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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Sum over all compositions of $n$ into exactly $k$ terms (overlaps of $SU(k)$ coherent states)

Given two $k$-dimensional vectors, $\vec{x}$ and $\vec{y}$, I would like to find, $$ d(\vec{x}, \vec{y}) = \left|\sum_{t_1 + t_2 + \cdots + t_k = n} \prod_{1\leq q \leq k} (x_q\times ...
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1answer
51 views

Why does the vector field $(\sin (\theta), - \cos(\theta), 0)$ indicate sideways motion?

If I study a physical system, such as a car, and let it drive forward a little bit, say a distance $m$, then I can draw out the right triangle and find the car's position at $(m\cos \theta, ...
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26 views

Meaning of Torricelli's Equation ($v^2=u^2+2as$)

The equation of motion $v^2=u^2+2as$ is usually presented as the particular formulation of the SUVAT system which doesn't involve t. It is derived from the others using some (perhaps well-motivated) ...
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37 views

partial differential equation applicational problem

As a Maths student with not much knowledge in physics, I dont understand how the "string" can be "cut" into half at x=L/2. Also, how many initial conditions(data) does this question have apart from ...
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1answer
18 views

Find out the angular speed in terms of time.

Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star: $u=\frac el\cos\theta+\frac 1l$, where $u$ is the reciprocal of the radial distance ...
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72 views

Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm ...
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3answers
106 views

infinitesimal intervals in physics

The density of states of a system in an interval $[E, E+dE]$ is given implicity by $dV = D(E)dE$ (Or I suppose explicitly, by $D(E) = \frac {dV}{dE}$, but we'll be integrating it anyway, so it doesn't ...
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32 views

Thermodynamics: find exit temp and velocity of air out of a nozzle?

I don't know if I can really ask a thermodynamics question here on this math site but I need help and this was the best site for when I needed help in math class. Concerning thermodynamics, I have ...
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1answer
99 views

Eigenvectors of almost-Toeplitz tridiagonal matrix.

I'm reading a book about semi-conductors, and when figuring out how dopants affect the energy-levels, one wishes to find the eigenvalues (and vectors) of a NxN tridiagonal matrix of the form ...
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1answer
24 views

re-arrange equation $L=2^{10(v-1)} v^2$

Is it possible to re-arrange this equation to make v the subject? $$L=v^2 . 2^{10(v-1)}$$ If so, what is the answer? If it helps (which by excluding zero it should)... $$0<v<1$$ I have tried ...
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1answer
58 views

How to find direction of velocity V2 to reach an object travelling at velocity V1, such that it takes least time?

If an object A is currently at point P1 moving with constant velocity V1, and there is another object, object B which currently at point P2 which can move with velocity v2, then what should be the ...
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1answer
33 views

Work done in a vector field

Say a particle is moving along a path $\gamma$ in a vector field, then the total work done by the force $\vec{F}$ on the particle is $\displaystyle \int_{\gamma}{\vec F}.d\vec{r}$. Say if this value ...
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2answers
98 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
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1answer
63 views

Can an integral of a function that is not well behaved be finite?

Consider the following integral which gives the time period of simple pendulum where $\theta_0$ is the initial inclination of pendulum with vertical. ...
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0answers
23 views

Projectile landing on an elevated point

A golfer is standing on a fairway and hits a shot to a green that is elevated 6.0 m above the point where she is standing. If the ball leaves her club with a velocity of 43 m/s at an angle of 40.0 ...
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0answers
24 views

Tangential and normal components of acceleration of a point moving along a curve

If a point is moving along a curve in polar coordinates, is the tangential component of its acceleration given by $r\left(d^2\theta \over dt^2\right)$ and the normal component by $r\left(d\theta \over ...
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0answers
70 views

Index notation confusion in tensor algebra

I have some confusions regarding index notation in tensor algebra. Let's assume $\vec{v}$ is a vector belonging to vector space $V$. Choosing a basis set $\{\vec{e}_\nu\}$, $\vec{v}=x^\nu\vec{e}_\nu$ ...
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1answer
44 views

Olbers' Paradox in an Euclidean universe with randomly located stars

This question is inspired by Olbers' Paradox. Imagine a universe $U$ shaped like $R^3$ with points $P$ randomly distributed in a Poisson fashion throughout, with density parameter $\lambda$ such that ...
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2answers
203 views

Setting up a differential equation to find time constant for RC-circuit

Problem: Calculate the time constant for charging the capacitor in the circuit shown in the figure. What is the maximum charge on the capacitor? Attempt at solution: Let current $I_1$ flow from the ...
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2answers
50 views

Water Density and Fluid Force (question below) [closed]

I've been trying to study the question and the answer below. Can someone tell me how to start this problem myself? I don't understand why they named one fourth of the circle equation the whole ...
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17 views

Approximations of the kind $x<<y$

I have an expression for a force due to charged particle given as $$F=\frac{kQq}{2L}\left(\frac{1}{\sqrt{R^2+(H+L)^2}}-\frac{1}{\sqrt{R^2+(H-L)^2}}\right)$$ where $R$, $L$ and $H$ are distance ...
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21 views

How do you calculate certain variables of two or more events that occur simultaneously compared to the same events happening subsequently.

Say you have two hoses, A and B, that fill up a pool of equal size at different rates. Hose A fills up a pool in 10 mins, hose B in 20 mins. Thus A = 1p/10m, B = 1p/20m. Lets say that Hose A filling ...
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1answer
75 views

Solutions to Kepler problem/Motion in central field

It is clear that particals in a repulsive central field given by a point mass at the origin are moving along hyperbolas, e.g. given by the expression ${x^2 \over a^2} - {y^2 \over b^2} = 1$ (after a ...
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4answers
85 views

How to solve $a \cos \alpha + b \sin \alpha = c$ for $\alpha$?

I'm solving a physics problem and I came down to solving an equation of the form $$a \cos \alpha + b \sin \alpha = c$$ Can someone help me to solve this? Thanks in advance!
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1answer
79 views

Vectors problem, Please Help!

A buoy is floating in water and is tied to a post. The water is creating a force of $3$ N on a bearing of $125^\circ$ and the wind is creating a force of $2$ N on a bearing of $230^\circ$ What force ...
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0answers
64 views

Attempt to solve the brachistochrone problem

I am attempting to solve the brachistochrone problem for fun. I don't have too much experience with differential equations and wanted to see if I am on the right path. My attempt Assumptions ...
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1answer
53 views

Integration of partial derivative $\frac{dL}{dq}$ with respect to $t$ where $q$ is implicitly a function of $t$

Is $\int_{t1}^{t2} \frac{\partial L}{\partial q}\delta{q} dt$ equal to $\left[\frac{\partial L}{\partial \dot{q}}\delta{q}\right]_{t1}^{t2} $ if $q$ implicitly depends on $t$ ? If not I ...
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2answers
19 views

Finding the derivative Maxwell Boltmann distribution

I need to basically show that the slope is $0$ when $\epsilon = kT$ $f(\epsilon)=\left(\frac{8 \pi}{m}\right)\left(\frac{m}{2 \pi kT}\right)^{\frac{3}{2}}\epsilon e^{\frac{-\epsilon}{kT}}$ So I ...
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30 views

According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general

I recently read Khinchin's derivation of Liouville's theorem. I was able to follow the math for the most part, however I was hoping for an intuitive understanding about why the form of the measure on ...
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23 views

Continuity of the divergence of a static electric field

Let $\rho:\Bbb R^3\to\Bbb R$ be a continuous charge density function. Define the electric field $\vec E:\Bbb R^3\to\Bbb R^3$ by $$\vec E(\vec r)=k\cdot\int_{\Bbb R^3}\rho(\vec{r}')\cdot\frac{\vec ...
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1answer
51 views

Velocity vector

We suppose that a ship, that is at the position $(1, 0)$ of a nautical map (with the North at the positive direction $y$) and it "sees" a rock at the position $(2, 4)$, is directed to North and is ...
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2answers
42 views

Angular velocity

The angular velocity $\omega$ of rotation of a rigid body has the direction of the rotaion axis and magintude equal to the rotation rate in rad per second. The orientation of $\omega$ is determined by ...
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1answer
52 views

Torque of a force

The torque $M$ of a force $\overrightarrow{F}$ as for the point $O$ is defined as the product of the magnitude of the force $\overrightarrow{F}$ and the perpendicular distance of the point $O$ and the ...
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0answers
43 views

I need help with divergence and gradient?

$$A_z = \mu{\frac{e^{-jBr}}{4\pi r}}∫I(z')e^{jBz'\cos\theta}dz'$$ Midway into my question, I want to compute: $$-j\left( \frac{\nabla(\nabla\cdot A) }{w\mu\varepsilon} \right).$$ Symbols like $ w, ...
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1answer
25 views

Can someone explain why acceleration is not negative in this problem?

Studying for a Physics exam and I'm going back through the homework and redoing the problems. I've come to a problem and am a little confused on the equations. The problem is: With what speed ...
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1answer
54 views

Solve this system of equations for acceleration

I'm dealing with a double Atwood's machine problem, and so far I got these $F=ma$ equations: $T - m_2g = m_2 a_2$ $T - m_3g = m_3 a_3$ $2T - m_1g = m_1 a_1$ $a_1 = -\frac{(a_2 + a_3)}{2}$ ...
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81 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) ...
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20 views

Are 1-D line sections of 2-D point source-invoked potential distributions positive definite?

Consider the 2-D potential distribution induced on the plane $y=0$ by a point source positioned at $(0, -y_0, 0)$ in the open halfspace below that plane. The material below the plane is assumed ...
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2answers
127 views

Calculating speed and distance travelled by an object using quadratic function

I'm actually getting stuck with a tricky part of a math problem using a quadratic function. An object propelled with an initial speed, has its altitude given after $t$ seconds by the quadratic ...
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1answer
30 views

time until gas is depleted cylinder

If one has a $10$ liter (water volume) tank of gas at $15$ Mpa, then how long until gas is depleted if gas is exiting the cylinder at a rate of $15$ liters per minute? ATM i have used ideal gas law, ...
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2answers
97 views

Can someone clarify the definition of flux?

I am confused by the concept of flux as used in vector calculus. Suppose I have a sphere. On the inside of this sphere is a spherically symmetric electric charge distribution. Now I want to find the ...
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2answers
50 views

How do I find torque on an axis when no weight or force is given?

I need some help with the following problem: Find the torque around the x axis of the triangle with vertices (0, 0), (1, 4) and (1, 0). Assume the density equals one. — Life of Fred Calculus, ...
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Question about DE related to physics; includes Hooke's Law and Newton's Second Law as well as system of DE equations and solutions, and a phase plane.

I mainly need help with part A, and a little bit on B and C. Thank you in advance for your answer or any comment or edit that helps!!! A second-order DE can be sometimes solved with clever ...
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27 views

Why is the difference of a stream function at two points the flow rate PER UNIT WIDTH?

For a velocity vector field $\bf{u}$$ = (u,w)$ in two dimensions ($x$ and $z$), we define the stream function $\psi$ to be; $$\psi(P_1) = \int_{P_0}^{P_1} \mathbf{u}\cdot\mathbf{n}dl$$ where $P_0$ is ...
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1answer
60 views

Force between parallel conductors using amperes law

Two parallel conductors are 0.3m long, and 0.15m apart. They each carry 2.5A of current in the same direction. Calculate the force between them. I did (2 * 10^-7) (2.5^2) / (0.15) and got the ...
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1answer
89 views

Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
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2answers
43 views

Why do we use the word “energy” for this integral?

I came across the following and I would like to know the motivation or idea behind using the word "energy". Suppose $u(x)$ is a $C^1$ function on some domain $\Omega \subset R^3$. Then, we can think ...
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1answer
45 views

Trying derive $a=v^2/R$ with complex numbers.

I'm trying to derive the radial acceleration formula for uniform circular motion: $a =v^2/R$. Here's what I've $\mathbf r(t)= Re^{-it/2}\mathbf e_1e^{it/2}$ $\implies \ddot {\mathbf r}(t) =\frac ...
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1answer
50 views

Wondering if my reasoning is correct for a tricky calculus limit word problem

and thanks for reading my post. The problem reads like this: If a book starts to fall off a building from 120 feet in the air and the position of the book after t seconds is given by s(t) = -16t2) + ...
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18 views

Does 'variance' have a physical meaning?

Let's say we have some probability density function $f(x)$, we can find the mean by solving $\mu = \int_{-\infty}^{\infty}xf(x)dx$. In a similar way, we can find the center of mass of an infinite wire ...