Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

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PDE basic traffic flow problem

I am analyzing a basic example of traffic flow presented here http://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf and have a question to the last transition in the traffic flow equation ...
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1answer
18 views

Converting speed vector base coordinate system to preserve destination

I have a vector that stores speeds for 3 degrees of freedom, like this: $v=[dx,dy,d\theta]$. Now, these speeds are obviously directional. The direction is measured in a global base frame. In this ...
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1answer
20 views

Straight Line Motion Problem (Revision)

I am studying for a physics exam using a textbook written by my professor for my course. In it there is a step-by-step solution to a problem, and I cannot figure out how to go from one step to the ...
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1answer
16 views

Laymans explanation of this Energy Transformation equation…

I know this will be basic but I'm trying to visualise this pendulum equation and it's left me a little stumped. If we have a mass on the end of a string and the mass is allowed to swing like a ...
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9 views

Quadratic eigenvalue equation for a Bloch problem

The Bloch problem in a tight-binding nearest neighbors form is following: $$ (\mathbf{a} c^{-1} + \mathbf{w} + \mathbf{a}^\dagger c) \phi = 0, ~ \mathbf{w} = \mathbf{w}^\dagger$$ $\mathbf{a}$, $\...
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17 views

Infinitesimal Strain Tensor in a Cubic Crystal

I'm currently working through Vectors and Tensor in Engineering and Physics, and there's a problem regarding the strain tensor that I'm having a bit of trouble with. Given a cubic crystal with zero ...
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80 views

Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
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2answers
39 views

Additive functions and measure theory

Key reference is the following: Hamel basis and additive functions Let's investigate real-valued functions $f(x)$ with the following (additive) property for all $\,a,b$ : $$ f(a+b)=f(a)+f(b) $$ It ...
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Solutions to Strum-Louiville equation continuous even with discontinuous coefficients?

In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Strum-Louiville equation $$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\...
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11 views

Can Fluctuation-Dissipation Theorem Apply to Magnetic Forces in Multi-Spin Systems [closed]

Let's say I have multiple spin systems (atoms in a protein) in a solution of water and the spin systems are all producing a magnetic field $\mathrm{B_{loc}}$ that affects nearby spin systems. Will the ...
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5 views

Proof of Exponential Decay Pattern of Time Correlation Functions for $\mathrm{B_{loc}(t)}$ in NMR Spectroscopy

For a given protein, I know that the NMR Spectroscopy magnet generates a field $\mathrm{B_o}$ and that the interactions with the spins in the local environment generates a much smaller field $\mathrm{...
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3answers
335 views

A limit in a Feynman “proof” about Fermat's Theorem.

As perhaps some of you already know, Richard P. Feynman, the famous physicist tried a non-orthodox (in his usual way, I suppose) proof of the Fermat's Last Theorem. He tried a probabilistic "proof" ...
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1answer
28 views

Integral of solid angle of closed surface from the exterior

Jackson derives Gauss's Law for electrostatics by transforming the surface integral of the electric field due to a single point charge over a closed surface into the integral of the solid angle, ...
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1answer
52 views

Rate vs velocity?

A bomber releases a bomb while flying vertically upwards at a velocity of 1500 ft/sec at a height of 10,000 feet. a) How long after it is released will it take the bomb to reach the ground. b) ...
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1answer
25 views

How does the sign of the acceleration depends on the direction of the distance choosen?

I am very confused as to how to determine directions in problems. For example I was doing the following problem A bomber releases a bomb while flying vertically upwards at a velocity of 1500 ft/...
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138 views

Closed form of $\int_0^1 \tan(\gamma\sqrt{1-x^2}) dx$

Some context: I'm studying the problem of nonperturbative pair creation from strong fields in quantum electrodynamics. For certain time dependent electric fields I can get some information about the ...
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1answer
20 views

Ray model in $3D$

Supposing we have a point source $p$ and multiple receiving points $r_i$ in $\Bbb R^3$ and there is a direct ray from $p$ to each $r_i$ and if there is a single mirror there is a single reflected ray ...
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1answer
35 views

Resistance of a frustum, or a pyramid cut from top

We know that Resistance is given by: $$R = \dfrac{\rho L}{A}$$ The figure of Resistance is like this, $L$ is perpendicular distance between $A_1$ and $A_2$ I assumed that the area change linearly, ...
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16 views

Maxwell–Boltzmann distribution average speed and second-order moment

Someone can give me a link where I can find the solution step by step of the following integrals. Otherwise,if there is someone so kind to solve them. $$dN(v)=4\pi N(\frac{m}{2\pi kT})^{\frac{3}{2}}...
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1answer
19 views

Computing new 3D coordinates given time and linear velocity

Assume that an object in a 3D space has a position $\displaystyle {x}, {y}, {z}$ and a linear velocity $\displaystyle v_{x}, v_{y}, v_{z}$ Can I predict the new $\displaystyle {x}, {y}, {z}$ position ...
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23 views

Clarification Needed:Equation Of Refracted Ray/Line

A ray of light is sent along the line $2x-3y=5$.After refracting across the line $x+y=1$ it enters the opposite side after turning by $15^0$ away from the line $x+y=1$.Find the equation of the ...
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1answer
18 views

Two Dimensional Motion using Vector Analysis

I came across the following question: Use vector methods to find the maximum angle to the horizontal at which a stone may be thrown so as to ensure that it is always moving away from the thrower....
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17 views

How to investigate the relationship between range and payload?

I am interested in learning about the relationship between range and payload for an electric aircraft. How do I use math to investigate the relationship between range and payload for an electric ...
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1answer
37 views

Finding the overlap between direction of distance in position space and direction of distance in velocity space

There are two objects A and B that can be described in position space and velocity space. The position space describes the instantaneous positions of the objects while the velocity space describes ...
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1answer
37 views

How to determine the optimal length away from the pivot point and angle to the radius to use the least amount of force to rotate the lever?

I'm writing an essay for school in physics and my topic is torque. My topic deals with the elbow joint and the tendon that attaches the radial bone to the biceps, the force that rotates the forearm ...
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1answer
20 views

What is the magnitude of the vector given below conditions?

Let vector B = 3m at 60 degree. Let vector C have the same magnitude as vector A. Vector C has a direction angle greater than of vector A by 25 degree. Dot product of A and B is 30. Dot product of B ...
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1answer
50 views

projectile motion find the angle

I would please like help on the following question related to projectile motion. An object is projected in a $60$ m/s velocity with a $\theta$ angle to the ground. If the object has $30^\circ$ ...
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79 views

Solutions to Laplace's equation between surfaces of a constant coordinate

In Physics (specifically electrostatics) it is often encountered that one must solve the Dirichlet problem for Laplace's equation \begin{equation} \nabla^2 \varphi(x_1,x_2,x_3) = 0 \end{equation} ...
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3answers
49 views

Solving Electric and Magnetic Fields for Charged Particle Path

I am using the Lorentz Force Equation and the electric-cross-magnetic field velocity equation] to solve for the E and B fields given the known path of a particle moving in 3D. So with that I have ...
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22 views

Quick question regarding the work of a dynamical system

I'm sitting in on a course in dynamics and I've noticed something very strange that I can't get my mind around. So when they do problems that involve springs, they are saying that the work is $$U_s=-...
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22 views

Moment of inertia in a four bar linkage

I'm trying to calculate the moment of inertia of a lumped mass on a four bar linkage with respect to the left pivot point. See the above image. At first, I thought it was just a matter of ...
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1answer
47 views

Trying to prove that geometric and algebraic interpretations of the dot product are equal.

I am a Classics major trying to teach myself physics. I am on summer vacation at the moment, and I am going through a book called Classical Mechanics by J. Taylor. I am on the first chapter, and I ...
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1answer
38 views

Geometric proof re: parabola focus and potential calculus connection

The image above is a proof that light traveling with an orientation perpendicular to the directrix into a parabola will be reflected to the focus of the parabola. (In the diagram, the purple dotted ...
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1answer
68 views

How can I solve $y''=\frac{a}{y^2}$ where a is a (positive) constant?

Actually, I found out a way to solve that, but I can't get rid of complex numbers. And it does not make sense when it comes to complex numbers as the original question that involves this differential ...
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1answer
74 views

Modeling particles moving through a chamber

Consider the following phenomenon. Particles each traveling with speed $v_i$ metres per second enter a chamber at a rate of $r$ particles per second. Upon entering the chamber, each particle begins ...
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1answer
33 views

angle between 2-D vectors (knowing inner product and length of vectorial product)

I'm struggling with the following exercise: Given the 2-D vectors $<A>$ and $<B>$, where their inner product $<A> \cdot <B> = -6$ and the magnitude of the vectorial product is ...
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21 views

Lyapunov exponent of multiple time series

I have multiple time series data that I have generated by varying the initial conditions infinitesimally. I now want to calculate the Lyapunov exponent to identify the sensitivity to initial ...
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26 views

Integral of magnetic field inside cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder, or a cylindrical shell of radii $R_1<R_2$, whose axis has the direction of the unit vector $\mathbf{k}$. For any point of ...
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1answer
30 views

Compute Christoffel symbols of a rotating cartesian coordinate system

Suppose we have a smooth manifold $(\mathbb{R}^3, \mathcal{O}_{\mathbb{R}^4}, \{(\mathbb{R}^3,x),(\mathbb{R}^3,y)\},\nabla,t)$ where $t:\mathbb{R}^3\rightarrow\mathbb{R}$ is such that $t(a,b,c,d)=a$, $...
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1answer
42 views

Energy of photons, and its relationship to frequency and wavelengths [closed]

I'm not fully grasping how $E=hv$ (Energy = Planck's constant $\times$ velocity) shows that the higher the wavelength, the lower the energy from the equation $c=v\lambda$ (speed of light = velocity $\...
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1answer
31 views

Calculate velocity $\nu$. [closed]

A lorry of mass $3.5\times10^4\text{ kg}$ attains a steady speed $\nu$ while climbing an incline of $1$ in $10$ with its engine operating at $175$ kW. Find $\nu$. $g=10ms^{-1}$. Neglect friction. The ...
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1answer
43 views

Kline calculus intuitive chapter 3 problem 14

I am starting to get frustrated as I am not able to comprehend these questions because they simply do not make sense to me. A body is dropped. At the instant when it has fallen $100$ feet, a ...
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1answer
53 views

shifting integration variable and taking derivative seemingly giving problem

I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ...
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1answer
30 views

Expression for nano scale crack propogation

The following image is of a nano-scale self replicating crack on a thin film/substrate pair. I want to construct a "position" function for the crack front's propagation. So far the only way I can ...
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Clarification about consequence of physics' first principle of thermodynamics

I'm reading the book Physics, by Tipler, and I'm confused at the following statement: [...] From the first principle of thermodynamics, $\Delta U = Q+ W$. Suppose an ideal gas is given heat while ...
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Deriving E=mc^2 from hollow box with mass M and photon

I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the ...
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pressure field calculation from velocity field

Velocity field $v = (kx, -ky, 0)$. Initial condition is that pressure is $P$ at origin. I have to find pressure field, which is easy in this case. My problem is the gravity. In part a of the ...
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1answer
22 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
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1answer
25 views

Understanding elastic collision between two rocks with unknown masses

I have this problem here that goes like this: Two curling rocks of equal mass, one red and the other yellow, are involved in a perfectly elastic, glancing collision. The yellow rock is initially at ...
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52 views

Work required to align pieces in a plane.

Given two piecewise continuous functions f(x) and g(x) and that $\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$ at all points, find the work used to shift each of the planar slolids ...