"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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Manifolds and magnetic potential

Assume we have a particle in $\mathbb{R}^3$, which we will subject to different fields independently. It will have some potential energy $U\in \mathbb{R}$ defined as some constant minus its kinetic ...
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20 views

Estimating curvature of oscillatory curve based on global constraints

I have a heuristic question about using global constraints of a problem to make estimates of local features of a curve, such as its curvature. Consider a suitably well behaved function on ...
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39 views

Meaning of this differentiation operators

I have been just reading this paper here: paper and was wondering how they carry out the differentiation in (4.9). In principle, this should be just the differentiation of 4.8 with the help of 4.7a. ...
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Center of Mass Coordinates

Can one view changing to center of mass coordinates http://www.worldforge.org/project/newsletters/July2002/LagrangianP3 as a kind of coordinate transformation in the same way one views changing ...
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1answer
33 views

Cross-Validation

Does anyone understand the paragraph below? The paragraph comes from Cross-valiation explanation at wikipedia. "It can be shown under mild assumptions that the expected value of the MSE for the ...
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1answer
201 views

Stoping time of light and photography [closed]

When taking a photo with a camera in manual mode, one needs to set the focus, the shutter speed and the exposure. The latter is controlled by adjusting the size of aperture, the circular opening that ...
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2answers
170 views

Why are 'differential operators on manifolds' differential operators?

It is clear what is meant by a differential operator on $\mathbb{R}^n$ (or some open subset). However, it is not clear to me why differential operators on smooth manifolds are defined the way they ...
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1answer
81 views

Bound on specific L2 function

Its stated in a paper by E. Lieb that this function is "clearly" in L2 $w(x)=|x|^{-1} - (g^2*|x|^{-1}*g^2)(x)$ with $g(x)=\xi^{3/4} \exp(-\pi \xi x^2/2)$ and therefore $||g||_2 =1$. It's ...
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35 views

Delta distribution - integration by parts of its differentiation

Some delta distribution physicist calculus. Assume there is given $$ \int_{\mathbb{R}^3} \sum_i f(\mathbf{x}) \delta^{(3)}(\mathbf{x}-\mathbf{a}_i) \ d^3x $$ with $f$ vanishing at infinity and ...
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43 views

Eigenvalues for the double-well potential

I am trying to find eigen-values, $E$, for the following differential operator: $$\left[ -\frac{1}{2}\frac{d^2}{dx^2} +L\left(x^2-a^2\right)^2\right]y(x) = E\,y(x) $$ where $L,a$ are two positive ...
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27 views

plane wave move out

I have a plane wave that is recorded by a set of receivers with x spacing between them (see the sketch). in the sketch (plane wave in black slant line, receivers are the little circles in black and ...
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15 views

Can multimodality exist in the absence of deterministic multistability?

I am not sure if this is the right SE to ask this question; it is about mathematical models for chemical reactions. I came across an article that says that multimodality of biochemical species can ...
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1answer
41 views

How to proof Frobenius Theorem in general?

The general Frobenius Theorem stating that Let $u_1,\dots,u_k$ be $k$ smooth linearly independent vector field on $M$. Let $$ W=\operatorname{Span}(u_1,\cdots,u_k) $$ Then $[u_i,u_j]\in W$ for ...
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83 views

Fourier-Laplace Transform of Heaviside Step function multiplied to Sine

In a Advanced Solid State lecture I encountered the following assertion- Fourier Transform of $\Theta(t)\sin(\omega_0 t)$ is ...
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2answers
42 views

Schrödinger-Operator on $L^2[0,2\pi]$.

In Reed-Simon Analysis of Operators they often talk about operators like $H = - \Delta +V$ as an operator on $L^2[0,2\pi]$ (like in Theorem XIII 88. What do they mean by that? Or is their a canonical ...
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30 views

Green's function the way George Green defined it

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...
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1answer
86 views

Uniform Circular Motion with Banked Road and Car

In Uniform Circular Motion, if a car is rounding a curve at a certain speed, and the angle of the road allows the car to drive around at that speed, that speed is called the "design speed." If the ...
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25 views

Is it possible to simulate fluid dynamics in a time-based and deterministic manner?

The Problem Domain I have a number of network-connected PCs. I want to be able to simulate and replicate the same simple fluid dynamics simulation (Eg Navier-Stokes), in real-time, between them. That ...
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1answer
87 views

Very strange “fact” regarding movement

Perplexing (for me at least) statement from the site: http://www.quora.com/Mathematics/What-are-some-of-the-most-counterintuitive-mathematical-results "Fact: You can have a car stand still for ...
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3answers
109 views

Compute the integral $\int_0^\pi \frac{\sin x}{x}dx$ .

How to compute precisely the integral $$\int_0^\pi \frac{\sin x}{x}dx$$ analytically? It is well-known that $$\int_0^{+\infty}\frac{\sin x}{x}dx=\frac{\pi}{2}.$$ One way to compute the above integral ...
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1answer
48 views

Solving Simplified Hamilton's Equation

I have a question on a project that I am working on. I have included a large amount of the background information so that all relevant information is included, however the question is as follows (it ...
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1answer
107 views

Learning Advanced Mathematics

I'm a 12th grade student and I've recently developed a passion for mathematics . Currently my knowledge in this particular area is comprised by : single-variable calculus , trigonometry , geometry , ...
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1answer
60 views

General relativity from a mathematics point of view

Goodmorning, I'm a university math student. I'm quiet familiar with differential geometry and I want to study the theory of general Relativity. I try to read some books, but all of these explain the ...
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1answer
40 views

What is the area under this graph? I am getting 2 different answers when using two different methods.

I found the area of the triangle using the formula first and got 2.25 Then found the area of the trapezium (Area of the whole graph) and subtracted the unshaded region and got 2.25 again. The third ...
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32 views

Heisenberg uncertainty principle in D-dimensional

For Heisenberg uncertainty principle in D-dimensional there is $d^2$ in the formula.where does this additional term comes compared with the case of one dimensional?
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50 views

how to transform a quadratic equation into a matrix form?

I have this type of equation: $$ - a^ {2} A - \eta ^{2} B - a \eta C - b^{2} A' - \eta' ^{2} B' - b \eta' C' - a \eta' D - b\eta E $$ The capital letters, $A, A', B, B', C, C', D, E$ are just the ...
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33 views

Root space of a Semi simple group an LVS?

A semi-simple Lie group has a Cartan Subalgebra ($H$) (CSA) -an LVS, Dual to this CSA LVS is root space($H^*$), which is set funtionals that map elements of CSA to real numbers and hence useful in ...
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1answer
20 views

definition of a smooth scalar potential

Have been asked to show that any flow described by a smooth scalar potential is irrotational. I know to show if a flow is irrotational curl of q = 0. But not too sure what is meant by smooth scalar ...
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1answer
40 views

A question about John Baez's definition of “stochastic Petri nets”.

John Baez, in his blog posts, introduces stochastic Petri nets as a Petri net that contains an additional function which maps each transition in the set of transitions $T$ to a real number. This ...
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47 views

Invalid use of the analytic continuation of the Riemann zeta function?

Watching this video on You Tube I got the impression that some sciences (in this case physics) use the analytic continuation of the Riemann zeta function without justification. Maybe this is just my ...
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45 views

Geodesic of symplectic manifold

Can I define geodesic on a manifold without Riemann structure? To be more specific, how can I define geodesic at symplectic manifold? Let's just look at simple case with symplectic form as ...
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125 views

Homotopy quantum field theories as functors

A homotopy quantum field theory is a symmetric monoidal functor $\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following ...
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1answer
40 views

Solutions to differential equation

Let $\left\{a,\lambda\right\}\subset\mathbb{R}$. Let the following differential equation for a function $x\left(t\right)\in\mathbb{R}^{\mathbb{R}}$ be given: $$ \boxed ...
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48 views

Calculate velocity after certain displacement

Given is a function $a(t)$ for the acceleration. Starting from an initial velocity $v_0$ I want to calculate the velocity $v_1$ after a certain displacement $s$. Is this calculation possible since ...
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104 views

Studying maths for physicists [closed]

I am looking forward to be a theoretical physicist ,to unify general relativity and quantum mechanics ( to make the theory of everything ) How should I study maths? Should I study proofs of ...
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67 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
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2answers
109 views

Sketching phase portraits [closed]

I am trying to answer this question: I would like to know how I go about drawing a phase portrait. All of the examples in my notes are simply the solution with no explanation, and this method of ...
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1answer
44 views

Changing variables for a partial differential equation

If I have the following systems of PDE \begin{align} u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)}=0\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)}=0, ...
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1answer
53 views

Direct Delta Function

I was reading a Mathematics for Physics book when I saw these exercises. By using the knowledge of direct delta function, show that: $\int_{-\infty }^{+\infty }f(x)\delta '(x-y)dx=-f'(y)$ ...
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1answer
37 views

Gauss Law and surface integral

Could somebody please explain to me the bottom line here. I don't understand how dS becomes r dtheta. I thought dS was supposed to be an outward pointing normal which is surely just r? I'm guessing ...
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Mathematics needed in the study of Quantum Physics

As a 12th grade student , I'm currently acquainted with single variable calculus, algebra, and geometry, obviously on a high school level. I tried taking a Quantum Physics course on coursera.com, but ...
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1answer
97 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
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1answer
30 views

Wronskian Bessel Equations

I need to compute the wronskian of $J_n$ and $Y_n$ (the Bessel functions of the first and second kinds). I've been able to find in many sources that it is $$W(J_n,Y_n)=\frac{\pi}{2x}$$, but I haven't ...
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2answers
22 views

How can you determine Mass / kg from the attached table

I spent the last two hours trying to figure out part D and I can't get my head around it... Part D has the formula mass/kg = Mass/kg = C / ΣC I'd imagine "C" is the number of C section of the ...
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Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
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209 views

Free lecture notes to Carl Bender's Mathematical Physics video lecture course?

I am just watching Carl Bender's Mathematical Physics video lecture course (about asymptotics and its application in physics) http://www.perimeterscholars.org/328.html which is great! Are there any ...
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35 views

Linearize a specific eqution

Is it possible to linearize this equation ? I tried without success .. ...
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2answers
95 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
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39 views

Determining how accurate an ellipse fit is

So I have an image of bacteria particles which are often shaped very irregularly with many grooves. Im trying to fit ellipses onto these particles so I can get a better, more smooth analysis of the ...
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1answer
82 views

How can the tension force be computed to test if a shape is moving or not?

Source Given the coordinates of n 3D joints (1kg each) connected by m rods. Assume rods have zero mass and joints with z=0 are fixed to the ground while others are free to move, will the shape be ...