"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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Trouble in understanding the explanation of Dirac-Delta Function by Feynman.

This is quoted from Feynman's Lectures' Normalization of the states in $x$: We return now to the discussion of the modifications of our basic equations which are required when we are dealing with ...
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1answer
43 views

Homogenous Linear Equations in the form of Determinants

In Arfken's "Mathematical Methods for Physicists" How did he arrive to: $x_1/x_3 = \frac{(a_2b_3-a_3b_2)}{(a_1b_2-a_2b_1)}$ Starting from:- $$ a_1x_1+a_2x_2+a_3x_3=0 ; \\ b_1x_1+b_2x_2+b_3x_3=0 ; ...
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Why does for the $16$ representation of $O(10)$ the term $16 \otimes 16 \otimes 16 \otimes 16 $ vanish?

Given the $16$ dimensional representation of the Lie group $O(10)$, why does the quartic term $$ 16 \otimes 16 \otimes 16 \otimes 16 $$ vanish? Surely I'm missing something very obvious, because ...
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37 views

How can I compute how many and which quartic invariants of a given group representation are linearly independent?

Given a representation $R$ of a group $G$ and the corresponding tensor product $$ R \otimes R = R_1 \oplus R_2 \oplus R_3 \oplus \ldots $$ how can I compute how many quartic terms $ \mathrm{O} (R^4) ...
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167 views

Is $\int |x\rangle \langle x|dx$ Mathematical?

I am enrolling in a Quantum Mechanics class. As we all know, the formulation of the basic ideas from QM relies heavily on the notion of Hilbert Space. I decide to take the course since it might help ...
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1answer
31 views

Work against friction is proportional to length of path

If, given that the frictional force is constant, one wants to show that the work done against friction is proportional to the length of the path, would this line of reasoning be correct? We can use ...
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1answer
32 views

Boundary value problem $y''(x)= \kappa^2 \left(y(x) + \frac{y(x)^2}{2} \right)$

A physical problem brought me to the following boundary-value problem $y''(x)= \kappa^2 \left(y(x) + \frac{y(x)^2}{2} \right)$ with $y(0)=0$ and $y(C)=-58$ for some $C>0.$ If there was no ...
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1answer
78 views

I can't find any formula to solve this differential equation.

$$\frac{dx}{dt} + x^2 = B + A\cdot e^{C\ln\big(\frac{x}{x_0}\big)+\ln(x_0)}, \quad x(t_0)=x_0$$ Can anyone please help me where I can start from this equation? I simplified a complicated equation ...
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1answer
54 views

How to rigorously understand continuous bases?

In Quantum Mechanics it is quite common to see the idea of a continuous basis of a Hilbert space. In truth if $\mathcal{H}$ is the state space of a quantum system and if $X : U\subset \mathcal{H}\to ...
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1answer
40 views

Hey, anyone who can help me with this calculus-problem?

I have something I wanna ask about. If a plane flies along curve 1/x^2 it shoots at a bunker at position (10,0). The shot will follow the tangent line of the curve. For what value x= a should the ...
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0answers
8 views

Is there a better way of showing that this maximises the range up the ramp?

Consider a system where a projectile is shot up a ramp. Let the ramp be inclined at some angle $\alpha$ and the projectile is shot at some angle $\theta > \alpha$ with fixed velocity $V$. If we ...
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0answers
8 views

My maths question is a Measurement question based on Finishing the quad Seating

Finishing the Quad Seating The Quad area has a new section on the full length of one side. The top of he seating is to have a wood finish which is o go to the full length of 26 meters. The piece ...
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1answer
57 views

derivative of tensor

Hi I am trying to simplify $$ A=\frac{1}{2}\left(\partial_j u_i+\partial_i ...
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1answer
52 views

Second fundamental form for surfaces (extrinsic curvature)

I'm trying to understand the Gibbons–Hawking term. There appears $K$- a trace of the second fundamental form, as they say. As far as I know, the second fundamental form is the following thing: ...
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2answers
71 views

Energy conservation for the wave equation

I recently encountered this problem in PDE class involving a concept I have never met, it states: $ u_{tt} - u_{xx} = 0 ; \space \space 0 < x < 1 ; \space \space t > 0 $ $ ...
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1answer
22 views

How to add Displacement using the Component Method?

How would I add the displacement using the Component method. So I am Given the Following: d = 20cm [N] d2 =50cm [S 35 E] d3 =100cm [ W 15 S] what I did was I first drew them and after that I added ...
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1answer
53 views

Noether's theorem and lie groups. A question related to the meaning of a lie group.

I'm doing a small research project on applications of Group theory and chose to investigate Noether's theorem. Evidently, Noether's theorem at its highest level does contain lots of elements of Lie ...
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0answers
36 views

Proving an equality with complex numbers

I have the following problem. I have two vectors $ \begin{bmatrix} \alpha_1\\ \beta_1 \end{bmatrix} $ and $ \begin{bmatrix} \alpha_2\\ \beta_2 \end{bmatrix} $ where the entries of the vector are ...
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2answers
47 views

How to verify the completeness of a given Hilbert space?

Hilbert space is defined as a complete inner product space. It is also said that a finite dimensional vector space with inner product is trivially complete. I have two questions. How can I verify ...
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0answers
25 views

Boundary conditions of Magneto-micropolar fluid flows

When we consider the 2D-Magnetohydrodynamic (MHD) equations, the boundary condition of mageto velocity is given by $B\cdot n=0$ and $curl B=0$ on $\partial\Omega$. The 2D magneto-micropolar fluid ...
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1answer
35 views

Finding electric flux given volume charge density

Question: Let $\rho_v = 8z(1 - z)$ C/m$^3$ for $0 < z < 1$ m, $8z(1 + z)$ C/m$^3$ for $-1<z<0$, and $0$ for $|z| > 1$. (a) Find $\vec{D}$ everywhere. (b) Sketch $\vec{D}_z$ vs. $z$, ...
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1answer
29 views

Calorimetry Question

If you are familiar with the formula q = (m) * (c) * (Delta Temperature) q = (Delta Heat) m = Mass c = Heat Capacity Delta T = Change in temperature The question is asking to solve with this info: ...
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634 views

Qualitatively, what is the difference between a matrix and a tensor?

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical ...
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1answer
68 views

Quantum Mechanics state space

In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm ...
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29 views

Does $\{f,g\}$ mean anything when neither $f,g$ are the hamiltonian of a system?

Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...
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93 views

Heat equation in 2d Circle polar coordinates

I was presented this problem in PDE class involving heat equation on unit circle in polar coordinates using separation of variables, giving the following heat equation problem: $ u_t = 9\Delta u ...
4
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1answer
66 views

An identity involving matrix norm and vector norm in relation to a density matrix in physics

This question is self-contained. For those who are interested, it arises from my study of the paper: K Lendi (1987), Evolution matrix in a coherence vector formulation for quantum Markovian master ...
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2answers
67 views

Double integral of symmetric function

Can someone please derive and explain how the LHS is equal to the RHS using the fact that the function $f$ is symmetrical with respect to time variables $t_1$ and $t_2$. Here $t$ is some constant. ...
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1answer
79 views

Physics related initial value problem (horizontal spring mass system)

Consider the horizontal spring-mass system where the spring-force is the only force acting on the mass. Suppose that a mass is initially at $x=x_0$ with an initial velocity $v_0$. Show that the ...
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0answers
34 views

50ft cable weighs 1lb/ft lifts 100lb box. Find work done by lifting box 50ft

I've been trying to work on this old exam problem that states: A 50ft cable that weighs 1 lb/ft is used to lift a 100 lb box up 50 ft. How much work is done? The problem lists that the correct ...
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1answer
63 views

Gauss' law in differential form for a point charge

I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that $ \nabla\cdot E=\frac{1}{\epsilon 0}\rho$, but when ...
2
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1answer
28 views

Circular Banked Track Friction [closed]

I've tried answering this question by resolving forces, then finding an expression for friction and inserting the given data so I can prove $F = 0$. However, I never get an answer of $0$. How do I ...
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1answer
45 views

Can an object be in freefall if it is traveling upward? [closed]

Can an object be in freefall if it is traveling upward? I'm thinking the answer is no?
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1answer
45 views

solving for initial velocity using the position vector

I am having trouble wrapping my head around this problem. The big picture is that i have to calculate the initial velocity v= needed for a soccer ball to cross a goal line. this is a homework ...
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2answers
38 views

Are four vectors in Special Relativity considered to be tensors?

In particular, I would like to know if the four velocity and the four acceleration are tensors.
2
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4answers
726 views

Where do the maximum and minimum values of velocity occur in the elliptic orbit [closed]

Where do the maximum and minimum values of velocity occur in the elliptic orbit? Why? Find the velocities. I need a detailed answer of this problem. Please help me. I know the answer. Maximum and ...
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0answers
31 views

What is the form that this commutation take?

[AB + BA,CD] is it [AB,CD] + [BA,CD]? More specifically I'm trying to find [xp + px,p*p] where p is p_x. I have tried [xp,pp]+[px,pp] = x[p,p]p + xp[p,p] + pp[x,p] + p[x,p]p + p[x,p]p + pp[x,p] + ...
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2answers
29 views

Multiplying vectors w/i, j, k [closed]

Three vectors are given by $a= 3.0i+3.0j–2.0k$, $b= -1.0i-4.0j+2.0k$, and $c= 2.0i+2.0j+1.0k$. Find (a) $a·(b\times c)$, (b) $a·(b+c)$, (c) $a\times(b+c)$. I know that you can multiply vectors to ...
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2answers
39 views

Recurrence relation with reciprocal in a circuit

The motivation for this recurrence relation is to find the total resistance in this circuit: Assuming that the capacitor has no resistance, with only one loop of the circuit, (let us suppose) the ...
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1answer
66 views

How to express this equation in terms of v?

I read from my physics textbook that the magnitude of a ripple voltage decreases if the capacitance is increased in a rectifier circuit, but the textbook didn't specify what the exact mathematical ...
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2answers
78 views

What is the connection between a metric and a manifold?

I am in process of reading a paper which contains something called a "Shahshahani Metric" which has uses in mathematical biology ...
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0answers
13 views

Convexity of the space of isospectral density matricies

Given the manifold $M$ of all complex, square matricies $\rho$ s.t. $\rho$ positive semidefinite $Tr(\rho)=1$ $\rho^{\dagger} = \rho$ consider the submanifold $N(\rho) = \{U\rho U^{\dagger} \ \ ...
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0answers
62 views

Placement of protons and neutrons in the nucleus

So, I'm creating a program that would represent a given atom (also different isotopes) in 3d view. I'd need some kind of formula to calculate the position of protons and neutrons to form a nucleus. ...
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2answers
170 views

Conformal Killing vectors fields on Minkowski spacetime

As is well known Minkowski spacetime (which is four dimensional vector space with scalar product $\eta _{\mu \nu}$ of signature $-+++$) is maximally symmetric, which manifests itself in presence of ...
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2answers
45 views

Center of mass of the three pennies?

Three pennies each of radius R and mass M attached at their edges. How to find the center of mass of the three pennies?
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1answer
36 views

Hamiltonian systems. Canonical Transformation.Wikipedia

I have some questions regarding Canonical transformation and hamiltonian systems. I will upload an image with the text from wikipedia: How can I obtain same results... I have no idea how they ...
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1answer
29 views

Covariant derivative notation?

I was reading up on covariant derivatives and came across this document. On the second page it says: We define a procedure called parallel transport by defining a vector $\vec A (\lambda)$ along ...
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2answers
390 views

Arnold's theorem on action-angles.

I changed the question slightly in its form to make it more readable. I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this ...
2
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0answers
12 views

Covariance of nonlinear sde

My problem is to compute the covariance of the following Ito process $$ dX_t=AX_t+\sum_{k=1}^{n}B_kX_tdW_k, $$ where $A,B_k$ are nonlinear operators defined on a complex separable Hilbert space $H$. ...
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1answer
27 views

Are there some simple way to remember Braid equation and Yang-Baxter equation?

The Yang-Baxter equation is $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ and the Braided equation is $R_{12} R_{23} R_{12} = R_{23} R_{12} R_{23}$. The indices in the equations are complicated. Are ...