DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com The mathematics tag covers non-applied pure mathematical disciplines that are traditionally not part of the mathematical physics ...
2
votes
4answers
445 views
Generalized functions in physics
Prior to the Dirac delta function, what other distributions functions where physicists using? I find it hard to motivate the theory of generalized functions with just the delta function alone.
-1
votes
1answer
253 views
Functional Derivative of Convolution
How to carry out the following functional derivative?
$$\frac{\delta F}{\delta n(r)}$$ where $$F=\int dr n(r) \int C(|r-r'|) n(r') dr'$$
is it simply:
$$2 \int dr' C(|r-r'|) ...
9
votes
4answers
3k views
How can area be a vector?
My professor told me recently that Area is a vector. A Google search gave me the following definition for a vector:
Noun: A quantity having direction as well as magnitude, esp. as
determining ...
2
votes
1answer
60 views
What interpretive difference is there between defining a function with or without a differential as a postfactor?
I have thought about this and looked for answers for a long time now, but I do not have any name or label for this problem, which is the reason for the long title of this question.
I have come across ...
4
votes
2answers
254 views
Combinatorial sum in a problem with a Fermi gas
I'm solving a problem involving a Fermi gas. There is a specific sum I cannot figure my way around.
A set of equidistant levels, indexed by $m=0,1,2 \ldots$, is populated by spinless fermions with ...
13
votes
9answers
3k views
What is the logarithm of a kilometer? Is it a dimensionless number?
In log-plots a quantity is plotted on a logarithmic scale. This got me thinking about what the logarithm of a unit actually is.
Suppose I have something with length $L = 1 km$.
$\lg L = \lg km$
It ...
10
votes
7answers
1k views
The philosophy behind the mathematics of quantum mechanics
My field of study is computer science, and I recently had some readings on quantum physics and computation.
This is surely a basic question for the physics researcher, but the answer helps me a lot ...
2
votes
4answers
208 views
Can we make a change of variables (for example to polar coordinates) into a divergent integral?
I know that if the integral is convergent we can always make a change of variable to make it better, however what happens with DIVERGENT integrals? can we make a change of variable into a divergent ...
6
votes
1answer
402 views
Is C60 really the “most spherical” fullerene?
In the late 80's and early 90's, Smalley and others made claims that the C60 fullerene bearing icosahedral symmetry was the most spherical molecule known, and perhaps the most spherical that could ...
2
votes
6answers
1k views
real world applications of Mathematics which use functions with singularities, not just as a matter of mathematical taste but for conceptual reasons
EDIT [for aptness of this question to this site, read 'real world applications' as 'applications in Physics']
The concept of function (of the form $f : \mathbb{R} \to \mathbb{R}$ ) has been used in ...
7
votes
6answers
384 views
Objects in Physics as a mathematician would see them
I'm a mathematician with hardly any knowledge of physics. Before I start reading volumes of physics books, I have a few questions that have been bugging me and that will help me start reading physics.
...
8
votes
1answer
6k views
Where does this equation originate from? (found in the Big Bang Theory)
Recently, I've been watching "The Big Bang Theory" again and as some of you might know, it's a series with a lot of scientific jokes in it - mostly about Physics or Mathematics. I understand most of ...
-2
votes
1answer
300 views
How to simplify e to power of j.t [closed]
I have exam tommorow and cant get this figured out :( dont blame me but please answer this question. I Want to simplify this term:
3 times ((e topowerof 5jt) + (e topowerof -5jt))
Thanks.
2
votes
2answers
308 views
How do you find conserved quantities for linear second order ODEs?
I have a differential equation of the form
$ \frac{d^2 y}{dt^2} + f(t) \frac{dy}{dt} + g(t) y = 0 $
where $f$ and $g$ are known functions of time.
Is there a systematic (or otherwise) way of ...
2
votes
2answers
267 views
a question on Lagrange's equation when the time derivative of the generalized co-ordinates is constant
Consider a system whose generalized co-ordinates are $q_i$ and is under the constraints $\dot{q_i} = K_i \forall i = 1,2,3,...$ where $K_i$ are constants. I have a problem in writing the Lagrange's ...
9
votes
3answers
391 views
infinite grid of planets with newtonian gravity
Assuming only Newtonian gravity, suppose that the universe consists of an infinite number of uniform planets, uniformly distributed in a two-dimensional grid infinite in both directions and not moving ...
2
votes
2answers
477 views
Helmholtz decomposition in the plane
Prove or disprove the following proposition:
For any smooth plane vector field $\mathbf{H}=\left(H_x,H_y\right)$, there exist scalar potentials $\phi$, $\psi$ such that
$H_x=\frac{\partial \phi ...
4
votes
1answer
1k views
Uniqueness of Helmholtz decomposition?
Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that
$$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$
...
0
votes
1answer
167 views
A question on smooth 1-manifolds
Consider two people living on two different smooth 1-manifolds $S$ and $T$ as shown in figure 1. The manifold $S$ is a bump function joining the points $A$ and $B$ and the manifold $T$ is formed by ...
3
votes
1answer
990 views
Why is the Yang–Mills existence and mass gap problem so fundamental?
Ref: Yang–Mills existence and mass gap
Can anyone please explain why the Yang–Mills Existence and Mass Gap problem is so important / fundamental to contemporary mathematics (and, presumably, ...
2
votes
4answers
2k views
How many digits of Pi are required in physics?
In other words: which physics experiment requires to know Pi with the highest precision?
10
votes
2answers
888 views
Applications of the Spectral Theorem to Quantum Mechanics
I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
2
votes
2answers
309 views
Diffusion across an interface and conservation of mass
I am reading a physiology book chapter (Mathematical Physiology, by Keener --Respiration chapter) about the gas exchange between capillaries and alveoli. It seems that this gas exchange can be modeled ...
0
votes
0answers
248 views
Fermat's Last Theorem [closed]
How is this Theorem useful, and how can it be used to advance mathematics, or science?
2
votes
1answer
340 views
What is the mathematical nature of space time quantization in string theory/super string theory?
I don't know much about string theory, apart from it being a theory of everything which brings QM, QED and nuclear forces and gravity under one single roof. I am curious to know from a mathematical ...
3
votes
0answers
451 views
How do I derive the critical temperature for bose condensation in two dimensions?
In class we derived the 3D case, but there's a step I don't understand:
$$ N = g \cdot {V \over (2 \pi \hbar)^3} \cdot \int\limits_{0}^{\infty}{1 \over{e^{\left( E_p \over{K_B T}\right)}-1}} d^3 p = ...
0
votes
6answers
318 views
Are the laws of physics applied mathematics? [closed]
This questions started with a question I had about gravity. If two objects of different weights fall to the earth at the same rate of acceleration, then it seems to me that gravity is in some ways ...
6
votes
2answers
1k views
What is the covariant derivative in mathematician's language?
In mathematics, we talk about tangent vectors and cotangent vectors on a manifold at each point, and vector fields and cotangent vector fields (also known as differential one-forms). When we talk ...
2
votes
1answer
318 views
Problem deriving kinetic energy from work
I'm currently reading a nice introductory book (german, could be translated as "Physics with a pencil"). The author works a lot with differential calculus and antiderivatives (integrals will be used ...
2
votes
0answers
308 views
A question about Dirac operator
The Dirac operator at 2 dimension can be written as
$$
D=\sum_{k=1,2}\sigma^{k}D_{k}=\left( \begin{array}{cc}
0 & \partial_{x}-i\partial_{y}-i(A_x-iA_y)\\
...
6
votes
5answers
502 views
Are the solutions in radicals of cubic and quartic of any use in physics?
We all know that there are analytic formulae to solve quadratic, cubic and quartic polynomial equations. But it seems to me that the only solution that widely used is physics is the solution of ...
2
votes
2answers
136 views
Mathematical Elegance in a Cosmological Theory Considered Necessary?
In general, it seems cosmological theories that encompass more and more of the phenomena of the universe are expected to be more and more mathematically elegant, in conception if not in detail.
Our ...
2
votes
1answer
236 views
Can the geodesic propagators in the Euclidean BTZ black hole can be written in terms of meromorphic functions on its conformal boundary?
I'm interested in knowing if ,in the context of AdS_{3}/CFT_{2}, we can (and how to) express the geodesic propagators on the bulk space of the Euclidean AdS_{3} black holes, in terms of meromorphic ...
3
votes
3answers
553 views
Can a functional derivative be calculated if we have a function of more than one variable?
Can a functional derivative be calculated if we have a function of more than one variable?
The functional derivative of, for example, $F[b(x)]=e^{\int_0^{x'} dx a(x,y) b(x)}$ is
...
7
votes
4answers
1k views
Number of dimensions in string theory and possible link with number theory
This question has led me to ask somewhat a more specific question. I have read somewhere about a coincidence. Numbers of the form $8k + 2$ appears to be relevant for string theory. For k = 0 one gets ...
9
votes
16answers
2k views
Can pure maths create new theories in physics or does the “idea” ALWAYS come before the math?
I am in a debate with a friend about the value of string theory in physics. He is concerned that we are wasting valuable intellectual and financial resources on a path that is fanciful and can't ever ...
1
vote
1answer
137 views
Time evolution of wave spectrum
A useful way of thinking (not only) oceanic waves is to consider them as a superimposition of linear modes: the elevation η of the sea surface is given by:
1: $\eta({\bf x}, t) = ...
1
vote
2answers
1k views
Solving a Young Laplace equation for a meniscus against a flat plate
This is more of a math question and one, furthermore, that I know the final answer to. What I am asking is more of a "how do I get there" question as this question was generated during a self study ...
22
votes
6answers
2k views
Formalizing Quantum Field Theory
I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or ...
4
votes
2answers
428 views
Existence and uniqueness of solutions for Einstein equations
Now that an equivalence of Navier Stokes and Einstein equations has been established, and it is known solutions to Einstein-Maxwell-Boltzmann exist and are unique, and it is known that Einstein ...
1
vote
3answers
907 views
Can vectors in physics be represented by complex numbers and can they be divided? [closed]
Below is attached for reference, but the question is simply about whether vectors used in physics in a vector space can be represented by complex numbers and whether they can be divided.
In ...
3
votes
1answer
439 views
Boundary conditions for Couette flow
I'm trying to reproduce a result from a paper (T. Thatcher, Boundary Conditions for Grad's 13 moment equations, equation (32), page 6), however, I haven't been able to do so. Hopefully someone can ...
12
votes
8answers
2k views
Crash course on algebraic geometry with view to applications in physics
could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an ...
2
votes
1answer
409 views
which areas in physics overlap with those of social network theory for the analysis of the graphs?
I am studying social networks in terms of graph theory and linear algebra. I know that physicists have published and worked alot in this field. This causes me to assume that there are sub-fields in ...
0
votes
0answers
138 views
Fourier analysis for waves [closed]
If we have 1D wave equation:
$$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$
we say that it's always possible to decompose the generic solution ...
13
votes
7answers
2k views
Applications of Algebraic Topology to physics
I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most ...
16
votes
7answers
2k views
Classical mechanics without coordinates book
I am a math grad student who would like to learn some classical mechanics. The caveat is I am not to interested in the standard coordinate approach. I can't help but think of the fields that arise in ...
2
votes
1answer
1k views
Spherical wave as sum of plane waves
How can we do this computation?
$\iiint_{R^3} \frac{e^{ik'r}}{r} e^{ik_1x+k_2y+k_3z}dx dy dz$
where $r=\sqrt{x^2+y^2+z^2}$ ? I think we must use distributions...
Physically, it's equivalent to ...
3
votes
2answers
240 views
Are gauge choices in electrodynamics really always possible?
If B is magnetic field and E electric Field, B=rotA, E= -gradV+dA/dt.There is Gauge's invariance, for the trasformation A'->A+gradL/dt, V'->V-dL/dt. Now, we can write:
-Coulomb Gauge : the choice of a ...
-2
votes
4answers
2k views
Meaning and application of convolution or deconvolution in physical sciences
In which real case scenarios a convolution or deconvolution operation is useful ?
