"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 ...

learn more… | top users | synonyms (1)

2
votes
0answers
19 views

Finding highest weight of a $gl(3)$ submodule of a $gl(4)$-module

We have $gl(3)\subset gl(4)$. I have a $gl(n)$ $V(3,2,1,0)$-module. I want to know two things: 1) What is the $gl(3,\Bbb C)\subset gl(4,\Bbb C)$ branching rule for the $gl(n)$ ...
1
vote
1answer
30 views

Eliminate all parameters from the differential equation $u_t-Au_x-Bu^3+Cu_{xx}=0$.

The question is to scale the equation $$u_t-Au_x-Bu^3+Cu_{xx}=0$$ to eliminate all parameters. where $D>0$ and $A,B$ are nonzero. I tried to substitute $U=U(x/L,t/t_0)$ into the equation ...
2
votes
1answer
26 views

Solving differential equation and obtain expressions for unknowns?

I have the following differential equation $my'' + \beta y' + mg = 0$ , with condition $y(0)=0$. I need to solve the equation and obtain expressions for the unknowns. I have attempted to use the ...
5
votes
2answers
156 views

How does curl relate to rotation?

The operation mathematically means $$(\nabla \times \vec A)\cdot\hat n = \lim_{\Delta S\to\ 0} \frac{\oint\vec A\cdot\ d\vec l }{\left | \Delta S \right |}$$ and the proof of this is quite logical. ...
2
votes
0answers
95 views

Solving recursions by calculating determinant of an infinite matrix

In this reference (pg. 4) and few others a specific parameter in a recursion formula is solved by setting the determinant of an infinite matrix to $0$. In this precise case we have $c_{n-1} - D_n c_n ...
0
votes
0answers
26 views

What is a weight vectors weight really?

What does a weight vector with weight $(\lambda_1,\cdots,\lambda_n)$ actually mean? Let $V$ be a $gl(n)$-module and I have that $v\in V$ is a weight vector if it is an eigenvector for all elements of ...
2
votes
2answers
64 views

How do you swap x/y to y/x?

I saw this video on Lorentz transformation and needed to refresh my memory a bit. If $$\frac{t}{t'}= \sqrt{1-\frac{v^2}{c^2}}$$ and $$\gamma = \frac {t'}t $$ How do you make this equal ? ...
0
votes
0answers
33 views

Stability of least isolated eigenvalue under positive perturbation

This would be a useful theorem. Have you seen it anywhere? $\mathbf{Theorem:}$ Suppose a self-adjoint operator $H_0$ on a Hilbert space has a simple isolated least eigenvalue $0$ with separation ...
1
vote
1answer
90 views

What does $\operatorname{supp}(A)$ mean?

I'm looking at a paper (specifically this one). In the paper, we have a positive operator $A$, and the operator $\operatorname{supp}(A)$ is supposed to be a projection operator. Does anybody know ...
2
votes
0answers
42 views

Affine geometry textbook

What's a good recommendation for a book on affine geometry at the undergrad level? I ask because I skimmed through the first bit of Vladimir Arnold's Mathematical Methods of Classical Mechanics and ...
1
vote
1answer
60 views

Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat?

Yesterday I learned that geometric relations between events can be characterized generally (and up to a common non-zero factor) in terms of their pairwise "Lorentzian distance, $d_{\ell}$", which ...
1
vote
2answers
85 views

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 ...
1
vote
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 ; ...
0
votes
0answers
17 views

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 ...
1
vote
0answers
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) ...
9
votes
2answers
197 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 ...
0
votes
1answer
39 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 ...
2
votes
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 ...
0
votes
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 ...
3
votes
1answer
58 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 ...
0
votes
1answer
41 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 ...
0
votes
0answers
10 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 ...
1
vote
1answer
64 views

derivative of tensor

Hi I am trying to simplify $$ A=\frac{1}{2}\left(\partial_j u_i+\partial_i ...
3
votes
1answer
62 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: ...
4
votes
2answers
88 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 $ $ ...
0
votes
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 ...
1
vote
1answer
62 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 ...
0
votes
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 ...
0
votes
2answers
61 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 ...
0
votes
0answers
28 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 ...
2
votes
1answer
42 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. $z$, ...
0
votes
1answer
36 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: ...
11
votes
3answers
682 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 ...
5
votes
1answer
80 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 ...
0
votes
0answers
30 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 ...
3
votes
0answers
128 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
votes
1answer
70 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 ...
-1
votes
2answers
80 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. ...
0
votes
1answer
125 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 ...
0
votes
0answers
42 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 ...
1
vote
1answer
69 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
votes
1answer
29 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 ...
0
votes
1answer
48 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?
0
votes
1answer
54 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 ...
2
votes
2answers
41 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
votes
4answers
814 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 ...
0
votes
0answers
32 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] + ...
0
votes
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 ...
1
vote
2answers
47 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 ...
0
votes
1answer
70 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 ...