"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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Using Feynman's Subscript Notation

I have a homework problem that wants me to calculate the force $\vec{F} = \vec{\nabla}_{\vec{X}}U + \frac{\mathrm{d}}{\mathrm{d} t} \left(\vec{\nabla}_{\dot{X}} U\right)$ where $U(\vec{X}, \dot{X}, ...
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154 views

Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm ...
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1answer
21 views

How to find out the asymptotic behavior of a Bessel function?

How can one find out the asymptotic behavior of a Bessel function? If we start from $z= 0$, we can get a Taylor series. But in physics, we have to know the asymptotic behavior of the solution at both ...
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1answer
59 views

Metric and Convariant Tensor

$g_{ij}$ is the metric tensor. Show that $g^{ij}$ which satsifies $g_{ij}g^{jk}=\delta_i^k$ is a covariant tensor of rank $2$. I am not sure how to show this? Does it instead mean to show that ...
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122 views

Find $t$ in $i = 50\sin\left(120\pi t -\frac{3\pi}{25}\right)$ where $i = 25$

An alternating current generator produces a current given by the equation $$i = 50\sin\left(120\pi t - \frac{3\pi}{25}\right)$$ where $t$ is the $\text{time}$ in $\text{seconds}$. (Q) Find ...
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55 views

Addition in linear vector spaces

In the definition of linear vector spaces, one of the axioms is that the addition must be commutative and associative. The addition of scalars and matrices are both commutative and associate. Can ...
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2answers
43 views

Center of mass Double Integral?

Can you help me with this problem? Find the center of mass of a lamina whose region $R$ is given by the inequality: $$|x|+|y|\le 1,$$ and the density in the point $(x,y)$ is : ...
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68 views

Calculate center of mass multiple integrals

Can you help me with this problem? Find the center of mass of a lamina whose region R is given by the inequality: and the density in the point (x,y) is : The region r is this one: Is this the ...
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1answer
153 views

Determining when these two waves separate

There's probably something really obvious I should be getting, but I haven't yet developed the intuition for working with the wave equation. Suppose we're given the wave equation $u_{tt} = c^{2} ...
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1answer
37 views

No non zero solution to E.V.P in $L^p$

Can you show that: If for some $1\leq p\leq \infty$ function $f\in L^p(\mathbb{R}^n)$ solves $\Delta f-\lambda^2 f=0$ then $f\equiv 0$. (This is essentially uniqueness of solution to homogenous ...
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61 views

Taking a stationary phase approximation of a multidimensional integral

I'm looking for a way to take a stationary phase approximation of an integral of the following form: $$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - ...
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27 views

A ball that is thrown upward.

A ball is thrown upwards from a point on the ground, with an initial velocity $v_0$. the ball is affected by earth's gravity, and force of fraction with air that depends on the velocity of the ball. ...
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2answers
17 views

Question about acceleration equation that is derived from place equation.

$a=4$ ${\bf R}(t)=7\sin(at)\hat{{\bf x}}+4e^{-8t}\hat{{\bf y}}+8t^{3}\hat{{\bf z}}$ how do I find the acceleration at time $t = 0.27778$ I know that the third derivative is: $ \vec {R^{(3)}(t)} = ...
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1answer
19 views

Third derivative of R(t).

How do I do third derivative of the following expression: $R(t) = 7sin(at)\hat x +4e^{-8t}\hat y + 8t^3\hat z$ $(a)$ represents acceleration my goal is to find what $a$ is equal to when $t=0.27778 ...
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1answer
50 views

Mathematical problem with solving a physics issue

The power absorbed by the BOX in the Fig.A is $p(t) = 2.5e^\left(-4t\right)$ W. Compute the energy and charge delivered to the BOX in the time interval $0 < t < 250$ ms. So let me show ...
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1answer
87 views

Tensor notation of a triple scalar product

I want to write the tensor notation for $$[a\dot\ (b\times c)]a=(a\times b)\times (a\times c).$$ What I got so far is: $$a \dot\ (b\times c)=a_i(\epsilon_{ijk}b_jc_k)=\epsilon_{ijk}a_ib_jc_k.$$ ...
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1answer
88 views

Tensors and rotation matrix

$a_{ij}$ is a rotation matrix that satisfies $\hat{e}'_i=a_{ij}\hat{e}_j$. Show that $\epsilon_{lmn}a_{mi}a_{nj}=\epsilon_{ijk}a_{lk}.$ Using the result from above, how can I show that ...
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1answer
109 views

Variances for K-Means clustering

Can somebody help me understand formulas with an example in the below image? The below image is about K-means clustering. The formulas are about calculations for the variance for within-clusters and ...
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1answer
36 views

Calculate this integral in $N$-dimensional space

I want to calculate the integral $$\int_{\mathbb{R}^N \times \mathbb{R}^N} \chi_{[0,E]}\left(\sum_{i=1}^N \frac{p_i^2}{2m} + \frac{m \omega^2 q_i^2}{2} \right) \,dp\, dq.$$ Now I should explain what ...
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1answer
296 views

Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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221 views

A Learning Roadmap to the “foundations” of Nonlinear Analysis (and certain specific topics)

I'm searching for throughout references that -- in the long term -- can help me gradually gain a solid background and firm foundations to understand the main methods and theorems to deal with ...
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3answers
47 views

Green's function impulse

Solve $y''(t)-a^2y(t)=\delta(t-t'), \ y(0)=0, \ y'(0)=0, \ t'>0.$ I am not sure how to solve an equation that has $\delta(t-t')$ as a solution. The book doesn't really elaborate further than ...
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1answer
73 views

Generating a rate equation from a paper

I'm going through equations in this paper Structure of Growing Networks with Preferential Linking, I was not able to understand how they derived equation $[3]$ by summing up equation $[2]$. eqn [2] ...
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3answers
90 views

Steepest descent method

I really don't understand how we generally choose the contour for the steepest descent method in complex analysis? I approximate the Fresnel integral $$ \int_{0}^{\infty}\cos{x^2}dx$$ and I found it ...
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57 views

Greens function method for Newtonian potential

this may be a silly question but, well you know when solving for the Poisson equation that gives the Newtonian potential, $\Phi$, (for a point mass, $M$, at the origin) $$\nabla^2 \Phi = 4\pi G ...
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75 views

Sorting out some integrals from physics

I'm doing some physics for a change, and I'm trying to sort things out a bit. From the definitions of mass, torque, momentum and angular momentum I've come up with the following integrals: ...
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2answers
96 views

Solving a PDE via method of characteristics

I'm interested in solving the following PDE via the method of characteristics: $$\frac{\partial f}{\partial t} - ax\frac{\partial f}{\partial p}+ bp \frac{\partial f}{\partial x} = 0,$$ with ...
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44 views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
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26 views

Questions about the formula for inductive reactance and $Z_t$

I am currently on the inductors unit in my Navy schooling and I have two questions about these formulas that I learned about. As I'm aware, the ability of an inductor to concentrate a magnetic field ...
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1answer
48 views

How do I find the relative coordinates of a picture of a plane in 3d space.

I have a box, with corners $A$ through $H$, as depicted above. I'll consider $B$ the origin of a coordinate system, with the $x$ axis in the direction through $C$, the $y$ axis through $A$ and the ...
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41 views

Discovering the mathematical nature of Nature - Galileo's inclined plane experiment

In 1638 Galileo published Two New Sciences, in which he described his inclined plane experiment. He discovered that the acceleration of gravity was uniform, and could be modeled mathematically by the ...
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34 views

Convert time derivative to a function of time

Physics: I am asking for help to derive a general expression for the total amount of energy lost as a function of time from a radiating object. I'll simplify my problem like this: Say for example ...
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1answer
46 views

Analytical solution for bound state energies of infinite well

I am trying to find bound state energies assuming infinite potential. I have been told it can be done by analytically solving Right Hand Side and Left Hand Side of an equation such as: ...
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1answer
38 views

Projectile Motion using cos and sin theta???

Golfball is struck to clear a tree 20m away and 6m high at an angle of elevation of 40degrees. Find the speed of the ball when it leaves the ground. I've created my displacement equation with i and j ...
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19 views

identity of equation

We have the equation ($\partial_{\mu}\partial_{\nu}$-$\eta_{\mu\nu}\Box$)$\phi=0$, where $\phi$ is a scalar field, $\Box=\partial_{\mu}\partial^{\mu}$ is a standart Dalamber operator, $\eta_{\mu\nu}$ ...
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1answer
27 views

arrange m balls in to n baskets

How can I write a given natural number into sum of required (m) natural numbers? Example: 10=2+8+0 here m=3 Let n_i be the values i:e 2,8,0 in the above example. I want to know whether any method ...
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1answer
27 views

Nested square brackets in tensor indices

I know that using square brackets on tensor indicies denote the anti-symmetric part $$ T_{[ab]} = \frac{1}{2} \left( T_{ab} - T_{ba} \right)$$ I now have to prove that $$ T_{a [[bc]d]} = T_{a ...
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21 views

Zeta function and heat kernel

It is easy to prove that zeta function $$\zeta_{\Lambda}(s)=\sum \frac{1}{\lambda_{n}^{s}}$$ and trace of heat kernel $$K_{\Lambda}(t)=\sum e^{-\lambda_{n}t}$$ satisfy the relashion ...
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1answer
83 views

Boundary conditions for the time-independent Schrödinger equation on the sphere

if you have a free Schrödinger operator on a sphere $-\Delta \psi(\theta,\phi) = E\psi(\theta,\phi),$ then the substitution $\psi(\theta,\phi) = f(\theta)e^{i n \phi}$ leaves you with the ...
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40 views

How to calculate Hill's discriminant?

I am currently reading this paper on Schrödinger operators see here. On page 6 and 7 they talk about Hill's discriminant and how this is connected with the spectral properties. They also show some ...
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1answer
38 views

Rotation about an axis by matrix multiplication

Suppose I have three axis of rotation vectors $\vec{v_1},\vec{v_2},\vec{v_3}$ and angle of rotation as vectors $\theta_1,\theta_2,\theta_3$. Take a vector $P$ then apply rotation around $\vec{v_1}$ ...
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1answer
30 views

how to calculate the phase shift in the formula that has sin in both side?

Given formula $asin ( x ) = b sin( x + \phi)$ where $a$ and $b$ are constants. I want to calculate $\phi$.
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1answer
56 views

physical meaning of heat equation

consider the heat equation $u_t=a(t)u_{xx}+f(x,t)$, $0<x<L$, $0<t<T$ subject to the initial condition $u(x,0)=g(x)$ and boundary conditions $u(1,t)=0,$ $u_x(0,t)+hu(0,t)=0$ where ...
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1answer
46 views

What happens when this turns to $dx$?

I have this equation: $$ ds^2=c^2dt^2-dx^2-dy^2-dz^2. $$ And I've also been given $$ x=x'\cos(\Omega t)-y'\sin(\Omega t), $$ which I need to substitute into the first equation. I've squared $x$ to get ...
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1answer
47 views

Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
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16 views

k-space tensor integral in statistical mechanics [duplicate]

k is the modulus of the vector k. Please help me to integrate the above tensor expression in the infinite domain of the vector k. I have tried to let u in the direction of kz and then transform the ...
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1answer
39 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
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2answers
68 views

Damped simple harmonic oscillator, phase space

I want to calculate and draw the phase space trajectory of this damped harmonic oscillator: $$\ddot{x}+\gamma\,\dot{x}+\omega^2x=0$$ for the two cases $\gamma=2\omega$ and $\gamma=\omega$. I'm ...
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1answer
36 views

Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
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181 views

Newton's Law of Cooling, age of Earth, weak math skills

I'm curious about a problem concerning the age of the earth, but I don't have the math skills to think properly about it. I've found the solution to Newton's Law of Cooling, and I can handle that ...