Derivative Definition and 1000 Threads

Proof: By definition of derivative, $$f'(a) = \lim_{x\rightarrow a}\frac{f(x) - f(a)}{x - a}$$ exists and is finite. Let ##(x_n)## be any sequence that converges to ##a##. By definition of limit, we have $$\lim_{x_n\rightarrow a} \frac{f(x_n) - f(a)}{x_n - a} = f'(a)$$. By definition of...

I'm reading a book on Classical Mechanics (No Nonsense Classical Mechanics) and one particular section has me a bit puzzled. The author is using the Euler-Lagrange equation to calculate the equation of motion for a system which has the Lagrangian shown in figure 1. The process can be seen in...

Suppose: - that I have a function ##g(t)## such that ##g(t) = \frac{dy}{dt} ##; - that ##y = y(x)## and ##x = x(t)##; - that I take the derivative of ##g## with respect to ##y##. One one hand this is ##\frac{dg}{dy} = \frac{dg}{dx}\frac{dx}{dy} = \frac{d^2 y}{dxdt}\frac{dx}{dy}##. On the other...

If $f(x)=7x-3+\ln(x),$ then $f'(1)=$ $a.4\quad b. 5\quad c. 6\quad d. 7\quad e. 8$ see if you can solve this before see the proposed solution

I've got here so far, but first of all I'm not sure if i did it right till the last line and second, if I've been right, i do not know what to do with the rest. should i consider each of levi-civita parentheses in the last line zero? and one additional question about the term in the first line...

The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.

I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***): $$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...

I'm having a hard time understanding how exactly to evaluate the expression} $$\partial_t \mathcal{T}\exp\left(-i S(t)\right)\quad \text{where}\quad S(t)\equiv\int_{t_0}^tdu \,H(u) .$$ The confusing part for me is that if we can consider the following: $$\partial_t \mathcal{T}\exp\left(-i...

If $(x+2y)\cdot \dfrac{dy}{dx}=2x-y$ what is the value of $\dfrac{d^2y}{dx^2}$ at the point (3,0)? ok not sure of the next step but $\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$

In cosmology we have a scale factor that depends only on time ##a(t)##. Now how can I solve this thing $$\frac{d}{da}(\dot{a}(t)^{-2}) = ?$$ Is it 0 ? Or something else ?

I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors. What people usually do is take the covariant derivative of the covector acting on a vector, the result being a scalar Invoke a product rule to...

My attempt so far: $$\begin{align*} (\nabla_X Y)^i &= (\nabla_{X^l \partial_l}(Y^k\partial_k))^i=(X^l \nabla_{\partial_l}(Y^k\partial_k))^i\\ &\overset{2)}{=} (X^l (Y^k\nabla_{\partial_l}(\partial_k) + (\partial_l Y^k)\partial_k))^i = (X^lY^k\Gamma^n_{lk}\partial_n + X^lY^k{}_{,l}\partial_k)^i\\...

I tried solving this question a few ways and this one logically made the most sense however I got it wrong and I am unsure of why. I first plugged in t=2 into p(t). p(2)=0.3(2)1/2+6.3 to obtain 6.724264069. I then found the derivative of D(p) which is D'(p)=-60000/p3. I plugged in...

I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth. F - [Force of gravity on rocket] = dp/dt F - (G * m_e * m_r / r2 ) = v * dm/dt + ma F = Force created by fuel (at time t) G = Gravitational Constant m_e = Mass of Earth m_r = Mass of...

Using product rule, we have: [d/dx] (πr^2)(h) = (πr^2)(1 ) + (2πr)(h) Why is the two there? V = 2 πrh+2πr^2 The derivative of h is 1, not 2. Please help!

Ok not sure if I fully understand the steps but presume the first step would be divide both sides deriving$$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$$offhand don't know the correct answer $\tiny{from College Board}$

Summary: The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector 'b'. Hello, I have the following Problem. The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector...

My theory is that dx = 1(x^0) = 1, which would mean d/dx(x^2) = 2(x^1)/1(x^0) = 2x/1 = 2x. I know that the derivative is literally the change in "y" over change in "x," but am confused as to what value the change in "x" has.

If I take the exponential function e^t and take the derivative, I think I get the same e^t. Even if I keep doing it over and over, second, third derivative, etc. My admittedly naive question, though, is this symmetric? Meaning...if I take the the integral of e^t, do I just get the reverse or...

In one of my textbooks about quantum mechanics, they mention a vehicle moving in a straight line along the x axis. With Newtons first law they take the second derivative from a which is d^2x/dt^2 and that should be equal to -∂V/∂x. What exactly does -∂V indicate? The complete equation...

Sean Carroll says that if we have metric compatibility then we may lower the index on a vector in a covariant derivative. As far as I know, metric compatibility means ##\nabla_\rho g_{\mu\nu}=\nabla_\rho g^{\mu\nu}=0##, so in that case ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu##. I can't see...

How are Savitzky–Golay second derivative different from Savitzky–Golay smoothing?

Matt and Hugh play with a tennis ball and a brick. Then they do some working out to derive the formula for the centripetal force (a = v^2/r) by differentiati...

Summary: Please see the attached problem and solution The answer is 1/5. I have tried various solutions and cannot get 1/5. What is my error? [Moderator's note: Moved from a technical forum and thus no template.]

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ... I need help in order to fully understand a remark of Apostol in Section 16.1 ... The particular remark reads as follows: Could someone...

I calculated an expression for the derivative of the inverse tan but I did not use the identity as suggested. Why did I need to use this identity. Did I do the problem correctly? I got the correct answer. I tried to do the derivative of the inverse sin the same way. I used the same figure 1 on...

I tried to find the derivative of the function V(P)= k/P which I found to be: V'(P) = kP-1 V'(P) = (1)(-1)(P)-1-1 = -1/(P2) And then I substituted in 1.30 into the derivative to obtain -0.5917 L/atm. And I am kind of confused how to actually find the derivative of this. I thought I was on...

My question is shown in Summary section. Please see the attached file.

I appreciate that this is perhaps a strange question but it's been bugging me a little. For instance, velocity is defined as the time derivative of position, so will always appear as the gradient a graph of x vs t. However, something like resistance as R = V/I is defined in terms of a ratio...

I don't understand my textbook, so I simply don't understand how to solve this math problem. I really appreciate some help!

So I just wanted to see if anyone could offer some suggestions. So in my mind this seems impossible, in the case of electric field a jump in time derivative of that field would indicated in my mind that electric charge was either introduced or removed from the system instantaneously which would...

Please help me understand this line from P&S, or point me towards some resources: Why is there another Lorentz transformation acting on the derivative on the RHS? Thanks

I'm currently watching lecture videos on QFT by David Tong. He is going over lorentz invariance and classical field theory. In his lecture notes he has, $$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##. He mentions he uses active...

Hello PF, here’s the setup: we have a geodesic congruence (not necessarily hypersurface orthogonal), and two sets of coordinates. One set, ##x^\alpha##, is just any arbitrary set of coordinates. The other set, ##(\tau,y^a)##, is defined such that ##\tau## labels each hypersurface (and...

Dear all, I'm having a small issue with the notion of Lie-derivatives after rereading Carroll's notes https://arxiv.org/abs/gr-qc/9712019 page 135 onward. The Lie derivative of a tensor T w.r.t. a vector field V is defined in eqn.(5.18) via a diffeomorphism ##\phi##. In this definition, both...

Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w)) Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform,d/dw (F{x(t)}), in terms of regular Fourier transform,X(w)=F{x(t)}. Definition Based Solution (not good enough): from...

So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. When I see ##\frac {\partial v^{\alpha}}{\partial x^{\beta}} + v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}##, I pretty easily see a...

PS: This is not an assignment, this is more of a brain exercise. I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition...

I hope I'm asking this in the right place! I'm making my way through the tensors chapter of the Riley et al Math Methods book, and am being tripped up on their discussion of geodesics at the very end of the chapter. In deriving the equation for a geodesic, they basically look at the absolute...

As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...

206 (day of year number) If $f(x)=\sin{(\ln{(2x)})}$, then $f'(x)=$ (A) $\dfrac{\sin{(\ln{(2x)}}}{2x}$ (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ (C) $\dfrac{\cos{(\ln{(2x)}}}{2x}$ (D) $\cos{\left(\dfrac{1}{2x}\right)}$ Ok W|A returned (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ but I didn't understand why...

I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error: I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative \begin{align}...

I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me? ##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\ I(R_j) =...

Many have probably seen an example of a function that is continuous at only one point, for example ##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, &...

Hi, let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##. In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a...

I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy. In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero...

We have a periodic function ##f: \mathbb{R} \rightarrow \mathbb{R}## with period ##T, f(x+T)=f(x)## The statement is the following: $$\frac{1}{T}\int_0^T f(x)dx =0 \implies \frac{1}{T}\int_0^T\frac{d}{dx} f(x)dx =0$$ Can you give me a hint on how to prove/disprove it? The examples I tried all...

Can someone clarify the use of semicolon in I know that semicolon can mean covariant derivative, here is it being used in the same way (is expandable?) Or is a compact notation solely for the components of?

Let's say I have a function whose derivative is (tan(x)-sin(x))/x. It is not defined for X=0 but as X approaches 0 the derivative approaches 0, so should I conclude that my function is not differentiable at X=0 or should I conclude that the derivative at X=0 is 0.

If given a function ##u(x,y) v(x,y)## then is it correct to write ##\frac{\partial }{\partial x}u(x,y)v(x,y)=\frac{u(x+dx,y)v(x+dx,y)-u(x,y)v(x,y)}{dx}##??