"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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2
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2answers
31 views

Is it possible to build a fiber bundle whose fibers are different? (Or we should not call it a fiber bundle?)

Suppose there is a fiber bundle $E$. The base space is $M$ so that $\pi:E\rightarrow M$ is the projection. By the definition, the bundle has a typical fiber $F$ such that the local trivialization over ...
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0answers
8 views

Thomson problem vs. maximizing sum of distance

Given $N$ equally charged points lying on the unit sphere ("electrons"), the Thomson problem consists of finding the configuration of these points such that the electrostatic potential energy $$ ...
0
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0answers
30 views

Calculating force per unit width

Question: A line source of strength $2πm$ is located a distance $a$ above a horizontal plate. Find the force per unit width on the plate, ignoring gravity and taking the pressure below the plate to be ...
2
votes
2answers
274 views

Is Physics really a rigorous subject? [on hold]

Though I can't give a precise definition of the term rigor (or better to say mathematical rigor) but intuitively in case of mathematics one may note that when we say that 'the proof is rigorous' we ...
1
vote
1answer
59 views

A difficult question on mathematical physics

Let $TQ^*$ be equipped with its standard symplectic structure and let $X_H$ be a Hamiltonian vector field which is tangent to the fibers of $\pi: TQ^* \to Q.$ I need to show that $$H=h \circ \pi = \pi ...
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0answers
18 views

Is my maths correct about Projectile Motion?

The initial launch velocity of a projectile is determined using R = 2$u^2$ / g. From this, can we calculate maximum height and time of flight if we know the initial launch velocity derived? I ask ...
0
votes
1answer
32 views

Two Body Orbit Problem [on hold]

I really need help urgently. What I've got are two different circles with their radius coming from a fixed center point. The two radius's which can be considered as a line are being rotated at a ...
0
votes
2answers
33 views

Does Runge Kutta need future state of system?

In order to use the RK methods, you need to know the state of the system at future time-steps which can be expensive to compute (e.g., in physics simulations). As a simple example I'll use RK-2: In ...
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0answers
35 views

Why does this graph produce a straight line? [duplicate]

When we graph the sin and cos of theta against the range of a projectile, we get a straight line. When we graph range against angle, we get a hyperbola. Why does the sin and cos of theta against ...
1
vote
2answers
67 views

What does adding $\sin\theta \cos\theta$ make my graph a linear relationship?

What is the point of adding sin n cos of theta when graphing range? e.g. I see on hyperphysics a graph of range vs sin n cos of theta and it makes the experimental data embody a linear relationship. ...
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0answers
17 views

HYDROSTATIC FORCE PROBLEM [closed]

The cross section of a dam has the shape of an isosceles trapezoid, 100 feet wide at the bottom, 50 feet wide at the top with a slant height of 75 feet. The dam is 800 feet long. Find the hydrostatic ...
-2
votes
4answers
168 views

Is $\nabla$ a vector?

The following passage has been extracted from the book "Mathematical methods for Physicists": A key idea of the present chapter is that a quantity that is properly called a vector must have the ...
1
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1answer
22 views

Pure Point Spectrum implies Spanning Eigenfunctions

If $H$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$, and the spectrum of $H$ is a pure point spectrum, i.e., the spectrum consists of discrete eigenvalues (perhaps with multiplicity ...
0
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3answers
38 views

How to check that this is an orthogonal linear map with $\det (A) = 1$, so it is a rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
2
votes
2answers
69 views

what is scattering theory?

I often read the the words "scattering theory", "scattering data", "scattering matrix", scattering XXX ... in my math lecture, but I realised that I am not able to define it correctly. A short search ...
0
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0answers
24 views

How to find the axis of the rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
1
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0answers
53 views

Neutron density PDE

On Mathews and Walker's book exercise (8-2) We are given that the neutron density n inside $U_{235}$ obeys the differential equation $$\nabla ^2u+\lambda u=\frac{1}{k}\frac{\partial{n}}{\partial{t}} ...
2
votes
1answer
76 views

Tough second order differential equation

I can't figure out this diff equation (in cylindrical coordinate). How can I solve it ? Any comments appreciated $$ \frac{1}{r}\frac{d}{dr}(r\frac{dE}{dr})+\frac{d^2E}{dz^2}+(\epsilon_0 ...
1
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0answers
70 views

Characteristic class integral: on what manifold does $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$ hold?

Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds? Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd ...
0
votes
0answers
25 views

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}, ...
3
votes
3answers
86 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 ...
0
votes
1answer
13 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 ...
0
votes
1answer
46 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 ...
2
votes
2answers
84 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 ...
0
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3answers
32 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 ...
0
votes
2answers
34 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 : ...
1
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3answers
52 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 ...
0
votes
1answer
52 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} ...
0
votes
1answer
29 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 ...
3
votes
0answers
31 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) - ...
0
votes
1answer
23 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. ...
0
votes
2answers
14 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)} = ...
0
votes
1answer
13 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 ...
0
votes
1answer
43 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 ...
2
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0answers
28 views

Help needed in understanding the question

Let $D:=\mathbb{C}^* \to \mathbb{C}^*$ be regarded as an open subset of $\mathbb{C}^2$ which is equipped with its standard (symplectic) $2-$form $$\omega_{std}=\frac{i}{2}(dz_1 \wedge d\bar{z_1}+dz_2 ...
0
votes
1answer
46 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: $$a \dot\ (b\times c)=a_i(\epsilon_{ijk}b_jc_k)=\epsilon_{ijk}a_ib_jc_k.$$ ...
1
vote
1answer
71 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 ...
2
votes
2answers
58 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 ...
0
votes
1answer
28 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 ...
1
vote
1answer
57 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 ...
0
votes
3answers
37 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 ...
1
vote
1answer
69 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] ...
1
vote
2answers
44 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 ...
0
votes
0answers
25 views

Lubrication Theory: Quick Question!

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
-6
votes
1answer
41 views

Calculating the mass of the earth [closed]

The formula I am using to calculate the mass of the earth is: M = ar2/G = 5.98 × 1024 kg. a being the acceleration of gravity (9.8 m/s squared), r being the radius of the earth ((6.4) *(10^6)), and ...
0
votes
3answers
47 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 ...
4
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0answers
55 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: ...
4
votes
2answers
85 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 ...
0
votes
0answers
29 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) = ...
0
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0answers
18 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 ...