Derivative Definition and 1000 Threads

Hi, if the force is the derivative of potential energy, does it mean that the force is equal to mg and with a constant gravity, it will be the same at any height? But in real life, F (or mg) would be different on the Earth's surface and 400 km above it (~8 m/s^2). So, this formula is used to...

Summary: How do I solve the derivative of (((x^{x})^{x})^{x})... How would I solve this derivative?

How do I compute the second derivative of an one dimensional array?

I need to evaluate ##\nabla_{\mu} A^{\mu}## at coordinate basis. Indeed, i should prove that ##\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})##. So, $$\nabla_{\mu} A^{\mu} = \partial_{\mu} A^{\mu} + A^{\beta} \Gamma^{\mu}_{\beta \mu}$$ The first and third terms...

I’d appreciate some help with a mathematical block that I’m sure is trivial to most of you. Given the expression (1): Take the derivative of E with respect to a, set to zero and solve for a. Answer is shown at the bottom. This is not homework; I’m following an account of the development of...

Hello, I am trying to solve the following problem: If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...

Hello! So, I was beginning to skim Kleppner and Kolenkow for an upcoming course I’m taking over the summer. I saw this on pg. 17 and was wondering if I’m making a silly mistake in understanding what the book is saying. When they take the derivative of the position function, isn’t the velocity...

I originally thought you’d have to use the chain rule to get h’, as in: f’(g(x))*g’(x). Plugging in 1 for x, I got an answer of 10. An online solution, however, said that you only had to get f(g(1)), which was f(-1), then look up f’(-1) in the table. Both approaches seem logical to me, but they...

I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says "We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation...

In a problem of an oscillating electric dipole, under appropriate conditions, one can find, for the potential vector calculated at the point ##\vec{r}##, the expression ##\vec{A}=\hat{k}\frac{\mu_0I_0d}{4\pi}\frac{cos(\omega(t-r/c))}{r}## where: ##\hat{k}## is the direction of the ##z-axis##...

I know that the Action has units Energy·time or Momentum·position. A second fact is that the derivative of the action with respect to time is Energy and similar with momentum-position, consistent with a units ie. dimensions check.Is it a coincidence that both are Noether conserved quantities...

When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?

see attached below; the textbook i have has many errors... clearly ##f_x## is wrong messing up the whole working to solution...we ought to have; ##\frac {du}{dx}=(9x^2+2y)+(2x+8y)3=9x^2+2y+6x+24y=9x^2+6x+26y##

I would like to know how to differentiate |sin(t)| to obtain d(|sin(t)|)/d(t). Thank you!

Lets consider T(\vec{p})=\frac{\vec{p}^2}{2m}=\frac{\vec{p}\cdot \vec{p}}{2m}. Then \frac{dT}{dt}=\vec{v}\cdot \vec{F}. And if we consider T=\frac{p^2}{2m} than \frac{dT}{dt}=\frac{1}{2m}2p\frac{dp}{dt} Could I see from that somehow that this is \vec{v}\cdot \vec{F}?

[Note: Link to the quote below has been pasted in by the Mentors -- please always provide attribution when quoting another source] https://www.feynmanlectures.caltech.edu/I_08.html Let s=16t^2 and we want to find speed at 5 sec. s = 16(5.001)2 = 16(25.010001) = 400.160016 ft. In the last...

i) Let ##\pi : E \rightarrow M## be a vector bundle with a connection ##D## and let ##D'## be the gauge transform of ##D## given by ##D_v's = gD_v(g^{-1}s)##. Show that the exterior covariant derivative of ##E##-valued forms ##\eta## transforms like ##d_{D'} \eta = gd_D(g^{-1}\eta)##. ii) Show...

Hi all, I am currently trying to prove formula 21 from the attached paper. My work is as follows: If anyone can point out where I went wrong I would greatly appreciate it! Thanks.

1. The laplacian is defined such that $$ \vec{\nabla} \cdot \vec{\nabla} V = \nabla_i \nabla^i V = \frac{1}{\sqrt{Z}} \frac{\partial}{\partial Z^{i}} \left(\sqrt{Z} Z^{ij} \frac{\partial V}{\partial Z^{j}}\right)$$ (##Z## is the determinant of the metric tensor, ##Z_i## is a generalized...

We know that the cofactor of determinant ##A##, is $$\frac{\partial A}{\partial a^{r}_{i}} = A^{i}_{r} = \frac{1}{2 !}\delta^{ijk}_{rst} a^{s}_{j} a^{t}_{k} = \frac{1}{2 !}e^{ijk} e_{rst} a^{s}_{j} a^{t}_{k}$$ By analogy, $$\frac{\partial Z}{\partial Z_{ij}} = \frac{1}{2 !}e^{ikl} e^{jmn}...

hello I own mathematica 10.02 it is virtually impossible to solve PDE's ,even with NDSolve,if the initial conditions contain a derivative I write Derivative[1,0] [0,x] == f[x] I mean the first t derivative of u[t,x] for x at t=0 is f[x] I own a book based on Mathematica 10.3 Even if a...

Here is the problem Here is my work on it. I thought I did it correct, but again, was told it was wrong.

Hello, I'm struggling with this for some time. So I have the function: f(x) = sqrt(1 - 1/x) The derivative of this function can be easily calculated. Now we define the function: F(x) = f(x)/f(x + dx) = sqrt(1 - 1/x)/sqrt(1 - 1/(x+dx)) I have a hard time to find F'(x) due to the presence of...

What's the derivative of deformation gradient F w.r.t cauchy green tensor C, where C=F'F and ' denotes the transpose?

I'm not sure if this is the correct forum to post this question, or should I post it in a math forum. But I was looking at some code when I found a 'strange' implementation to compute the derivative of a function, and I wanted to know if any of you has an idea of why such an implementation is...

Consider these two examples: D[ Re[ Exp[ I*t ] ], t ] D[Re[Exp[I*t]],t] /. t-> 0.5 Mathematica seems to get stuck differentiating the "Re[ ]" function after (rather naively) applying the chain rule. This is a trivial example, but we might have a more complicated function defined like...

Let's say we have a function ##M(f(x))## where ##M: \mathbb{R}^2 \to \mathbb{R}^2## is a multivariable linear function, and ##f: \mathbb{R} \to \mathbb{R}^2## is a single variable function. Now I'm getting confused with evaluating the following second derivative of the function: $$ [M(f(x))]''...

Im confused about a certain part of solving an equation. So I used the hyerbola formula to find the answer but I think I did the math wrong. X^2-y^2=c^2 X=1 Y= (2x^5-1)^2 I did the calculations as you can see in the picture but I know I messed up on the square root part. When you square one...

Hi Guys Sorry for the rudimentary post. I am busy with a numerical solution to a mechanics problem, and the results are just not as expected. I am re-checking the mathematics to ensure that all is in order in doing so I am second guessing a few things Referring to the attached scan, is the...

Is the following true? ##\left| \frac{d\vec{u}}{d t} \right| \overset{?}{=} \frac{d |\vec{u}|}{d |t|}##

$$h_x=y$$ $$h_y=x$$ Substituting the coordinates of a given point: $$y'=-\frac {y} {x}$$ $$y'=-\frac {1} {2}$$ A unit vector: $$\frac {1} {\sqrt{5}, \frac {2} {\sqrt{5}}$$ $$D_\vec u h(2,1) = \frac {1} {\sqrt{5}, \frac {2} {\sqrt{5}} \cdot \vec (1,2)$$ $$D_\vec u h(2,1) = \frac {5} {\sqrt{5}}$$

Summary:: According to Yale’s University PHYS: 200: v*(dv/dt) = d(v^2/2)/dt Could someone explain how has he reached that conclusion? He claims to be some standard derivation rules, but I can’t find anything about it. As much as I can tell: (dv/dt)* v = v’ * v = a* v thanks! [Moderator's...

I am trying to find the equations of motion of the angular momentum ##\boldsymbol L## for a system consisting of a particle of mass ##m## and magnetic moment ##\boldsymbol{\mu} \equiv \gamma \boldsymbol{L}## in a magnetic field ##\boldsymbol B##. The Hamiltonian of the system is therefore...

ok this is pretty straightforward to me, my question is on the order of differentiation, i know that: ##\frac {d^2y}{dx^2}=####\frac {d}{dt}.####\frac {dy}{dx}.####\frac {dt}{dx}## is it correct to have, ##\frac {d^2y}{dx^2}=####\frac {d}{dt}##.##\frac {dt}{dx}##.##\frac {dy}{dx}##? that is...

Hello everyone, in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...

I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative 1) for scalar function; $$\nabla_a\phi \equiv \partial_a\phi~~(1)$$ 2) for vectors; $$\nabla_bV^a = \partial_bV^a + \Gamma^a_{bc}V^c~~(2)$$ $$\nabla_cV_a = \partial_cV_a -...

Q: See f(t) in graph below. Does the graph of g have a point of inflection at x=4? There is a corner at x=4, so I don't think there is a point of inflection. Does a point of inflection exist where f''(x) does not exist? The solution says there is a point of inflection, could anyone explain why...

The action considered is \begin{align*} S[\Phi] = \int_M dt d^3 x \sqrt{-g} \left( -\frac{1}{2}g^{ab} \partial_a \Phi \partial_b \Phi - \frac{1}{2}m^2 \Phi^2\right) \end{align*}I can "see" unrigorously that variation with respect to ##\partial_t \Phi(x)## it is going to be\begin{align*}...

Hello, I am trying to calculate the partial derivative of a convolution. This is the expression: ##\frac{\partial}{\partial r}(x(t) * y(t, r))## Only y in the convolution depends on r. I know this identity below for taking the derivative of a convolution with both of the functions only...

This proof has three steps and is very similar to (if not the same as) that other proof I posted here.(1) Prove the existence of a ball centered around ##a## with the property that ##f'## evaluated at any point in the ball is positive. (2) Prove that the right end-point of this ball is bounded...

I know the integral of a P(V) eqn gives an eqn for work. I was wondering if taking the derivative of a P(V) eqn gives an eqn for change in pressure?

Heisenberg equation of motion for operators are given by i\hbar\frac{d\hat{A}}{dt}=i\hbar\frac{\partial \hat{A}}{\partial t}+[\hat{A},\hat{H}]. Almost always ##\frac{\partial \hat{A}}{\partial t}=0##. When that is not the case?

I have some doubts with respect on how the functional derivative for the kinetic energy in density functional theory is obtained. I have been looking at this article in wikipedia: https://en.wikipedia.org/wiki/Functional_derivative In particular, I'm interested in how to get the...

Hi guys, I'm having trouble computing a pass 1 to 106.15. It's in the pictures. So, what a have to do is the derivative of ##f## with respect to time and coordinates. Then I need to rearrange the terms to find the equation 106.15. I am using the following conditions. ##r## vector varies in...

First, we shall mention that it is known that the covariant derivative of the metric vanishes, i.e ##\nabla_i g_{mn} = 0##. Now I want tro prove the following: $$ \nabla_i A_k = g_{kn}\nabla_i A^n$$ The demonstration I encounter takes advantage of the Leibniz rule: $$ \nabla_i A_k = \nabla_i...

[Moderator's note: Thread spun off from previous thread due to topic change.] This thread brings a pet peeve I have with the notation for covariant derivatives. When people write ##\nabla_\mu V^\nu## what it looks like is the result of operating on the component ##V^\nu##. But the components...

Hi all, I am having some problems expanding an equation with index notation. The equation is the following: $$\frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$ I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply. Any hint on this would...

Hi all, I was wondering is if the following partial derivative can be computed without a specific ##u(t,x)## $$\partial_tu\big[(t,x-t\kappa V)\big]$$ I was thinking it can't be done, because we could have $$u_a(t,x)=tx \Rightarrow \partial_tu\big[(t,x-t\kappa...

(a) ##f(x)## is continuous only at ##x=3##: 1- If ##x\in\mathbb Q##, ##f(x)=9## at ##x=3##; around, there is ##\mathbb Q## 2- If ##x\in \mathbb R\setminus \mathbb Q##, this is the set of irrational numbers. Intuitively, if ##x## was in ##\mathbb R##, ##x^2## and ##6(x-3)+9## would meet at...

If the sign on the sign diagram of f" changes from positive to negative or from negative to positive, this means the critical points of f" is non-horizontal inflection of f But what about if the sign does not change? Let say f"(x) = 0 when ##x = a## and from sign diagram of f", the sign on the...