Derivatives Definition and 1000 Threads

suppose there is a vector field V on a manifold M V generates a flow on M suppose \gamma(t) is an integral curce in this flow now there is another vector field W on M why not define the lie derivative of W with respect to V as the limit of the divide (W(\gamma+\delta...

I am having bit of a problem proving Eq. (0.2): (0.1) \text{ } G_{\omega}(t-t^\prime)=\theta(t-t^\prime) \frac{e^{-i\omega(t-t^\prime)}}{2\omega}+\theta(t^\prime-t) \frac{e^{i\omega(t-t^\prime)}}{2\omega} (0.2) \text { }\left (-\partial^2_{t}-\omega^2 \right ) G_\omega...

I appologise in the lack of distinction between curly d's and infinitesimals! All derivatives are partial and anything outside of brackets is an infinitesimal. also, I sincerely apologise for any dodgy terminology, but I am for the most part self taught (regarding calculus) :/ (also, 0 is my...

I was thinking how do I differentiate the domain of functions... Suppose I have a function: f(x) = \left\{\begin{matrix} x^2 -1, \;\; |x| \leq 1\\ 1 - x^2, \;\; |x| > 1 \end{matrix}\right. And I need to derive it: f'(x) = \left\{\begin{matrix} 2x, \;\; |x| \leq 1\\ -2x...

I'm reading about lateral derivatives... I know that a function is said derivable on a point if the lateral derivative coming from left and the lateral derivative coming from right are both equal at that point. f'(a) exists only if both f'+(a) and f'-(a) exists and are equal... right? Ok...

Is it possible to take a derivative with respect to a function, rather than just a variable? I'll give a simple example of how I imagine such a thing would work to try to explain... \frac{d}{d(sin(x))}\left(sin^2(x)\right) = 2 sin(x) Can you take a derivative this way? Also, can you...

It is common lore to write lagrangians in field theories in the form L(t)=\int d^{3}x\mathcal{L}(\phi_{a},\partial_{\mu}\phi_{a}). Nonetheless, is there any particular reason for doing that? Why do we neglect higher order derivatives? Does it mess around with Lorentz invariance or something...

Confusion with understanding derivatives "in respect to..." I have been teaching myself Calculus for the past 2 weeks or so and I've just barely started learning Implicit Differentiation. There's a few things I have trouble understanding, which this is probably a simple concept that I am...

Homework Statement Find f'(c) and the error estimate for: f(x)= \sqrt{x^{2}+1} Homework Equations The error is given by: E(\Delta x) = \frac {1}{2}M \Delta x and f''(c) \leq M The Attempt at a Solution So the first derivative is: f'(x) = \frac...

Homework Statement See first figure. Homework Equations N/A The Attempt at a Solution See second figure. I defined direction of the line by which the two planes intersect as, \vec{d} and found that the point they are asking about is when, t=1 and I'm stuck here. This...

Homework Statement Find f'(c) and the error estimate for the limit: f'(c) = \lim_{x \to 0^{+}} \frac {f(c+\Delta x) - f(c)}{\Delta x} I just included that to show that we are working with one (right) sided limit the actual problem is: f(x) = \frac {1}{x} \;\; with\;\; c = 3...

Homework Statement Find the Laplacian of F = sin(k_x x)sin(k_y y)sin(k_z z) Homework Equations \nabla^2 f = \left( \frac{\partial}{\partial x} +\frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot \left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} +...

The definitions of a harmonic function u are: It has continuous 1st and 2nd derivatives and it satisfies \nabla^2 u = 0. Is the second derivative equal zero consider continuous? Example: u=x^2+y^2 ,\; \hbox{ 1st derivative }=u_x + u_y = 2x+2y,\; \hbox{ 2nd derivative }=u_{xx} + u_{yy} +...

Homework Statement If f is differentiable at a, prove the following: \lim_{h,k \to 0^+} \frac{f(a+h)-f(a-k)}{h+k} = f'(a) Homework Equations N/A The Attempt at a Solution At the moment, I don't have a complete proof worked out, but I was wondering if someone could comment on the...

Homework Statement If I'm asked to find the acceleration at t=2 s I can just put the X with two dots on top of it parentheses(2 s)? X(2) = what ever I calculate it being equal to that's all I have to put right the X with two dots indicated the derivative of x( distance) the two dots...

When do derivatives in the sense of distributions and classical derivative coincide? Of course f needs to be differentiable. What else? Any reference?

Hi, Everyone: I was never clear n this point: given that z is a single complex variable, how/why does it make sense to talk about z having partial derivatives.? I mean, if we are given, say, f(x,y); R² -->Rⁿ then it makes sense to talk about...

Homework Statement Prove that for all y(x)=ax^2+bx+c where a is a constant !=0 and x is a real number that \frac{y'(x2)^2- y'(x1)^2}{(x2-x1)} = 2y''(x)Homework Equations I don't know what to put here in mathematics but here... y(x)=ax^2+bx+c y'(x)=2ax+b y''(x)=2aThe Attempt at a Solution I...

Hi, As per Clariut's theorem, if the derivatives of a function up to the high order are continuous at (a,b), then we can apply mixed derivatives. I am looking at http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives and I cannot understand in the example for non-symmetry, why the...

Homework Statement Sketch the graph of each function. List the coordinates of where extrema or points of inflection occur. State where the function is increasing or decreasing, as well as where it is concave up or concave down. 9. f(x)=x3-12x Homework Equations The Attempt at a...

If z is complex, the following rules are true, right? \frac{d}{dz}z^n = nz^{n-1} \frac{d}{dz}\ln{g(z)} = \frac{1}{g(z)} \frac{d}{dz}g(z) \frac{d}{dz}e^{g(z)}=e^{g(z)}\frac{d}{dz}g(z) These are of course the same rules as for real variables. When do I need to be careful about...

Homework Statement Could some mathematically minded person please check my calculation as I am a bit suspicious of it (I'm a physicist myself). This isn't homework so feel free to reveal anything you have in mind. Suppose I have two functions \phi(t) and \chi(t) and the potential V which...

Hi there, I am trying to refine my continuum mechanics, which I learned as an enginner. I need to get a better undertstanding of the differences between upper and lowerc onvected derivative, and Jaumann derivative, as well as Lie derivative. I am not far, hopefully, from having gained a...

Hi guys, I would like to ask you where you spot the mistake in the derivatives of the loglikelihood function of the cauchy distribution, as I am breaking my head :( I apply this to a Newton optimization procedure and got correct m, but wrong scale parameter s. Thanks! LLF =...

Given a smooth manifold with no other structure (like a metric), one can define a derivative for a vector field called the Lie derivative. One can also define a Lie derivative for any tensor, including covectors. Incidentally, with antisymmetric covectors (differential forms) one can define...

1. f(x)= ax^2 = ae^TR, nez Prove f '(x) = a(n) x^(n-1)2. n does not equal 03. I don't even understand it

Let f (x) = (x^2 − 1)^n . Prove (by induction on r) that for r = 0, 1, 2, · · · , n, f^ (r) (x)(the r-th derivative of f(x)) is a polynomial whose value is 0 at no fewer than r distinct points of (−1, 1). I'm thinking about expanding f(x) as the sum of the (n+1) terms, then it's easier to...

Hi I've just been learning about how to get first derivatives implicitly and I think I'm getting it. Then the book comes onto calculating second derivatives implicitly and I don't know how to handle the dy/dx terms you might have in your equation from the first implicit differentiation...

Consider the partial dierential equation, (y4-x2)uxx - 2xyuxy - y2uyy = 1. We will make the substitution x = s2 - t2 and y = s - t, to simplify (a) Solve for s and t as functions of x and y the farthest point i got to was x = s^2 - t^2 = (s+t)(s-t) = y(s+t) y = s - t s+t = x/y i...

I heard that in classical field theory, terms in the Lagrangian cannot have more than two derivatives acting on them. Why is this? In quantum field theory, I read somewhere that having more than two derivatives on a term in the Lagrangian leads to a violation of Poincare invariance. Is this...

Hi all I hve two Q I want the explaine how to solve Q1 :(A) Find an equation for tangent to curve y = X^3 - 4X + 1 at the point (2,1) (b ) What is the range of values of the curve's slope Number ( A ) I can solve it but ( B) I face problem to solve...

Q Find all older derivatives y = X^4/2 = 3/2X^2 -X y4 = 2X^2 - 3 x ^ 1 - 0 y4 = 4X^1 - 3 -0 y4 = 7

Not sure I understand exactly what this question is asking. This is obviously a volume in R3 and so how do you get a tangent inside a volume? Or is it just along the plane y = 2 intersecting the volume? Also, what is a parametric equation...? Thanks for the help: Question: The ellipsoid 4x^2...

My question revolves around the following derivative: for z(x,y) *sorry I can't seem to get the latex right. ∂/∂x (∂z/∂y) What I thought about doing was using the quotient rule to see what I would get (as if these were regular differentials). So, I "factored out" the 1/∂x, then did...

Hi I was just reading about that total derivatives in the Lagrangian does not give any contributions in perturbation theory but that they can play role in non perturbative regimes. But there was no statement WHY that is so? Does anyone have an idea and reading advices? I have the most...

Homework Statement express (\frac{\partial u}{\partial s})_{v} in terms of partial derivatives of u(s,t) and t(s,v) Homework Equations The Attempt at a Solution I'm pretty stuck with this problem. I know that dv = (\frac{\partial v}{\partial s})_{t} ds + (\frac{\partial...

Homework Statement f(x) = 5e^(2x+1) Homework Equations Chain rule, power rule and constant multiplies rule The Attempt at a Solution f(x) = 5e^(2x+1) = 5f(x) e^(2x+1) f(u) = e^x f'(u) = e^x g(x)= 2x+1 g'(x) = 2 5f'(x) = 2e^2x+1 =10e^2x+1 Is that...

1. fxy 2. fyx Are the above 2 derivatives equal, in general. Please explain if you know the answer. Regards, -sgsawant

In the midst of https://www.physicsforums.com/showthread.php?t=403002", I came upon a rather interesting, though probably elementary, question. Analagous to the fundamental theorem of calculus, is there a formula or theorem concerning the expression \frac{\partial}{\partial...

Homework Statement An object is traveling along a linear path according to the equation s(t) = 4t^3 - 3t^2 + 5 where t is measured in seconds and s(t) measured in meters. 1. How fast is the object moving at t = 4 seconds? 2. What is the position of the object when it stops...

Homework Statement Sketch the graph of x ^ (4/9) * e ^ (-x) Homework Equations None. The Attempt at a Solution My y' = -x ^ (4/9) * e ^ (-x) ( 1 - 4/9x ^ 1/9). I keep on getting a reaaally long derivative for y'' and thus cannot place it on my sign table. Could someone please...

Homework Statement Find the slope of the tangent line to the curve of intersection of the vertical plane x - y + 1 =0 and the surface z = x2+y2 at the point (1, 2, 5) Homework Equations Gradients, Cross products The Attempt at a Solution I'm pretty lost here. I think I have to...

learning calculus here. got differential calculus, though it is a little foggy, and most of integral calculus, which is a little foggier. also using very unpolished precalc background, though i did give most of it a once-over. i have many questions which i can't think of, but of the top of my...

Hello, I have a function in discrete domain f:\mathbb{Z}\rightarrow \mathbb{R}, and I assume that f is an approximation of another differentiable function g:\mathbb{R}\rightarrow \mathbb{R}. In other words f(n)=g(n), n\in \mathbb{Z}. When one wants to approximate the first derivative of g...

Homework Statement lim (e^(7x)-1)/x^2 x-->0 The Attempt at a Solution I typed in "does not exist" and it was wrong.

When I am taking a partial derivative of an equation with respect to theta_dot, then theta is constant, right? What if I am taking partial derivative with respect to theta, will theta_dot be constant? In this case, theta_dot = omega (angular velocity), but I must keep equation in terms of...

Hey I have been asked to find the first and second derivatives of lx-al-lx+al I have, for the first derivative got, sign(x-a)-sign(x+a) and for the second, i have: 2(delta)(x-a)-2(delta)(x+a) am i right in both cases? I also have to draw them 'schematically' how do i do this?

Not a homework question, but It will help me none the less, In my book it says \frac{d}{dt} \int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx is equivalent to \int_{-\infty}^{\infty} \frac{\partial}{\partial t}|\Psi(x,t)|^2 dx I understand how It becomes a partial derivative, since I'm...

Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian...

Homework Statement a) (f ° g)′(−2) = ? b) (g ° f)′(2) = ? Homework Equations f(−2) = −3, g(−2) = −4, f(2) = 3, g(2) = −3, f ′(−2) = −1, f ′(−4) = −2, f ′(2) = 5, g ′(−2) = 1, g ′(2) = 2, g ′(3) = −4. The Attempt at a Solution I have no idea how to do it every thing I've tried...