Derivative Definition and 1000 Threads

I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...

Mod note: Moved from a homework section 1. Homework Statement N/A Homework Equations f(x + Δx,y) = f(x,y) + ∂f(x,y)/∂x*Δx The Attempt at a Solution Sorry this isn't really homework. We were given this equation today in order to derive the Taylor expansion formula in two variables and I'm not...

I see that derivative of y with respect to x is just like the ratio of y over x. But, Why Un (the formula to find nth term) is not the derivative of Sn (the sum of sequence formula) ?? For example, 1 2 5 10 -> y = x2+1 +1...

i'm dumb, sorry

I understand that ψ goes to zero as x goes to infinity. Is it also true that dψ/dx must go to zero as x goes to infinity?

http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx On this page the author makes it very clear that: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ provided ##\frac{dx}{dt} \neq 0##. In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying...

Homework Statement I did an experiment to test the conservation of mechanical energy in an oscillating pendulum. As part of the analysis I had to find the pendulum's vertical position with time using the formula: y = L-sqrt(L^2-x^2) where L was the pendulum's length (L=1 m). Then for the next...

In mathematical parlance, we say "take the derivative of a function f" to indicate that we are computing a new function, which maps slopes, that derives from f. However, in physics, we say "take the derivative of velocity". However, velocity is a quantity, not a function. What does it mean to...

How can I figure out ##\partial_\mu x^2## on the manifold ##(M,g)##? I thought that it should be ##2x_\mu##, but I think I'm wrong and the answer is ##2x_\mu+x^\nu x^\lambda \partial_\mu g_{\nu\lambda}##, right?! In particular, it seems to me, we can't write...

I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary? I'm not sure about it but I believe since the Lie derivative is...

Homework Statement A determinant a is defined in the following manner ar * Ak = Σns=1 ars Aks = δkr a , where a=det(aij), ar , Ak , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δkr is...

I'm struggling to get started with the proof that an open interval D containing x0 exists such that f'(x) ≠ 0 for all x∈D, given f'(x0)≠0. It seems like it should be easy but I've been scratching around for an hour now and have gotten nowhere, could anyone give me some advice to help me along?

Homework Statement Consider the fermionic part of the QCD Lagrangian: $$\mathcal{L} = \bar\psi (\mathrm{i} {\not{\!\partial}} - m) \psi \; ,$$ where I used a matrix notation to supress all the colour indices (i.e., ##\psi## is understood to be a three-component vector in colour space whilst...

Homework Statement for ##0<\alpha,\beta<2##, prove that ##\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]## Homework Equations Mean value theorem: ##f'(c)=\frac{f(b)-f(a)}{b-a}## The Attempt at a Solution I got the answer for the question but I have made an assumption but I don't know if...

Never mind. I got this one. Couldn't figure out how to delete the post though.

Let f(x) = tan(2x) - cot(2x) defined on x∈]0,π/4[ Prove that derivative of f(x) is 16/1-cos(8x) What I did was: 2 * Sin^2(2x) + 2 * Cos^2(2x) / Cos^2(2x) + Sin^2(2x) If I factor the 2, I reach: 2 * (Sin^2(2x) + Cos^2(2x) / 1+cos(4x)/2 + 1-cos(4x)/22 * 1/ 1 = 2? What went wrong?

Hello all, I'm having a minor annoyance in proving an identity. The identity is the following \star\text{d}\star A_p = \frac{(-)^{p(D-p+1)-1+t}}{(p-1)!}\nabla_\mu A^\mu_{\,\, \mu_1 \cdots \mu_{p-1}}\text{d}^\mu_1\wedge \cdots \wedge \text{d}^\mu_{p-1} I'm stuck at the first step of proving...

Homework Statement [/B] Consider the following action: $$\begin{align}S = \int \mathrm{d}^4 z \; \bar\psi_i(z) \, (\mathrm{i} {\not{\!\partial}} - m)_{ij} \, \psi_j(z)\end{align}$$ where ##\psi_i## is a Dirac spinor with Dirac index ##i## (summation convention for repeated indices). Now I would...

Homework Statement a. k(t) = (sqrt(t+1))/(2t+1)b. y = (3^(x^2+1))(ln(2))The Attempt at a Solution For the first problem, I know I use the quotient rule for derivatives (L)(DH)-(H)(DL)/((L)^2) which would go to: ((2t+1)(1/(2sqrt(t+1)) - (sqrt(t+1))(2))/((2t+1)^2) I get stuck here, maybe it's...

Homework Statement Various values of the functions f(x) and g(x) and their derivatives are given in the table below. Find the derivative of f(x+g(x)) at x=0. at x=0 f(x)=5 f'(x)=2 g(x)=1 g'(x)=3 at x=1 f(x)=7 f'(x)=3 g(x)=-2 g'(x)=-5 2. Homework Equations Chain ruleThe Attempt at a Solution...

This stems from considering rigid body transformations, but is a general question about total derivatives. Something is probably missing in my understanding here. I had posted this to math.stackexchange, but did not receive any answers and someone suggested this forum might be more suitable. A...

Hi - I know the final result for the n'th derivative, I am looking though at getting an expression for the 1st derivative of f(z). From $ f({z}_{0}) = \frac{1}{2\pi i} \oint_{c} \frac{f(z)}{z - {z}_{0}}dz $ we get $ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =...

I just came across this in a textbook: ## (\partial_{\mu}\phi)^2 = (\partial_{\mu}\phi)(\partial^{\mu}\phi) ## Can someone explain why this makes sense? Thanks.

i have solved the following one but not sure...anyone give me the solve..i want to be sure.. nth derivative of {e^ax * Sin(ax+b)}

Suppose we have a Kalman filter. We have a position sensor, for example GPS. We use the filter to estimate position. However in all examples I see higher derivatives in the state vector: speed, acceleration and sometimes jerk. There is no sensor that calculates these values directly, so they...

For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?

What does v equals dx/dt mean? I interpret v as: the limiting value as a vanishingly small value for time t (dt) goes to 0. Or lim as dt-->0 of dx/dt.

I need to find the derivative with respect to time of the magnetic flux (dΦB/dt). I have a time of .0085 seconds, and a magnetic flux of .0008 Wb. I am a little hazy on my calc skills.

I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality δ(dq/dt) = d(δq)/dt Where q is some coordinate, and δf is the first variation in...

I am trying to find the derivative of x^x using the limit definition and am unable to follow what I have read. Can someone help me understand why lim [(x+h)^h -1]/h as h ---> 0 = ln(x). This part of the derivatio

Homework Statement Define f(x,y) = x+2y, w = x+y. What is ∂f / ∂w? Homework EquationsThe Attempt at a Solution f = w+y so: ∂f/∂w = ∂(w+y)/∂w = ∂w/∂w + ∂y/∂w = 1 + ∂y/∂w. But I'm really not sure if this is right and if it right so far, I can't figure out what ∂y/∂w should be...

Say I have a position vector p = e(t) p(t) Where, in 2D, e(t) = (e1(t), e2(t)) and p(t) = (p1(t), p2(t))T And if I conveniently point the FIRST base vector of the frame at the particle, I can use: p(t) = (r1(t), 0)T I want the velocity, so I take v = d(e(t))/dt p(t) + e(t) d(p(t))/dt...

Homework Statement $$y=a\cdot x\cdot ln\left(\frac{b}{x}\right)$$ The derivative should be 0 (to maximize), what's x? Homework Equations $$(ln\:x)'=\frac{1}{x}$$ $$(x^a)'=ax^{(a-1)}$$ $$(uv)'=u'v+v'u$$ The Attempt at a Solution $$\dot y=a \left[ ln \left( \frac{b}{x} \right)-x\frac{x}{b}x^{-2}...

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...

Why is Hamiltonian defined as 1st derivative with respect to time ? From the units of energy (kgm2s-2) I would expect it to be defined as 2nd derivative with respect to time. (I'm reading http://feynmanlectures.caltech.edu/III_11.html#Ch11-S2)

So i am studying GR at the moment, and I've been trying to figure out what the derivative (not covarient) of the mixed metric tensor $$\delta^\mu_\nu$$ would be, since this tensor is just the identity matrix surely its derivative should be zero. Yet at the same time $$\delta^\mu_\nu =...

Can relative maximum and minimum points exist when a function is defined at say x=c, however the derivative does not exist or tends to infinity? Ie the graph of. F (x)= |x|, for x=c=o. If I am correct the relative minimum is at o, can it also be the abs minimum? I recalled the theorem by...

Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less...

Using the standard equation of a circle x^2 + y^2 = r^2, I took the first and second derivatives and obtained -x/y and -r^2/y^3 , respectively. I understand that the slope is going to be different at each point along the circle, but what does not make sense to me is that the rate of change of...

Homework Statement Growth is observed for a cubic crystal. Initially the height of the cube is 1 cm . the surface of the cube increases at a rate of 6 cm2 / hour. Question: Calculate dh/dt Homework Equations ds/dt = ds/dh * dh/dt The Attempt at a Solution ds/dt = 6 cm2 / hour ds/dh = h*h =...

According to this link: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx The second derivative test can only be applied if ##f''## is continuous in a region around ##c##. But according to this link...

Homework Statement Suppose we have an equation, ex + xy + x2 = 5 Find dy/dx Homework Equations Now I know all the linear differentiation stuff like product rule, chain rule etc. Also I know partial differentiation is differentiating one variable and keeping other one constant. The Attempt at...

When we obtain the velocity vector for position vector (r, θ, φ) Why do we take the time derivative of the radial part in the 3D Spherical Coordinate system only? Don't we need to consider the polar angle and azimuthal angle part like (dr/dt, dθ/dt, dφ/dt)?

Hello. In the proof of uniqueness of ( multi-variable ) derivative from Rudin, I am a little stuck on why the inequality holds. Rest of the proof after that is clear .

Homework Statement f(x,y)=cos(xy)+ye^{x} near (0,1), the level curve f(x,y)=f(0,1) can be described as y=g(x), calculate g'(0). Homework Equations N/A Answer is -1. The Attempt at a Solution If you do f(0,1)=cos((0)(1))+1=2, do you have to use linear approximation or some other method?

Hi, friends! Let the quantity ##I\boldsymbol{\omega}## be given, where ##I## is an inertia matrix and ##\boldsymbol{\omega}## a column vector representing angular velocity; ##I\boldsymbol{\omega}## can be the angular momentum of a rigid body rotating around a static point or around its -even...

What's the matter: So, I think I have some skills when it comes to differentiation after taking calculus 2 last semester, but when it starts to intertwine with physics, and interpreting physical phenomenon through equations, It appears I could use some help. Anyway, the problem that I got hung...

Hi all, first post here. I'm a junior Physics/Math double major at UMass Amherst, playing with some problems over the summer. I'll get right into it. A rope with constant tension T is deflected through the angle 2\theta_{0} by a smooth, fixed pulley. What is the force on the pulley? It is...

As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation a*(a/b)sin(pi*x) The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can...

Today, I had a class on Complex analysis and my professor wrote this on the board : The Laplacian satisfies this equation : where, So, how did he arrive at that equation?