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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2
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1answer
238 views

Tensors in math and physics

I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me ...
6
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2answers
874 views

Mathematical significance of the “Dirac conjugate”

Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ ...
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4answers
4k views

Projectile Motion - Finding initial velocity

You need to send a bowling ball up exactly $205$ meters so someone at the top of the Washington Monument can take a picture of it "hanging" in space. The bowling ball will be shot from a mortar tube ...
3
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1answer
354 views

Is there a non-variational derivation of Snell's law from Fermat's principle?

Every proof I've seen of Snell's law from Fermat's principle uses some sort of variational argument, mostly involving variational calculus. Niven's wonderful book, Maxima and Minima Without Calculus ...
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1answer
294 views

Calculating Humidity, Wet Bulb Temp and Dew Point [closed]

I have been trying to simulate a climate system, but I have hit a wall. I am trying to calculate Humidity. But to calculate Humidity you need the Wet Bulb Temperature, but to calculate the Wet Bulb ...
2
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1answer
513 views

Solve integral by Mellin-Barnes representation

We start with the following integral $$I:=\int_0^\infty\mathrm{d}\alpha\,(a+\alpha)^{-\lambda}$$ which is easily evaluated to ${\frac {{a}^{-\lambda+1}}{\lambda-1}}$ by direct integration over ...
3
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2answers
980 views

Resolving vectors into components

The problem is to determine the components of $F_2$. Problem image Method 1 Method 2 My question is why do I receive different answers?
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2answers
2k views

Can somebody help and explain me with angular and linear Velocity textproblems?

So we got this as homework today, and I just don't understand it; how to start or how to do it. Explanations and/or setups would be great :) A wheel with ...
1
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0answers
147 views

Hermitian conjugation and representations of the Lorentzian Clifford algebras

The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma ...
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0answers
288 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
0
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2answers
1k views

Convert this figure to watts , where W = 1 J/s , and then estimate the average per capita energy consumption rate in watts. [closed]

The world consumes energy at the rate of about $466 EJ$ per year, where the joule$(J)$ is the SI energy unit. Convert this figure to watts$(W)$ , where $W= 1 J/s$, and then estimate the average per ...
2
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1answer
87 views

An integral for the Earth's insolation

Consider the function $$ [-\pi/2,\pi/2] \ni \theta \mapsto s_\beta(\theta) = \int_0^{2\pi} \sqrt{ 1 - \left(\cos \theta \sin \beta \cos \gamma - \sin \theta \cos \beta \right)^2} \, d \gamma $$ for ...
9
votes
1answer
250 views

Help computing an integral for Green's function of a discrete Laplacian on a square lattice

I need to calculate the following integral: $$ \int_0^1 \int_0^1 \frac{1-\cos(2 \pi k_1 x) \cos(2 \pi k_2 y)}{4 \sin(\pi k_1)^2 + 4 \sin( \pi k_2)^2} dk_1 dk_2 $$ I have tried to use some contour ...
1
vote
3answers
8k views

Find the component of $\vec{a}$ along $\vec{b}$?

I'm trying to do homework for my physics class, and it says I should find 'the component of $\vec{a}$ along the direction of $\vec{b}$'. The vectors are: $\vec{a}$ = 7.1i + 8.9j $\vec{b}$ = 5.8i + ...
3
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0answers
177 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
1
vote
2answers
345 views

How to get the derivative of a physical formula?

I've heard that if $U_{ind} \neq $ constant, you must use the following formula: $ U_{ind} = N \times \dfrac{d \phi}{dt}$ I however don't know how to get the derivative of this formula? In math I ...
2
votes
2answers
350 views

How to calculate the mass of a layer of an atmosphere?

In a problem I'm working on, I need to derive what fraction of molecules in the Earth's atmosphere are found in the troposphere (the bottom layer extending from the surface to approx. 12 km). I have ...
4
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0answers
136 views

Gaussian Integral with non-polynomial exponent

I am currently trying to evaluate this Integral: $$\int\limits_{u_0}^{u_1} \exp\left[-\angle(H(u),N)^2\right]du$$ Where ...
7
votes
4answers
312 views

Is length adimensional when space is not flat?

Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...
2
votes
1answer
444 views

Center of Mass of “Combinations” of 1-Dimensional Objects.

Center of mass for one-dimensional objects is given by $\displaystyle\frac{\int x \, dm}{M}$ or $\displaystyle\frac{\int x \rho \, dx}{M}$, where $\rho$ is density. Now, the center of mass of a rod ...
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1answer
596 views

Finding the initial velocity using calculus

I throw a stone at 20 degree, when the stone falls to the ground, it reaches 100m further. Using CALCULUS methods, find the initial velocity of the stone.
4
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1answer
203 views

Existence Energy of Wave Equation

I was just going trhough some properties of the wave equation, including the energy of the wave equation given by $E(t)=\int_{-\infty}^{\infty}u_t^2+c^2u_x^2 dx$, i.e the sum of kinetic and potential ...
7
votes
2answers
630 views

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding ...
3
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1answer
206 views

Why are higher order differentials $dr^2$ and $ dr^3$ ignored here?

I'm doing a problem in physics, but it's the math part I'm curious about: Charge density is defined by $\rho = \frac{dQ}{dV}$, then $Q = \int_{V}^{} \rho \text{d}V$ The problem is dealing with a ...
3
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1answer
455 views

Energy of wave equation decreasing

I have problems checking that the energy $E(t)=\frac{1}{2}\int_I(u_t^2+c^2u_x^2)dx$ on an open interval $I\subset \mathbb R$, such that $u(0,x)=0$ and $u_t(0,x)=0$ for $x\in\mathbb R\setminus I$ is ...
2
votes
0answers
391 views

How to convert a hologram into an image?

Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...
11
votes
3answers
7k views

Dirac Delta Function of a Function

I'm trying to show that $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$ Where $a_{i}$ are the roots of the function $f(x)$. I've tried to proceed by ...
2
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0answers
1k views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
0
votes
2answers
832 views

Finding the angle and velocity to hit a target in t seconds

My goal is to compute an angle + velocity combination to hit a target at point $(x,y)$ in exactly $t$ seconds. (Uniform gravity, no drag, no wind) I know that the general formula for trajectory is: ...
2
votes
1answer
897 views

Electric Potential of an off axis charge (Legendre Generating Function)

An insulated disk, uniform surface charge density $\sigma$, of radius R is laid on the xy plane. Deduce the electric potential $V(z)$ along the z-axis. Next ...
2
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1answer
515 views

Can the differentiating and squaring process in the cochlea explain a reported dichotic stimulation experiment?

On this math.stackexchange on url What is Octave Equivalence? in an answer on the related ( octave equivalence ) question is stated: Mathematically, this signifies that the mammalian cochlea ...
2
votes
1answer
240 views

concise review of Maxwell's electromagnetic equations for math students

I am a graduate student in applied mathematics and I am looking for a concise introduction to Maxwell's equations / basic principles of electromagnetism. (I have enjoyed the book by Purcell, ...
2
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1answer
2k views

Earth-Sun distance equation

I was studying solar geometry and I read the next equation $$r=r_{0}\left(1+0,017\sin\left[\frac{2\pi(d_{n}-93)}{365}\right]\right) $$ where $r_{0}$ is the average distance between the Earth and the ...
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vote
1answer
345 views

Help with integral for electric potential

I need help evaluating the following integral $$\frac{ \sqrt 2 \sigma}{2 \epsilon_0} \int_0^R \frac{r \,dr}{ \sqrt{(z- \frac{ rh}{R} ) ^2 + r^2} }$$ This integral pertains to the Electric potential ...
2
votes
1answer
194 views

P-adic Numbers and Eternal Inflation

In October(??) 2011, Leonard Susskind gave a talk and with few other people wrote a paper about P-adic numbers and measure problems(??) in cosmology. Has there been any recent talks, papers, ...
2
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4answers
6k views

Why are these angles equal for object on inclined plane?

This is a common setup for kinematics problems in physics. My geometry is rusty and I want to understand this very simple idea. I am having trouble understanding why the angle $\theta$ formed by ...
1
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2answers
127 views

Why are expressions such as $\operatorname{ln}(T)$ used in thermodynamics where $T$ is not dimensionless?

In all thermodynamics texts that I have seen, expressions such as $\operatorname{ln}T$ and $\operatorname{ln}S$ are used, where $T$ is temperature and $S$ is entropy, and also with other thermodynamic ...
1
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1answer
2k views

Surface area , center of mass, and moment of inertia of paraboloid

I have done a bunch of work and simply wish to check that it makes sense. I have a hollow parabola of height b and base radius b ($ z = \frac{x^2 + y^2}{b}$ bounded by z = b) 1) surface area of ...
2
votes
2answers
80 views

Hint for integral

Could someone provide a hint as to why $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$ where $b$ is a constant, $i$ is $\sqrt {-1}$, implies that $$2\int d^3x \,\,x_ia_j(\vec ...
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2answers
30k views

solve for time given distance, initial velocity, and acceleration

I've been having trouble sorting this one out. I need to compute the time it will take for a vehicle traveling along a straight line to reach a particular point. I have the initial velocity ($v_i$), ...
12
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2answers
2k views

Cross product and pseudovector confusion.

So called pseudovectors pop up in physics when discussing quantities defined by cross products, such as angular momentum $\mathbf L=\mathbf r\times\mathbf p$. Under the active transformation $\mathbf ...
2
votes
1answer
164 views

The PDE $u_t = -\Delta^2 u -\Delta u + f$

Does the PDE $u_t = -\Delta^2 u -\Delta u + f$ have a physical use or meaning? I am asking specifically about the the Laplace term after the biLaplace term.. is it unusual or "unnecessary" in some way ...
2
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1answer
2k views

How is this second form of the Euler-Lagrange equation arrived at?

The Euler Lagrange equation $\frac{\partial F}{\partial q}-\frac{d \frac{\partial F}{\partial \dot{q}}}{d t}=0$ can also be put in the form $\frac{\partial F}{\partial t}-\frac{d (F- ...
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0answers
60 views

Simple equation misunderstanding

Im trying to use an equation on this page http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics The angle of lean, $\theta$, can easily be calculated using the laws of circular motion $$ ...
4
votes
2answers
666 views

$SU(2)$ Representation of $SO(3)$

I've often seen it written that $SU(2)$ is a "two-valued representation" of $SO(3)$ (in theoretical physics books mainly). I have a major conceptual issue with this however. I know there is a Lie ...
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0answers
328 views

Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?

This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
1
vote
2answers
979 views

Updating a quaternion orientation by a vector of euler angles

I'm trying to understand why this formula works to update an orientation with an angular velocity represented as a vector of rotations in $radians/{second}$. I understand that two quaternions ...
1
vote
1answer
472 views

Discontinuity of double-layer potentials

I'm currently reading about solutions to boundary-value problems for Laplace's equation, and I'm a bit confused with regards to the discontinuity properties of double-layer potentials. So the text ...
26
votes
2answers
1k views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
4
votes
4answers
37k views

Derivative of position is velocity and of velocity is acceleration?

This is more of a personal question (i.e. not school related), but how has it been proven that the derivative of position is velocity and derivative of velocity is acceleration? I did some Google ...