Derivative Definition and 1000 Threads

Hello there, Recently I encountered a type of covariant derivative problem that I never before encountered: $$ \nabla_\mu (k^\sigma \partial_\sigma l_\nu) $$ My goal: to evaluate this term According to Carroll, the covariant derivative statisfies ##\nabla_\mu ({T^\lambda}_{\lambda \rho}) =...

Hello, I wish to verify that the following pair ofcurves meet orthogonally. \[x^{2}+y^{2}=4\] and \[x^{2}=3y^{2}\] I recognize that the first is a circle, and the second contains 2 lines (y=1/3*x and y=-1/3*x). I thought to get an implicit derivative of the circle, and to compare it to the...

It is well known that the product rule for the exterior derivative reads d(a\wedge b)=(da)\wedge b +(-1)^p a\wedge (db),where a is a p-form. In gauge theory we then introduce the exterior covariant derivative D=d+A\wedge. What is then D(a ∧ b) and how do you prove it? I obtain D(a\wedge...

I am currently enrolled in a statistics course, and the following is stated in my course book with no attempt at an explanation: Suppose that f is the probability density function for the random variable (X,Y), and that F is the distribution function. Then, f_{X,Y}(x,y)=\frac{\partial^{2}...

If we consider function ##z=z(x,y)## then ##dz=(\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy##. If ##z=const## then ##dz=0##. So, (\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy=0 and from that \frac{dx}{dy}=-\frac{(\frac{\partial z}{\partial...

Homework Statement Homework Equations $$F(x)=\int_a^x f(x),~~F'(x)=f(x)$$ The Attempt at a Solution In F'(x), x is at the end of the domain a-x, so, in my function ##~\cos(x^2)~## i also have to take the end of the domain, and it's 2x, so F'(x)=cos(4x2), but it's not enough. The answer is...

Show that the derivative of $\ln(x)$ is $1/x$.

Homework Statement f(x) = Log(cosh(x-1)), find f'(x). Homework EquationsThe Attempt at a Solution f'(x) = [1/cosh(x) - 1] * d/dx [cosh(x) - 1], => f'(x) = sinh(x) / [cosh(x) - 1] Although, my marking scheme says the answer should instead be; cosh(x) + 1 / sinh(x). Can someone explain where...

I'm studying boundary layers. I am confused by what I am reading in this book. The book says the friction force (F) per unit volume = $$\frac{dF}{dy}=\mu\frac{d^2U}{dy^2}$$ They say $$\frac{dU}{dy}=\frac{U_\infty}{\delta}$$ This makes sense to me, delta is the thickness in the y direction...

Prove that if f is a differentiable function on R such that f(1) = 1, f(2) = 3, f(3) = 3. There is a c $$\in$$ (1 , 3) such that f'(c) = 0.5 I think the mean value theorem should be used, but I can't figure out how to prove such value exists

Homework Statement I have given two graphs which i try to show in the picture here. The question into find u'(1) and v'(5) Homework Equations So the relevant equations here are the Product Rule and the Quotient Rule, which I know and is not the big problem here. I think (but imnot sure) the...

On my exam, we had to find the derivative of 4^x. This is what I did Y=4^x lny=xln4 y=e^xln4 and then finding the derivative for that I got, (xe^(xln4))/4 My professor said that it was wrong and even after I told her what I did to get the answer. She told me the answer was (4^x)ln4 . Which I...

Homework Statement In our physics course, we were studying one dimensional waves in a string. There, our teacher stated that the kinetic energy in a small piece of a string is dK=\frac{1}{2}μdx\frac{\partial y}{\partial t}^2 were μ is the linear density of the string, so he claimed that...

Prelude Consider the convolution h(t) of two function f(t) and g(t): $$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$ then we know that by the properties of convolution $$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$ Intermezzo We also know that...

Hi. I have this problem with differentiating vectors. Primarily because I have no experience at all (or whatsoever) about differentiating vectors. I am at a total loss here. I even have no idea regarding the difficulty of this (thus the [ I ] prefix). Please help me. How did the two equations...

As far as I understand it, the Lorentz factor ##\gamma(\mathbf{v})## is constant when one transforms between two inertial reference frames, since the relative velocity ##\mathbf{v}## between them is constant. However, I'm slightly confused when one considers four acceleration. What is the...

https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-b-implicit-differentiation-and-inverse-functions/session-15-implicit-differentiation-and-inverse-functions/MIT18_01SCF10_Ses15b.pdfso derivative of arctan is 1/(x^2+1) and this is obvious...

I have a very fundamental question about the linear variational method (Huckel theory). It says in any textbook that the variational method provides energy upper bound to the actual energy of a wavefunction by using test wavefunction. \varepsilon = \frac{\sum_{i,j}^{n}C_{i}C_{j}H_{ij}...

In quantum mechanics, the velocity field which governs phase space, takes the form \begin{equation} \boldsymbol{\mathcal{w}}=\begin{pmatrix}\partial_tx\\\partial_tp\end{pmatrix} =\frac{1}{W}\begin{pmatrix}J_x\\J_p\end{pmatrix}...

Hi PF! When doing a force balance in fluid mechanics, ##\sum F = D_t(mV)##. This equation typically results to the Navier-Stokes equations. I'm reading a the following problem: For small particles at low velocities, the first (linear) term in Stokes’ drag law implies ##F = kV##, where ##k##...

I was reading a research paper, and I got stuck at this partial differentiation. Please check the image which I have uploaded. Now, I got stuck at Equation (13). How partial derivative was done, where does summation gone? Is it ok to do derivative wrt Pi where summation also includes Pi...

Is the derivative of a function everywhere the same on a given curve? Or is it just for a infinitesimally small part of the curve? Thank you for the answer.

Hi, With respect to derivative notation... d/dx(y) //1st derivative of y w.r.t x d/dx (dy/dx) = d^2y/dx^2 //2nd derivative d/x (d^2y/dx^2) = d^3y/dx^3 //3rd derivative If you continue finding derivatives in this way, why do the d's increment in the...

I have the following equations: \left\{ \begin{array}{l} x = \sin \theta \cos \varphi \\ y = \sin \theta \cos \varphi \\ z = \cos \theta \end{array} \right. Assume \vec r = (x,y,z), which is a 1*3 vector. Obviously, x, y, and z are related to each other. Now I want to calculate \frac{{\partial...

Hey! :o I want to find the first and second derivative of the function $$\psi (\lambda )=f(\lambda x_1, \lambda x_2)$$ where $f(y_1, y_2)$ is twice differentiable and $(x_1, x_2)$ is arbitrary for fix. I have done the following: $$f(g(\lambda), h(\lambda)) : \\...

Hello PF, 1. Homework Statement I've been having problems with the deriative of a function, although I thought I've done everything right, my solution doesn't match with the right solution. I have no clue what (or if) I've done anything wrong, or simply don't know the tricks I was supposed to...

Homework Statement Question has been attached to topic. Homework Equations Chain rule. The Attempt at a Solution $$\frac {dy}{dt} . \frac{dt}{dx} = \sqrt{t^2+1}.cos(π.t)$$ $$\frac{d^2y}{dt^2}.(\frac{dt}{dx})^2 = 2 $$ $$\frac{d^2y}{dt^2}.(t^2+1).cos^2(π.t)= 2 $$ and for the t=3/4...

As mentioned in the section on complex functions (The Pantheon of Derivatives - The Direction), the main parts of defining a differentiation process are a norm and a direction. So to extend the differentiation concepts on normed vector spaces seems to be the obvious thing to do. Continue...

I want to gather the various concepts in one place, to reveal the similarities between them, as they are often hidden by the serial nature of a curriculum. There are many terms and special cases, which deal with the process of differentiation. The basic idea, however, is the same in all cases...

Homework Statement Take the Covariant Derivative ∇_{c} ({∂}_b X^a) Homework Equations ∇_{c} (X^a) = ∂_c X^a + Γ_{bc}^a X^b ∇_{c} (X^a_b) = ∂_c X^a_b + Γ_{dc}^a X^d_b - Γ^d_{bc} X^a_d The Attempt at a Solution Looking straight at ∇_{c} ({∂}_b X^a) I'm seeing two indices. However, the b is...

I know that ##D_{\vec{v}} f = \nabla f \cdot \vec{v}## is the directional derivative. My question is why must the vector ##\vec{v}## be a unit vector? I am sure there is an obvious answer, but my book doesn't really explain it.

Quick question. I know that if we have a curve defined by ##x=f(t)## and ##y=g(t)##, then the slope of the tangent line is ##\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}##. I am trying to find the second derivative, which would be ##\displaystyle \frac{d}{dx}\frac{dy}{dx} =...

Homework Statement How to find absolute min/max of f(x)=x^3+3x^2-24x+1 on[-1,4] I need to find the absolute min and absolute max. Homework EquationsThe Attempt at a Solution I first took the derivative, reduced, and set it to equal 0 to find crit numbers. x^2+2x-8=0 Factored. (x+4)(x-2) = 0...

I've been thinking about it since yesterday and have noticed this pattern: We have, the first order derivative of a function ##f(x)## is: $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$ The second order derivative of the same function is: $$f''(x)=\lim_{h\rightarrow...

Homework Statement From differentiating Planck's distribution and setting it equal to 0 I've reached the equation below. But now I'm asked to estimate the solution for a/λ. It's suggested that we try to do it graphically/trial and error as it's tricky to do analytically. I'm wondering how I...

Homework Statement The problem is in the attached file. The part I need a little help with is part b. Homework Equations and attempt at a solution[/B] For part a, I got h(8) = 2, h'(6) = -2, and h''(4) = -2. For part c, I found that the integral from 0 to 5 is 7, so I multiplied 7 by 7 to...

Homework Statement Hi guys, I am having real trouble with the function 10ii) I can take the derivatives, but I feel like I am missing something, with what I have done. I set $f_x=0$and $f_y=0$ but really can't seem to find away to solve, i keep getting (0,0) which when I plug into wolfram it...

Homework Statement f(z)=2x^3+3iy^2 then it wants f '(x+ix^2) The Attempt at a Solution So I take the partial with respect to x and i get 6x^2 then partial with respect to y and I get 6iy, then I plug in x for the real part and x-squared for the imaginary part, then I get f '...

Homework Statement The problem asks to find g'(2), g''(2), and g'''(4). Homework Equations and attempt at solution[/B] The derivative of g(x) is just the function f(x). So g'(2) = f(2) = -2. I'm not sure how to find g''(2) and g'''(4). I understand that g''(2) is f'(2), but how do I find...

So let's assume an object moves along a straight line relative to some fixed origin. Clearly we can study this motion with the help of a position vs. time graph which shows how the position varies as time goes on. Now, as far as I understand, the slope of this graph at any time t gives the...

If p is a function of x which is a function of t and you evaluate delta_p/delta_t as delta_t goes to zero, it should be possible that delta_p/delta_t equals delta_p/dx (or dp/dx) before reaching dp/dt. Is it possible to find an expression for t where this happens? Hm.. maybe when t = x^-1(dx) ...

Homework Statement Find the differential Homework Equations Chain rule : dy/du=dy/du*du/dx Product rule: f(x)g'(x) + g(x)f'(x) The Attempt at a Solution I have tried to move the radical to the top of the equation by making it into an exponent (x^2+1)^-1/2. I then used the product rule and the...

Homework Statement Hi guys, I am have a problem with the question displayed below: [/B] Its 6.1 ii) I am really not sure how I am suppose to approach this. I am new to partials, so any advice would be great. Homework EquationsThe Attempt at a Solution So far I have: $$\frac{\partial ^2...

Homework Statement Suppose ##f(x) = x^5 + 2x + 1## and ##f^{-1}## is the inverse of function f. Evaluate ##f^{-1}(4)## solution: 1/7 Homework Equations ##(f^{-1}(x))=\frac{1}{f'(f^{-1}(x))}## The Attempt at a Solution I attempted to use my calculator's solve function to get the solution of...

Homework Statement Hello, I need help finding the derivative. The question wants me to find the equation of the tangent line to the curve y=\dfrac{6}{1+e^{-x}} at point (0, 3). I'm unclear on when to use the chain rule at certain areas. Homework Equations Product Rule: f(x)g'(x)+f'(x)g(x)...

I'm have trouble understanding a fundamental question of a derivative. So a derivate gives me a tangent line at any given point on a function. this makes sense for me for a function y=x^2 because the derivative is y'=2x which is a straight line function. But what about y=x^3 where the...

Homework Statement Our textbook, Fundamentals of Complex Analysis, (...) by Saff Snider says on page 135 that by choosing some suitable branch for the square root and the logarithm then one can show that any such branch satisfies the equation below. The homework/task is to find all such branch...

Homework Statement [/B] Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is: Pn = Pn-1-√Pn-1 Homework Equations [/B] Considering that the...

Homework Statement Given ## d \vec r = dr \hat r + r d \theta \hat {\theta} + r \sin \theta d \phi \hat {\phi}.## Find ## d \hat r , d \hat {\theta} , d \hat {\phi}. ## Homework Equations I know that ## d \hat {e_j} = \omega^i_j \hat {e_i} ## and that ## \omega_{ij}=- \omega_{ji} ## and ## 0 =...

Hi. I want to solve \frac{\partial x^{\nu}}{\partial x^{\mu} + \xi ^{\mu}}, knowing that \frac{\partial x^{\nu}}{\partial x^{\mu}} = \delta ^{\nu}_{\mu}. How can I do this?