Definition Definition and 1000 Threads

I have a rather general question about the definition of entropy used in most textbooks: S = k ln Ω, where Ω is the number of available microstates. Boltzmann wrote W rather than Ω, and I believe this stood for probability (Wahrscheinlichkeit). Obviously this is not a number between 0 and 1, so...

I know the logic of proving/disproving mathematical statements, I learned it by reading books, texts regarding to the matter, lots of exercises ( in the subject of how to do mathematical proofs and in the subject of proving/disproving statements in my math courses [ e.g. Linear Algebra, Real...

Hi, I was working through the following problem and I am getting confused with the solution's definition of the dual. Problem: Given the optimization problem: minimize ## x^2 + 1 ## s.t. ## (x - 2) (x - 4) \leq 0 ## Attempt: I can define the Lagrangian as: L(x, \lambda) = (x^2 + 1) + \lambda...

According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...

abstraction levels in programming define different approaches with a varying degree of detail for representing, accessing and manipulating data. What I understand by abstraction is that we hide what is not necessary. Isn't that.

Hi, a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of...

Hello everyone, I'd like to share a doubt I am currently struggling with. So we know that ΔU=−W, where ΔU is the difference of potential energy and Wthe work done by the force to move the body from point A to point B. When analyzing this for the gravitational force, since we have U=−GmM/R, with...

So the part in italics "an operation performed on one quantity which when performed on unity produces the other." I do not understand. Can anyone help me understand what this means? I know how to multiply fractions, but this explanation is confounding to me.

Hello. Considering this DE; $$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2 Now the solution for the initial value should be C = -44; And for x(t) I get ; $$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$ Now to get the biggest domain of definition I did this; $$ -44...

Hi, although there is a lot of discussion here in PF, I'd like to ask for a clarification about the definition of 'spatial x direction' in the context of flat or curved spacetime. Consider a set of free-falling gyroscopes (zero proper acceleration) passing through an event A with different...

Hello! I have some troubles with the definition of the so called super Lie module. In Alice Rogers' textbook "Supermanifolds theory and applications" definition goes as follows Suppose that ##\mathbb{A}## is a super algebra and that #\mathfrak{u}# is a super Lie algebra which is also a super...

Mathematicians will use the term "elementary functions," often in the context of integration wherein some integrals cannot be expressed in elementary functions. The elementary functions are usually listed as being arithmetic, rational, polynomial, exponential, logarithmic, trigonometric...

Hi all, I would like to understand the definition of finite size correction, radiative correction and weak magnetism correction, with their impacts on the beta spectrum. I'm not a physics student, thus I would like to seek for a help about the simple explanation that can be understand by...

* The general formula for the magnetic moment of a charge configuration is defined as ##\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r##* For an electron it's said that the correct equation relating it's spin and magnetic moment is is ##\vec{\mu} =g\frac{q}{2m}\vec{S}## * It's...

The curl is defined using Cartersian coordinates as \begin{equation} \nabla\times A = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}. \end{equation} However, what are the...

Why do we want to always deal with single valued functions? In the classical treatment a function is a rule which assigned to one number another number. In the modern sense, it is a rule which assigns to each element in a set called the domain an element (one element) in a set called the range...

Hi all, I was trying to understand an equation where: axial charge, ##G_A=g^P_A(Z_+-Z_-)+ g^N_A(N_+-N_-)## What is the meaning of ##Z_+## , ##Z_-##, ##N_+## and ##N_-##? From the article I read, axial charge will be zero when the nuclei has zero spin. What if I have Germanium 73 which has...

How is the order of a partial differential equation defined? This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0## And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial...

I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...

I'm writing some notes for myself (to read in my rapidly approaching declining years) and I'm wondering if this statement is correct. I"m not sure I am posting this question in the right place. "Summary: The matrix representations of isometric (distance-preserving) subgroups of the general...

The definitions of them seem like arbitrary choices or an abuse of notation. I wonder what the reasons behind the definitions are. Thanks. PS. My instructor said such defs simplify the process of solving modular equations.

Hello PF. I am thinking about the power spectrum when observing X-rays. We are trying to obtain the power spectrum by applying a window function ##w(t)## to a light curve ##a(t)## and then Fourier transforming it. I have seen the following definition of power spectrum ##P(\omega)##. Suppose...

Please could you help me find a rigorous mathematical definition of sampling as it is used in mathematical statistics? Let ##X:\Omega\rightarrow\mathbb{R}## be a random variable and ##X_{1},...,X_{n}## is a statistical sample. What is its mathematical meaning in probability theory? Are we...

1)In my book , there is a definition of fermi energy as topmost filled level in the ground state of an N electron system. This definition holds only for absolute zero,right? If it is not absolute zero,fermi energy is the energy at which the probability of a state being occupied is 50 percent...

Hi, PF, I've got a quote, and a philosophical question Figure shown suggests that a function ##f(x)## can have local extreme values only at ##x## points of three special types: (i) Critical points of ##f## (points ##x## in ##\mathfrak{D}(f)## where ##f'(x)=0##) (ii) Singular points of ##f##...

Hi, starting from this post Basic introduction to gravitation as curved spacetime I would ask for a clarification about Rindler coordinates. From this wiki entry Rindler coordinates I understand that the following transformation (to take it simple drop ##y,z##) $$T = x\sinh{(\alpha t)} ...

[Moderator's note: Thread spun off from previous one due to closure of the previous thread.] I have been thinking about this off and on, and though late to the thread, want to propose another way of looking at this that can be presented both at B level or A level. I post here at B level, and...

I see much confusion about the "origin" of the universe. When discussing what caused the big bang, we need to describe the conditions of the universe before the big bang. For me, the big bang occurred in the universe. The universe existed prior to the big bang and provided the energy to fuel...

If the book had said that electrical potential energy is the negative of work done by electrical force on a charge, then the definition would be very clear and easy to understand. So, why should the book give this confusing definition instead.

for undergraduate students how to explain this?

Hi, I am used to plot data with the GNUplot software. This time I need to add an image to my plot. I did with the following line 'my_image.png' binary filetype=png center=(3.25,-9.1) dx=0.007 dy=0.0125 w rgbimage notitle but the final resolution of the picture is pretty bad compared to the...

I'd appreciate some clarification of this passage in the paper Spinning test particles in general relativity by Papapetrou, The definition is easy enough to understand, but what's the motivation? ##X^{\alpha}## are the coordinates of points on the worldline whilst ##x^{\alpha}## are...

Hi! I read this definition of Stellar Parallax "It is expressed quantitatively by one-half the angle subtended by the Earth's diameter E1E2 perpendicular to the line joining the star and the sun (see Fig. 2-10)." (Source Alonso and Finn: Volume 1). But, I was unable to understand how they...

This is from my notes: Point D is called ultimate tensile strength and defined as highest possible within this material. So it means that point D should be at the highest point of the graph (more like absolute maximum in math)? Because it seems that from the graph point D is not at maximum...

What is the definition of volume inside a black hole? we know the grr element of Schwarzschild metric is negative inside event horizon, so how to define a volume inside event horizon? if there is no definition of volume, is there the definition of density?

Mentor's note: Moved from HW for a better fit Homework Statement:: A joule is defined as "the energy transferred to an object when a force of one Newton acts on that object in the direction of its motion through a distance of one meter." Explain this definition. Relevant Equations:: 1J = 1N.m...

Hey guys, I have two questions: 1) I thought absolute electrode potential is galvani potential difference at the interface. However, it is given by this equation in John Bockris - Modern Electrochemistry: $$ E(abs) = ^M\Delta^S\phi - \mu_e^M/F $$ First term is galvani potential difference on...

E.g. all real numbers could be a domain but not necessarily, etc. Am I right?

In https://mathworld.wolfram.com/InnerProduct.html, it states "Every inner product space is a metric space. The metric is given by g(v,w)= <v-w,v-w>." In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand, "As for every normed vector space, an inner product space is a metric...

Apostol defines limit for vector fields as > ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)## means that : ##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0## Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##

Hello everyone, I was curious about how do we define a physical quantity and mathematical relation between them. For example: Work done is defined as product of force along the displacement and displacement i.e W=Fcos(theta)d. Now this definition is given in books straightforward but my...

Consider a simple shear x=kY, y=Y, z=Z. How is Poynting effect defined? If the normal stress along y is 0 but that along x isn't 0, is that also a sort of Poynting effect?

I would like to know the meaning and/or background for the username you have chosen. Username stories are interesting. Do you agree?

In lecture notes at a university (I'd rather not say which university) the following definition for Hermitian is given: An operator is Hermitian if and only if it has real eigenvalues. I find it questionable because I thought that non-Hermitian operators can sometimes have real eigenvalues. We...

Is it reasonable to define the n-dimensional holes of a topological space X as the non-zero Homology/Homotopy classes of X? Can we read these as obstructions to continuously shrinking a simple closed curve * to a point within the space? *I understand this is what we mean by a cycle.

I have the following definition: $$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$

I have the following definition: $$\lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$

Summary:: I wish this video (and YouTube in general) was around when I took intermediate level mechanics as an undergraduate physics student: I wish this video (and YouTube in general) was around when I took intermediate level mechanics as an undergraduate physics student:

In my notes on general relativity, hypersurfaces are defined as in the image. What confuses me is that if f=constant, surely the partial differential is going to be zero? I'm not sure if I'm missing something, but surely the function can't be equal to a constant and its partial differential be...

Context Boltzmann first defined his entropy as S = k log(W). This seems to be pretty consistently taught. However, the exact definitions of S & W seem to vary slightly. Some say S is the entropy of a macrostate, while others describe it as the entropy for the system. Where the definition of...