Derivative Definition and 1000 Threads

The derivative for the parametric equations ##x=f(t)## and ##y=g(t)## is given by ##\frac{dy}{dx}=\frac{\Big(\frac{dy}{dt}\Big)}{\Big(\frac{dx}{dt}\Big)}## The proof of the above formula requires that ##y## be a function of ##x##, as seen in...

Homework Statement To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0## where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)## Homework Equations I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...

I can's understand the fact about the equation i can't prove the equation from the first attachment to the second attachment pls help. Sorry for bad english

For a harmonic function of a complex number ##z##, ##F(z)=\frac{1}{z}##, which can be put as ##F(z)=f(z)+g(\bar{z})##and satisfies ##\partial_xg=i\partial_yg##. But this function can also be put as ##F(z)=\frac{\bar{z}}{x^2+y^2}## which does not satisfy that derivative equation! Sorry, I...

I have a derivative of a function with respect to ##\log \left(r\right)##: \begin{equation*} \frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2...

(V(s)_{||})^\mu = V(s)^\mu + s \Gamma^\mu_{\nu \lambda} \frac{dx^\nu}{ds} V(s)^\lambda + higher-order terms (Here we have parallel transported vector from point "s" to a very close point)Hi, I tried to make some calculations to reach the high-order terms for parallel transporting of vector...

Homework Statement Currently I am doing a problem of finding Wigner function for Fock States, while I got this derivative, I found the derivative, but I am not sure my answer is correct. please verify whether my answer is correct or not. $$ \frac{∂^2n}{∂β^n∂(β*)^n} exp(-|β|^2-4|α||β|) $$ β...

I am trying to prove that the above is true when performing the change of variable shown. Here is my attempt: What I am not quite understanding is why they choose to isolate the partial derivative of ##z## on the right side (as opposed to the left) that I have in my last line. This ultimately...

Homework Statement Hi everyone, I'm currently working on year 12 maths and am able to answer questions in the maths book for the various rules of differentiation (chain, product, quotient) and can determine which questions should be answered using which rules. But in the maths book, the...

hi, I tried to take the covariant derivative of riemann tensor using christoffel symbols, but it is such a long equation that I have always been mixing up something. So, Could you share the entire solution, pdf file, or links with me? ((( I know this is the long way to derive the einstein...

Production function Q(K,L) without equation However partial derivatives are given Partial derivatives: Q(K,L) = (K^2 - KL + L^2)/(K+L) + 4K . ln(K+L) Derivative to K Q(K,L) =( K^2 + L^2) / (K+ L) Dervative to L A. Calculate the derivative in point (10,L) If I am correct...

This is (should be) a simple question, but I'm lost on a negative sign. So you have ##D_m V_n = \partial_m V_n - \Gamma_{mn}^t V_t## with D_m the covariant derivative. When trying to deduce the rule for a contravariant vector, however, apparently you end up with a plus sign on the gamma, and I'm...

Given a polynomial ##f(x)##. Suppose there exists a value ##c## such that ##f(c)=f'(c)=0##, where ##f'## denotes the derivative of ##f##. Then ##f(x)=(x-c)^mh(x)##, where ##m## is an integer greater than 1 and ##h(x)## is a polynomial. Is it true? Could you prove it? Note: The converse is true...

In Carroll, the author states: \nabla^{\mu}R_{\rho\mu}=\frac{1}{2} \nabla_{\rho}R and he says "notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due to metric compatibility." I'm not seeing this very clearly :s What's the reasoning...

Homework Statement In the problem, I should provide proof for the statement, where ##f^{(n)}(x)## denotes the ##n##th derivative of the function ##f(x)##: $$ f(x)g^{(n)}(x) = \sum_{k=0}^n (-1)^k \binom{n}{k} \frac{d^{n-k}}{dx^{n-k}} \left[ f^{(k)}(x)g(x) \right] $$ Homework Equations The...

Homework Statement Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)## Homework Equations ##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ## ##D_vU^ρ = ∂_vU^ρ +...

OK, I lied a bit. It's not JUST the derivative of an integral. It's the derivative of a cosine of an integral. Solving the problem of the motion of a simple pendulum under a gravitational field using the lagrangian, I came into this mess (which I don't know if it's right)...

Hi! As the title says, what is the derivative of a matrix transpose? I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one). Any advice?

Hello, I am trying to calculate the following integral: \[\int \frac{f'(x)}{f^{2}(x)}dx\] I suspect is has something to do with the rule of f'(x)/f(x), with the ln, but there must be more to it than that. can you assist please ? Thank you !

Homework Statement [/B] Consider the equation z=6x8ln(x) where z and x are functions of t.If dx/dt=5 when x=e calculate dz/dt. Homework Equations [/B] Do I have to rearrange the equation to do this?The Attempt at a Solution

Hey there! 1. Homework Statement I've been given the operators a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator: a=x+\partial_x and a^\dagger=x-\partial_x with...

I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions: 1) Wald first defines...

I came across a simple equation in classical mechanics, $$\frac{\partial L}{\partial \dot{q}}=p$$ how to derive that? On one hand, $$L=\frac{1}{2}m\dot{q}^2-V$$ so, $$\frac{\partial L}{\partial \dot{q}}=m\dot{q}=p$$ On the other hand...

When doing calculus, we typically say that we "take the derivative of a function ##f(x)##." However, rigorously, ##f(x)## is not a function but rather the value of the function ##f## evaluated at ##x##. Thus, in order for this wording to be correct shouldn't we have to write something like...

I have an equation that looks like At, r = Aφ, r If I know that Aφ = r4 , then how do I find At ? I believe that the above equation is equivalent to: ∂/∂r (At) = ∂/∂r (Aφ) , correct? Then substitute the value Aφ and we have ∂/∂r (At) = ∂/∂r (r4) And then to get At I take the integral on...

What is the derivative (with respect to t) of $\displaystyle \begin{align*} y = 16\,\left[ \sinh{(7\,t)} \right] ^3 \cosh{(7\,t )} \end{align*}$? One way to do this is to apply the product rule. To do this, we need to know the derivative of each factor. $\displaystyle \begin{align*}...

consider x is between the interval [a,b] would it be correct to say that the derivative of a definite integral F(x) is f(x) because as dx approaches zero in (x + dx), the width of ALL "imaginary rectangles" would closely resemble a line segment which approximates f(x)? therefore change in area...

Homework Statement As part of a problem, I need to derive the EOM for a generalized Lagrangian. Before I get there, I'm trying to refresh myself on exactly how these derivatives work because the notation is so bizarre. I am trying to follow a simple example I found online: Start with...

Homework Statement http://imgur.com/MSkNkno Homework EquationsThe Attempt at a Solution I know that I would have to take the derivative of H(x) which is G(x)+G'(x)x so then I would need G'(x) which I figured would be f'^-1(x) but I'm not sure about that. Doing that I got a value of 16 which...

So this is a problem for microeconomics, but should follow under general calculus: The point is x=(x1(p1,p2,u),x2(p1,p2,u)) where u is a constant on the function u(x1,x2). p1 is the price of x1 and p2 for x2. I'm supposed to show that (dx1/dp2)=(dx2/dp1). I've been given the info that for the...

Homework Statement If ##\lim_{x\rightarrow a} \frac{f(x^3)-f(a^3)}{x-a} = -1##. then f'(1) = ... A. -1 B. -1/3 C. 1/3 D. 1 E. 2 Homework Equations [/B] ##f'(x) = \lim_{h->0}\frac{f(x+h)-f(x))}{h}##The Attempt at a Solution [/B] I really have no idea about this problem. Please help me.

I am trying to prove the following: $$3d\sigma (X,Y,Z)=-\sigma ([X,Y],Z)$$ where ##X,Y,Z\in\mathscr{X}(M)## with M as a smooth manifold. I can start by stating what I know so it is easier to see what I do wrong for you guys. I know that a general 2-form has the form...

I do not get some points of this proof about the time derivative of a unit vector $\hat{u}$ (costant magnitude) which is following a precession motion. The picture is the following. I want to prove that $$\frac{d\hat{u}}{dt}=\vec{\Omega}\wedge \hat{u}.$$ I'm ok with almost all the proof except...

The formula T = -(ħ/2m)∇2 implies that T is proportional to the second spatial derivative of a wavefunction. What is the origin of this dependence? In classical mechanics, T = p2/2m. Is it also the case in classical mechanics that p2/2m is proportional to a second spatial derivative? I...

I am trying to explain to someone the formal notion of a limit of a function, however it has made me realize that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any...

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...

For a conservative force \vec{F}=-\vec{\nabla} U \implies dW=-\vec{\nabla}U \cdot d\vec{s} Where d\vec{s} is the infinitesimal vector displacement. Does the following hold? -\frac{\partial U}{\partial \vec{s}}=-\vec{\nabla} U \cdot d\vec{s}=d W, i.e. the infinitesimal work is minus the...

$$ƒ = b^n$$ $$ b,n,I ∈ ℤ $$ Condition: Upon choosing a base value b.. $$ n | b^n ≤ I $$ (n is determined based off the value of b to yield the highest ƒ without going over I) $$1<b<L , L<<I$$ where I is some large number, and L is also sufficiently large such that we want to avoid going...

I understand that the derivative of a function ##f## at a point ##x=x_{0}## is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where ##\Delta x## is a small change in the argument ##x## as we "move" from ##x=x_{0}## to a neighbouring...

Homework Statement Suppose ##D_if(P) = 2## and ##D_jf(P) = -1##. Also suppose that ##D_uf(P) = 2 \sqrt{3}## when ##u = 3^{-1/2} \hat i + 3^{-1/2} \hat j + 3^{-1/2} \hat k##. Find ##D_vf(P)## where ##v = 3^{-1/2}(\hat i + \hat j - \hat k)##. Homework EquationsThe Attempt at a Solution...

So to find the x values of the stationary points on the curve: f(x)=x3+3x2 you make f '(x)=0 so: 3x2+6x=0 x=0 or x=-2 Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2) so: 6(0)+6=6 6(-2)+6=-6 so the maximum has an x value of -2 and the minimum has an x value...

Homework Statement Page 16 (attached file) \frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ \frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ H = mean curvature of surface Σ A = the second fundamental of Σ ν = the unit normal vector field along Σ φ = the scalar field on three manifold M φ∈C^{∞}(Σ)...

Hi For a sphere: x = r*cos(a)*sin(o) y = r*sin(a) z = -r*cos(a)*cos(o) where r is radius, a is latitude and o is longitude, the directional derivative (dx,dy,dz) is the jacobian multiplied by a unit vector (vx,vy,vz), right? So i get: dx = cos(a)*sin(o)*vx - r*sin(a)*sin(o)*vy +...

I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a connection and how it relates to the covariant derivative. As I understand it a connection ##\nabla...

https://www.physicsforums.com/attachments/5396 I assume the derivative cancels the intregal but the $x^2$ ?

Homework Statement I am designing a MIMO communication system, with input signal s, channel H and transform matrix T. The received signal is corrupted by noise. Homework Equations [/B] The received signal is r = Hs+n And then it is transformed (compressed) by: y = Tr And then its...

Homework Statement ∂z/∂x of ycos(xz)+(4xy)-2z^2x^3=5x[/B] Homework Equations n/a The Attempt at a Solution ∂z/∂x=(5+yz-4y+6z^2x^2)/(-yxsin(xz)-4zx^3)[/B] Is this correct? Just trying to make sure that's the correct answer. I appreciate the help. I can post my work if need be. Thanks

I'm very interesting in functions of the nature: f(x) = x^{x} f(x) = x^{x^{x}} and so on. I believe these are called tetrations? Regardless, I sought to generalize the nth derivative of f(x)=x^x and it is proving to be difficult. First I tried just repeatedly differentiating until I could...

Hello, Given a Lie group G and a smooth path γ:[-ε,ε]→G centered at g∈G (i.e., γ(0)=g), and assuming I have a chart Φ:G→U⊂ℝn, how do I define the derivative \frac{d\gamma}{dt}\mid_{t=0} ? I already know that many books define the derivative of matrix Lie groups in terms of an "infinitesimal...

Hey all, for a function approximation program t run fast enough i need to solve for where the function (represented by a NDDP) is at a minimum (necessary trust me), althogh I have no idea how to go about differentiating it, i tried to break it up from its's general formula (the pi operators and...