Derivatives Definition and 1000 Threads

I am given the following: $$u = (x,t)$$ $$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$$ and $$E = x + ct$$ $$n = x - ct$$ I need to solve for $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}{\partial t^2}$$ using the chain rule.How would I even...

I'm can't seem to figure out how to functionally differentiate a functional such as Z(J)= e^{\frac{i}{2} \int \mathrm{d}^4y \int \mathrm{d}^4x J(y) G_F (x-y) J(x)} with respect to J(x) . I know the answer is \frac{\delta Z(J)}{\delta J(x)}= -i \int \mathrm{d}^4y J(y) G(x-y) but I'm struggling...

Hi all, I have had the following question in my head for quite a while: Thermodynamic potentials written in differential form look like $$dU = TdS - PdV$$ and we can obtain equations for say, temperature by doing the following partial $$T = \frac {\partial U}{\partial S} |_V$$ Does this mean...

This is a calculus of variations problem from Boas chapter 9. I seem to be misunderstanding something with differentiation. Given $$F=(1+yy')^2$$ then $$\frac {\partial F} {\partial y'}=2(1+yy')y$$ and $$\frac {\partial F} {\partial y}=2(1+yy')y' .$$ Now this one I am not so confident...

I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as ' a ' instead of ' a / b ' . ------ b

Hi, I am curious about: $$x^m \delta^{(n)}(x) = (-1)^m \frac {n!} {(n-m)!} \delta^{(n-m)}(x) , m \leq n $$ I understand the case where m=n and m>n but not this. Just testing the left hand side with m=3 and n=4 and integrating by parts multiple times, I get -6. With the same values, the...

At the end of a long proof I came across something in tensor calculus that seems too good to be true. And if something seems too good to be true ... The something is that a second order partial derivative vanishes if one of the parts in the denominator is in the same reference frame as the...

I am reading I am reading Spacetime and Geometry : An Introduction to General Relativity -- by Sean M Carroll and have arrived at chapter 3 where he introduces the covariant derivative ##{\mathrm{\nabla }}_{\mu }##. He makes demands on this which are \begin{align} \mathrm{1.\...

Hi all, I have ran into some mathematical confusion when studying the aforementioned topic. The expression for retarded time is given as $$t_R = t - R/c$$ ##R = | \vec{r} - \vec{r'} |##, where ##\vec{r}## represents the point of evaluation and ##\vec{r'}## represents the source position. I...

Looking at Munkres "Analysis on Manifolds", it says for $A\subset R^n, f: A \rightarrow R^m$ suppose that $A$ contains a neighborhood of $a$. Then $f$ is differentiable at $a$ if there exists an $n$ by $m$ matrix $B$ such that, $\frac{f(a+h)-f(a)-Bh}{\left| h \right|}\rightarrow 0$ as...

Homework Statement Homework EquationsThe Attempt at a Solution I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates. I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am...

Hi forum. I'm trying to prove a claim from Mathematical Analysis I - Zorich since some days, but I succeeded only in part. The complete claim is: $$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists...

Suppose we have to deal with the question : $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}=?\frac{\partial}{\partial y}\frac{\partial}{\partial x}$$ This seems true for independent variables. But if at the end x and y are linked in some way like $$x=f(t),y=g(t)$$ this is no more the...

Homework Statement For the following function ##f(x)=\frac {2x^2} {x^2-4}##, find the following: a) The x and y intercepts b) the horizontal and vertical asymptotes c) the first and second derivatives d) any local maximum or minimum points e) the intervals of increasing and decreasing f) any...

x*abs(y)*(y+x^2+x)=f(x,y) so, on normal points they are tangent vectors on some point in the chosen direction. how about in critical points, where there shouldn't be any on a geometrical standpoint? can i say they exist if i get them with the definition? or the result i get has no value? like...

Hi. In infinite volume a normalizable wavefunction → 0 as r or x,y,z→ 0 but do all the derivatives and higher derivatives → 0 as well ? Thanks

Hello. Glad to meet you, everyone I am studying the [Mathematical Methods for Physicists; A Comprehensive Guide (7th ed.) - George B. Arfken, Hans J. Weber, Frank E. Harris] In Divergence of Vector Field, I do not understand that How to transform the equation in left side into that in right...

I've just learned about the covariant derivatives (##\nabla_i## and ##\delta/\delta t##) and I have a doubt. We should be able to say that $$ J^i = \frac{\delta A^i}{\delta t} = \frac{\delta^2 V^i}{\delta^2 t} = \frac{\delta^3 Z^i}{\delta^3 t} $$ where ##J## is the jolt. This...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.4: The Total Derivative ... ...I need help in order to fully understand Theorem 12.3, Section 12.4 ...Theorem 12.3...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.2: The Directional Derivative ... ...I need help with part of Example 4, Section 12.2 ...Section 12.2, including the...

The answer for derivative of y=5tanx+4cotx is y'=-5cscx^2. But how come on math help the answer is 5sec^2x-4csc^2x? I have a calculus test coming up and I really would appreciate if someone could explain! - - - Updated - - - Oh nvm I see my mistake!

It is mentioned in Reif's book, statistical physics, that trough dimensional analysis it can be shown that: $$\frac{1}{\beta} = kT $$ where ##\beta## equals ##\frac{\partial \ln \Omega}{\partial E}## and k is the Boltzmann constant. I don't quite see how to reach this result, can anyone give me...

The building of theoretical mechanics can be constructed using only the first and the second derivatives (those of coordinates in case of kinematics: velocity and acceleration and those of energy in case of dynamics: force and gradient thereof). It is obviously unavoidable if one wants to deal...

I am integrating the below: \begin{equation} \psi(r,v)=\int \left( \frac{\frac{\partial M(r,v)}{\partial r}}{r-2M(r,v)}\right)dr \end{equation} I am trying to write ψ in terms of M. Please, any assistance will be appreciated.

O'Neill's Elementary Differential Geometry, in problem 3.4.5, asks the student to prove that isometries preserve covariant derivatives. Before solving the problem in general, I decided to work through the case where the isometry is a simple inversion: ##F(p)=-p##, using a couple of simple vector...

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...

I understand the concept of derivatives but when it comes to integrals and their uses I do not understand what they do and where you use them.In derivatives you can understand how a function changes but in integration everything is so illogical.Can someone explain me the use of integrals in...

Homework Statement Show that u(x, y) = y/π ∫-∞∞ f(t) dt / ((x - t)2+y2) satisfies uxx + uyy = 0. Homework Equations Leibniz' Rule The Attempt at a Solution I'm not even sure Leibniz' Rule can be applied here since there seems to be a discontinuity in the integrand when x=t and y=0. When I...

Homework Statement Hello I am given the equation: ut - 2uxx = u I was given other equations (boundary, eigenvalue equations) but i don't think I need that to solve this first part: The book says to get rid of the zeroth order term by substituting u = exp(t)V(x,t). I tried to but I can't find...

Homework Statement This is derivation 2 from chapter 8 of Goldstein: It has been previously noted that the total time derivative of a function of ## q_i## and ## t ## can be added to the Lagrangian without changing the equations of motion. What does such an addition do to the canonical momenta...

Homework Statement When trans-2-methylcyclohexanol is subjected to acid-catalyzed dehydration, the major product is 1-methylcyclohexene. However, when trans-1-bromo-2-methylcyclohexane is subjected to dehydrohalogenation, the major product is 3-methylcyclohexene. The attempt at a solution...

I'm trying to better understand how people refer to symmetry in Physics and Differential Geometry. In "Exterior Differential Systems and Euler Lagrange Partial Differential Equations," by Bryant, Griffiths and Grossman, it seems a vector field is a symmetry of a Lagrangian if the Lie derivative...

Hey! :o I want to verify that $$w(x,t)=\frac{1}{2c}\int_0^t\int_{c(t-\tau)-x}^{x+c(t-\tau)}f(y,\tau)dyd\tau$$ is the solution of the problem $$w_{tt}=c^2w_{xx}+f(x,t) , \ \ x>0, t>0 \\ w(x,0)=w_t(x,0)=0, \ \ x>0 \\ w(0,t)=0 , \ \ t\geq 0$$ For that we have to take the partial derivatives of...

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...

I subtracted the ##\mu##-th component of the Lie Derivative of a Vector ##U## along a vector ##V## from the ##\mu##-th component of the Covariant derivative of the same vector ##U## along the same vector ##V## and I got ##(\nabla_V U)^\mu - (\mathcal{L}_V U)^\mu = U^\nu \partial_\nu V^\mu -...

Find the directions in which the directional derivative of f(x,y) = x^2+ xy^3 at the point (2,1) has the value of 2. What I have done so far which I am not sure how to continue: partial derivative of fx = 2x + y^3 and fy = 3xy^2 gradient vector, <fx,fy> at (2,1) = <5,6> Let u = <a,b>...

So, I'm now studying thermodynamics and our teacher proved some time ago the following mathematical result: If f(x,y,z)=0, then (∂x/∂y)z=1/(∂y/∂x)z But today he used this relation for a function of four variables. Does this result still hold, because I'm not really sure how to prove it. If...

Hello everyone, I am trying to solve this wee problem regarding partial derivatives, and not sure how to do so. The following image shows level curves of some function \[z=f(x,y)\] : I need to determine whether the following partial derivatives are positive or negative at the point P...

Homework Statement A homeowner wishes to enclose a rectangular garden with fencing. The garden will be adjacent to his neighbour’s lot. There will be fencing on all four sides. His neighbour will be paying for half the shared fence. a) What should the dimensions of the garden be if the area is...

A point is moving on the graph of 3x^2 + 4y^3 = xyWhen the point is at P = (1/7, 1/7) its y-coordinate is increasing at a speed of 3 units per second. What is the speed of the x-coordinate at that time and in which direction is the xcoordinate moving?

Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of D&K's notation for directional derivatives ... ... D&K's definition of directional and partial...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of D&K's notation for directional derivatives ... ... D&K's definition of directional and partial...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of D&K's notation for directional and partial derivatives ... ... D&K's definition of directional and...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of D&K's notation for directional and partial derivatives ... ... D&K's definition of directional and...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with another aspect of the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with another aspect of the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read as...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Proposition 2.3.2 ... ... Duistermaat and Kolk's Proposition 2.3.2 and its proof read as...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of Example 2.2.5 ... ... Duistermaat and Kolk's Example 2.2.5 read as follows: In the above text by D&K we...