they...

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...