Derivative Definition and 1000 Threads

Given a finite-dimensional normed linear space ##(L,\lVert \cdot \rVert)##, is there anything that suggests that at every point ##x_0 \in L##, there exists a direction ##\delta \in L## such that that ##\lVert x_0 + t\delta \rVert \geqslant \lVert x_0 \rVert## for all ##t \in \mathbb{R}##?

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Section 3.1 Partial Derivatives and Directional Derivatives ... I need some help with Example 3 in Chapter 3, Section 1 ... Example 3 in Chapter 3, Section 1 reads as follows:In the above text we read...

Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted ##\phi _ { t } ( p ) .## Now consider $$t \mapsto a _ { t } \left( \phi _ { t } (...

Hello! Do the derivatives change sign under C, P or T transformation. For example, for the photon vector field we have, under C, ##A_\mu \to -A_\mu##. Do we also get ##\partial_\mu \to -\partial_\mu ##? And what about P and T? Thank you!

Homework Statement In calculus, I learn that the derivative of the inverse function is g'(x) = 1/ f'(g(x)) Homework Equations So.. The Attempt at a Solution Can someone give me an example of where I need to know this, or is this just a math exercise. Is there a relatively simple physics...

Is it true that if ##f## is differentiable at ##a## that ##f'(a) = \lim_{h\to a}\frac{f(a+h) - f(a)}{h} = \lim_{h\to a}\frac{f(a-h) - f(a)}{-h}##. That is, can the sign of ##h## be flipped. I've seen this a few times and it seems a bit dubious.

Homework Statement Find the value of h'(0) if: $$h(x)+xcos(h(x))=x^2+3x+2/π$$ Homework Equations Chain Rule Product Rule The Attempt at a Solution I differentiated both sides, giving h'(x)+cos(h(x))-xh'(x)sin(h(x))=2x+3 Next I factored out and isolated h'(x) giving me...

I am writing some automatic differentiation routines for Taylor series, and would like to verify my results for the value and first six derivatives of ##sinh## and ##cosh## evaluated at ##\pi /3##, and also ##tanh##, and ##sec^2##, evaluated at ##\pi / 4##. I have attempted to use this site to...

Homework Statement [/B] Find the directional derivative of the function at the given point in the direction of the vector v. $$g(s,t)=s\sqrt t, (2,4), \vec{v}=2\hat{i} - \hat{j}$$ Homework Equations $$\nabla g(s,t) = <g_s(s,t), g_t(s,t)>\\ \vec{u} = \vec{v}/|\vec{v}|\\ D_u g(s,t) = \nabla...

Hi all, I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit. I'm not a mathematician by training, so there must exist some terminology which...

Homework Statement Find out the quotient derivative i.e. the derivative of polynomial upon polynomial and then find the minima and maxima.[/B] ##W\left(z\right)=\frac{{4z+9}}{{2-z}}## Homework Equations ##\left( \frac{f}{g} \right)' = \frac{f'\,g - f\,g'}{g^2}## The Attempt at a Solution...

I have read that the integral of d3x ∇(ψ*ψ) is zero because the total derivative vanishes if ψ is normalizable. Does this mean that the integral of d3x ∇(ψ*ψ) is ψ*ψ evaluated at the limits where ψ is zero ? Thanks

We define the differential of a function f in $$p \in M$$, where M is a submanifold as follows In this case we have a smooth curve ans and interval I $$\alpha: I \rightarrow M;\\ \alpha(0)= p \wedge \alpha'(0)=v$$. How can I get that derivative at the end by using the definitions of the...

Homework Statement Using the chain rule, find a, b, c, and d: $$\frac{\partial}{\partial x'} = a\frac{\partial}{\partial x} + b\frac{\partial}{\partial t}$$ $$\frac{\partial}{\partial t'} = c\frac{\partial}{\partial x} + d\frac{\partial}{\partial t}$$ Homework Equations Chain rule...

If ##\theta## is angular displacement, does ##\frac{d^2\theta}{dt^2} = (\frac{d\theta}{dt})^2##? Proof?

Homework Statement Take ∂2E/∂t2 E(r,t)=E0cos((k(u^·r−ct)+φ) in which u^ is a unit vector. Homework Equations d/dx(cosx)=-sinx The Attempt at a Solution I had calc 3 four years ago and can't for the life of me remember how to differentiate the unit vector. I came up with...

Homework Statement I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##. Homework Equations $$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$ The Attempt at a Solution [/B] So...

Homework Statement If I have the following expansion f(r,t) \approx g(r) + \varepsilon \delta g(r,t) + O(\varepsilon^2) This means for other function U(f(r,t)) U(f(r,t)) = U( g(r) + \varepsilon \delta g(r,t)) \approx U(g) + \varepsilon \delta g \dfrac{dU}{dg} + O(\varepsilon^2) Then up to...

If ##f'(0) = 0## and ##n## is the smallest natural number such that ##f^{(n)}(0)\neq 0##, then the higher-order derivative test states the following: 1. If ##n## is even and ##f^{(n)}(0)>0##, then ##f## has a local minimum at ##0##. 2. If ##n## is even and ##f^{(n)}(0)<0##, then ##f## has a...

In these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes10.pdf, at the end of page 4, it is mentioned: (3) V(x) contains delta functions. In this case ψ'' also contains delta functions: it is proportional to the product of a...

In Rudin, the derivative of a function ##f: [a,b] \to \mathbb{R}## is defined as: Let ##f## be defined (and real-valued) on ##[a,b]##. For any ##x \in [a,b]##, form the quotient ##\phi(t) = \frac{f(t) - f(x)}{t-x}\quad (a < t <b, t \neq x)## and define ##f'(x) = \lim_{t \to x} \phi(t)##, if the...

Homework Statement if ## f(x) ={\int_{\frac{\pi^2}{16}}^{x^2}} \frac {\cos x \cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz## then find ## f'(\pi)## 2. The given solution Differentiating both sides w.r.t x ##f'(x) = {-\sin x {\int_{\frac{\pi^2}{16}}^{x^2}} \frac{\cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz }+{...

Homework Statement Assume that you want to the derivative of a vector V with respect to a component Zk, the derivative is then ∂ViZi/∂Zk=Zi∂Vi/∂Zk+Vi∂Zi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm Now why is it that I can change m to i and i to j in ViΓmikZm?

If we are representing the basis vectors as partial derivatives, then ##\frac{\partial}{\partial x^\nu + \Delta x^\nu} = \frac{\partial}{\partial x^\nu} + \Gamma^\sigma{}_{\mu \nu} \Delta x^\mu \frac{\partial}{\partial x^\sigma}## to first order in ##\Delta x##, correct? But in the same manner...

When a classical field is varied so that ##\phi ^{'}=\phi +\delta \phi## the spatial partial derivatives of the field is often written $$\partial _{\mu }\phi ^{'}=\partial _{\mu }(\phi +\delta \phi )=\partial _{\mu }\phi +\partial _{\mu }\delta \phi $$. Often times the next step is to switch...

I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$ Any insight would be greatly...

Hi all! I was messing around with the equation for time dilation. What I wanted to do was see how the time of a moving observer ##t'## changed with respect to the time of a stationary observer ##t##. So I differentiated the equation for time dilation ##t'## with respect to ##t##: $$\frac {dt'}...

I'm reading a pdf where it's said that the function ##f: \mathbb R \longrightarrow \mathbb{R}^2## given by ##f(x) = \langle \sin (2 \pi x), \cos ( 2 \pi x) \rangle## is not one-to-one, because ##f(x+1) = f(x)##. This is pretty obvious to me. What I don't understand is that next they say that the...

Hello,I was wondering. Is the exponential function, the only function where ##y'=y##. I know we can write an infinite amount of functions just by multiplying ##e^{x}## by a constant. This is not my point. Lets say in general, is there another function other than ##y(x)=ae^{x}## (##a## is a...

According to David Morin (link: https://books.google.com/books?id=Ni6CD7K2X4MC&pg=PA636), the time-derivative of the Lorentz factor is (##c=1##): ##\dot{\gamma} = \gamma^3 v \dot{v}##, and the four-acceleration: ##\mathbf{A} = (\gamma^4 v \dot{v}, \gamma^4 v \dot{v} \mathbf{v} + \gamma^2...

Could someone explain to me how from this graph you can deduce that ##\tan(\theta) = \frac {df} {dx}##. Thanks

In my classical mechanics course, the professor did a bit of algebraic wizardry in a derivation for one of Kepler's Laws where a second derivative was simplified to a first derivative by taking the square root of both sides of the relation. It basically went something like this: \frac{d^2...

I am aware that the negative derivative of potential energy is equal to force. Why is the max force found when the negative derivative of potential energy is equal to zero?

1. The problem statement, all variables, and given/known data Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##. Homework Equations Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...

Homework Statement I am unsure as to how the partial derivative of the basis vector e_r with respect to theta is (1/r)e_theta in polar coordinates Homework EquationsThe Attempt at a Solution differentiating gives me -sin(theta)e_x+cos(theta)e_y however I'm not sure how to get 1/r.

##\int d^4 x \sqrt {g} ... ## if I am given an action like this , were the ##\sqrt{\pm g} ## , sign depending on the signature , is to keep the integral factor invariant, when finding an eom via variation of calculus, often one needs to integrate by parts. When you integrate by parts, with...

As part of my work, I'm making use of the familiar properties of function minima/maxima in a way which I can't seem to find in the literature. I was hoping that by describing it here, someone else might recognise it and be able to point me to a citation. I think it's highly unlikely that I'm the...

Can someone point me some examples of how the Lie Derivative can be useful in the General theory of Relativity, and if it has some use in Special Relativity. I'm asking this because I'm studying how it's derived and I don't have any Relativity book in hand so that I can look up its application...

Hey! :o Let $I=[a,b]$, $J=[c,d]$ compact intervals in $\mathbb{R}$, $g,h:I\rightarrow J$ differentiable, $fI\times J\rightarrow \mathbb{R}$ continuous and partial differentiable as for the first variable with continuous partial derivative. Let $F:I\rightarrow \mathbb{R}$. I want to calculate...

Homework Statement The problem is attached as pic Homework Equations ∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...

Homework Statement Is it possible to accurately approximate the speed of a passing car while standing in the protected front hall of the school? Task: Determine how fast cars are passing the front of the school. You may only go outside to measure the distance from where you are standing to the...

Homework Statement Let ##f: \mathbb{R} \rightarrow \mathbb{R}## a function two times differentiable and ##g: \mathbb{R} \rightarrow \mathbb{R}## given by ##g(x) = f(x + 2 \cos(3x))##. (a) Determine g''(x). (b) If f'(2) = 1 and f''(2) = 8, compute g''(0). Homework Equations I'm not aware of...

I was taking notes recently for delta y/ delta x and noticed there's more than one way to skin a cat... or is there? I saw the leibniz dy/dx, the triangle of change i was taught to use for "difference" Δy/Δx, and the mirror six ∂f/∂x which is some sort of partial differential or something...

https://en.wikipedia.org/wiki/Euler's_formula (1) eix = cos(x) + isin(x) (2) eixidx = (-sin(x) + icos(x))dx (3) eix = (-sin(x) + icos(x)) / i (4) eix = cos(x) + isin(x) Just lost in circles. Just for fun.. post a solution for x.

For a DEQ like this: y = y( x ) a y'''' + b y''' + c y'' + d y' + f y = g( x ) where a, b, c, d, f are constants. I would think it would be called a "constant coefficient DEQ", but a DEQ like this would also be called this a y y'' + b ( y' )2 = g( x ) but I am only interested in...

Homework Statement Homework EquationsThe Attempt at a Solution I am trying to repair my rusty calculus. I don't see how du = dx*dt/dt, I know its chain rule, but I got (du/dx)*(dx/dt) instead of dxdt/dt, if I recall correctly, you cannot treat dt or dx as a variable, so they don't cancel...

Homework Statement Hi I am looking at part a). Homework Equations below The Attempt at a Solution I can follow the solution once I agree that ## A^u U_u =0 ##. However I don't understand this. So in terms of the notation ( ) brackets denote the symmetrized summation and the [ ] the...

Homework Statement We have the equation for gravity due to the acceleration a = -GM/r2, calculate velocity and position dependent on time and show that v/x = √2GM/r03⋅(r/r0-1) Homework Equations x(t = 0) = x0 and v(t = 0) = 0 The Attempt at a Solution v = -GM∫1/r2 dt v = dr/dt v2 = -GM∫1/r2...

Homework Statement You are applying for a ##\$1000## scholarship and your time is worth ##\$10## an hour. If the chance of success is ##1 -(1/x)## from ##x## hours of writing, when should you stop? Homework Equations Let ##p(x)=1 -(1/x)## be the rate of success as a function of time, ##x##...

Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...