Derivative Definition and 1000 Threads

Homework Statement Given $$L = \left(\nabla\phi + \dot{\textbf{A}}\right)^2 ,$$ how do you calculate $$\frac{\partial}{\partial x}\left(\frac{\partial L}{\partial(\partial\phi / \partial x)}\right)?$$ Homework Equations By summing over the x, y, and z derivatives, the answer is supposed to...

Homework Statement "Suppose ##f:(a,b) \rightarrow ℝ## is differentiable at ##x\in (a,b)##. Prove that ##lim_{h \rightarrow 0}\frac{f(x+h)-f(x-h)}{2h}## exists and equals ##f'(x)##. Give an example of a function where this limit exists, but the function is not differentiable." Homework...

Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...

I'm trying to show that the theta term in the QCD Lagrangian, ##\alpha G^a_{\mu\nu} \widetilde{G^a_{\mu\nu}}##, can be written as a total derivative, where ##\begin{equation} G^a_{\mu\nu} = \partial_{\mu} G^a_{\nu} - \partial_{\nu}G^a_{\mu}-gf_{bca}G^b_{\mu}G^c_{\nu} \end{equation} ##...

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on $$\mathbb{R}^n$$" ... ... I need some help with another aspect of Definition 9.1.3 ... Definition 9.1.3 and the relevant accompanying text read as follows...

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" ... ... I need some help with another aspect of Definition 9.1.3 ... Definition 9.1.3 and the relevant accompanying text read as follows: In the...

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on $\mathbb{R}^n$" I need some help with an aspect of Definition 9.1.3 ... Definition 9.1.3 and the relevant accompanying text read as follows...

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with an aspect of Definition 9.1.3 ... Definition 9.1.3 and the relevant accompanying text read as follows: At the top of the above...

Derivative of a Vector-Valued Function of a Real Variable - Junghenn Propn 9.1.2 ... I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.1.2 ...

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.1.2 ... Proposition 9.1.2 and the preceding relevant Definition 9.1.1 read as follows: In the above...

What function equals the negative derivative of itself? f(x) = -f'(x)

I am continuing to work through Lessons on Particle Physics. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf I am on page 22, equation (1.5.58). The authors are deriving the Hermitian conjugate of the Dirac equation (in order to construct the current). I am able to...

What are the current values for Friedman's scalar and its first derivative with respect to time? Thanks.

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's definition of "differential" ... Kantorovitz's Kantorovitz's definition of "differential" reads as follows...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's definition of "differential" ... Kantorovitz's Kantorovitz's definition of "differential" reads as follows: Is the...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 3 on pages 65-66 ... Kantorovitz's Example 3 on pages 65-66 reads as follows...

What is the covariant derivative of the position vector $\vec R$ in a general coordinate system? In which cases it is the same as the partial derivative ?

Homework Statement Problem 1.7 in Griffiths "Quantum Mechanics" asks to prove $$\frac{d\left \langle p \right \rangle}{dt}=\left \langle -\frac{\partial V}{\partial x} \right \rangle$$ Homework Equations Schrödinger equation The Attempt at a Solution I was able to arrive at the correct...

Homework Statement This is a silly question,but i have a problem.How do we solve derivative of -x using first principle of derivative. I know that if derivative of x w.r.t x is 1 then ofcourse that of -x should be -1. Also it can be solved by product rule taking derivative of -1.x .Homework...

On Riemannian manifolds ##\mathcal{M}## the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0##...

Homework Statement Taking derivative of 3 term product. ## \frac{d}{dx} (3x^3 y^2 y'^2) ## Homework Equations I read that (abc)' = (ab)c' + (bc)a' + (ca)b' The Attempt at a Solution ## 9x^2(y^2y'^2) + 6x^3yy'^2 + 6x^3y^2y' ## is this correct ?

Homework Statement How do we find the derivative of function: y= √[(1-sinx)/(1+sinx)] This is the exercise problem from my textbook. I have not covered chain rule yet. So please you basic derivative rules to solve it. Homework Equations Here is the answer of derivative given in my textbook...

As part of the work I'm doing, I'm evaluating a contour integral: $$\Omega \equiv \oint_{\Omega} \mathbf{f}(\mathbf{s}) \cdot \mathrm{d}\mathbf{s}$$ along the border of a region on a surface ##\mathbf{s}(u,v)##, where ##u,v## are local curvilinear coordinates, and where the surface itself is...

Homework Statement Hi I'm having a trouble with finding min value of given function: f(x) = sqrt((1+x)/(1-x)) using derivative.First derivative has no solutions and it is < 0 for {-1 < x < 1} when f(x) is given for {-1 < x <= 1}. For x = - 1 there is a vertical asymptote and f(x) goes to +...

For the polar equation 1/[√(sinθcosθ)] I found the slope of the graph by using the chain rule and found that dy/dx=−tan(θ) and the concavity d2y/dx2=2(tanθ)^3/2 This is a pretty messy derivative so I checked it with wolfram alpha and both functions are correct (but feel free to check in case...

Homework Statement Find the derivative of ##f(z)=\frac{1.5z+3i}{7.5iz-15}## Homework EquationsThe Attempt at a Solution I had no difficulty using the standard derivative formulas to find the derivative of this function, but the actual result, that the derivative is zero, is confusing. For real...

1. Homework Statement Hi, I have done part a) by using the expression given for the lie derivative of a vector field and noting that if ##w## is a vector field then so is ##wf## and that was fine. In order to do part b) I need to use the expression given in the question but looking at a...

Homework Statement Apologies if this is a stupid question but just thinking about the see-saw rule applied to something like: ## w_v \nabla_u V^v = w^v \nabla_u V_v ## It is not obvious that the two are equivalent to me since one comes with a minus sign for the connection and one with a plus...

In Hartle's Gravity we have the covariant derivative (first in an LIF) which is: ##\nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}## As the components of the tensor ##\bf{ t = \nabla v}##. But, it's not clear which components they are! My guess is that...

Hi, it may be interesting question but what is the d (dy)/dx (y is function of x)? I think it is nearly zero but if it is 0 then how can we prove it?

Homework Statement Use the relation ##\langle \vec e^a, \vec e_b \rangle = \delta^a_b## and the Leibniz rule to give an expression for the derivative of a dual frame vector ##\frac{\partial \vec e_b}{\partial x^a}## in terms of the connexion components. Homework EquationsThe Attempt at a...

Homework Statement $$y = x.\log_e {\sqrt{x}}$$ Homework Equations f(x) = g(x) h(x) f ' (x) = g ' (x) . h (x) - h ' (x) . g(x) The Attempt at a Solution $$y = x .\log_e {\sqrt {x}}$$ $$y '(x) = 1.ln \sqrt{x} + \frac{1}{2} $$ the right answer is $$ y ' = \log_{10} {\sqrt{x}} + \frac{1}{2} $$...

Homework Statement Compute ##\partial_n f## where ##n## is normal to ##f##, and ##f## lies in the ##x-z## plane and is parameterized by $$x(s) = \frac{1}{c} \sin (c s);\\ z(s) = \frac{1}{c} (1-\cos (c s)) $$ Homework Equations ##\partial_n f = \nabla f \cdot \hat n## The Attempt at a Solution...

I forgot where I came across this and why I got so determined to figure it out but I wanted to ask about this d/dx(v^2) business. My question is to solidify my understanding of the chain rule with physics equations (sorry for crap terminology). Therefore, I know I use it and do the math as...

If energy is ihw and p is ihk, can force be written as derivatives of these? Might the fundamental forces just be some patterned change in the change of the wave functions of Dirac's equation? Edit: the title should be "Time derivative of ihk" but I can't edit the title.

Hi all I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts: $$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$ so that $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...

<Moderator's note: Moved from the homework forum.> Homework Statement The following equation is one of a few equations that describe a plasma model. The left hand side is the part I am having trouble with in that I can't seem to visualise what (V dot delV) actually does. My physics teacher has...

I find directional derivatives confusing. For example if there is a change in a direction and if this direction have both x and y components should not the change be calculated as square root of squares, i.e the pythogores theorem? Would you please provide a simple demonstration showing the...

Can someone explain it to me step by step?

We first need to remember that two lines are perpendicular when their gradients multiply to give -1. Now, the gradients of the tangents at the points where $\displaystyle \begin{align*} x = 1 \end{align*}$ and $\displaystyle \begin{align*} x = -\frac{1}{2} \end{align*}$ can be found by...

I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is. Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...

Traditionally, derivatives are taught as a function that have... "Whole" transitions. Take the following example: If you have the function f(x) = x^2, we find that f'(x) = 2x, and that f''(x) = 2. In other words, it has a first and a second derivative. But what would it even mean to take a...

Hello . I have problems with two exercises . 1.\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} } Here, I have to write when this limit will be exist. 2.\lim_{(h,k) \to (0,0) } \frac{2hk}{(|h|^a+|k|^a) \cdot \sqrt{h^2+k^2} } Here, I have to write for which a \in \mathbb{R}_+ this...

Hey, Homework Statement I was working on a kinematics experiment using Tracker to do a video analysis. I obtained a graph of displacement against time for the body under constant acceleration and the software also gives me the rms error between the parabolic trend line and the data points...

Homework Statement The entire problem is in the attached picture. I have been checking and double checking for about an hour, found solutions online which agree with my solution, but I cannot find any answer beside -3.697 m/s which is marked wrong by the computer program. Homework Equations Is...

Hi all I'm having trouble understanding what I'm missing here. Basically, if I write the Ricci scalar as the contracted Ricci tensor, then take the covariant derivative, I get something that disagrees with the Bianchi identity: \begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla...

find the value of $$df^{-1}/dx at $x=f(a)$$ $$f(x)=x^3-6x^2-3$$ $$x \ge 4$$ $$a=3$$ ok the inverse would be $$x=y^3-6y^2-3$$ but don't see how to isolate $y$ or if we need to

Hi, so I am trying to find the derivative of this function. \[ \frac{dP}{dt}=rP(K-P), \] r and K are positive constants describing the natural growth rate and carrying capacity of the population, respectively. I was trying to find the derivative, and I suppose that I am supposed to apply the...

The covariant derivative in standard model is given byDμ = ∂μ + igs Gaμ La + ig Wbμ Tb + ig'BμYwhere Gaμ are the eight gluon fields, Wbμ the three weak interaction bosons and Bμ the single hypercharge boson. The La's are SU(3)C generators (the 3×3 Gell-Mann matrices ½ λa for triplets, 0 for...

Hello, I am dealing with a function f(x) involving various complicated trig functions. I took the first derivative of the function f(x), which is doable, and set it to zero to find the at which value x the function f(x) would be maximum. However, once I have the derivative of f(x), it does not...