Definition Definition and 1000 Threads

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" ... ... I need some help with another aspect of Definition 9.1.3 ... Definition 9.1.3 and the relevant accompanying text read as follows...

From Rosenlicht, Introduction to Analysis: Definition: Let E, E′ be metric spaces, let p0 be a cluster point of E, and let f(complement(p0)) be a function. A point q ∈ E" is called a limit of f at p0 if, given any e > 0, there exists a δ > 0 such that if p ∈ E , p < > p0 and d( p, p0) < δ...

HI, I have read about tensors and generally understand the concept. One core thing about Vector/Tensor as I understand it is that its magnitude and direction should not change with change in coordinate system. I get that when I write vector and also when I use matrix transformation. However...

Please bare with me. Most of you know I actually don't have a great math background. In any case I'm going way back and filling in some very basic math that I have long forgot. I have some questions about terms in a polynomial. Here is an example $$3x^5+7x^3-5$$ 1. From my book 3 and 7 are...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as follows: I am...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of D&K's definition of disconnectedness/connectedness ... ... Duistermaat and Kolk's definition of...

Hi, I have a question in definition of reflection/transmission coefficients in TE/TM modes. Let's see TE polarization case. The reflection coefficient for the magnetic field is defined as: However the transmission coefficient for the magnetic field is defined as: Now, let's see the TM mode...

Homework Statement Hi, The below website states the definition of Mach number, which is a quantity that expresses how compressible a fluid is. https://physics.info/turbulence/ M = sqrt [ ( inertial resistance in the fluid ) / ( compressional resistance in the fluid ) ] = v / c...

Hello, I am trying to find a Magnetostatic(3D) solution for my design. My design consist of a coil embedded in Ceramic material developed by LTCC technology. I am below queries: 1. Is it important to define a material"Insulating"? How it will affect the Inductance of the system and the...

I am not even sure anymore of what "anarchist" means. I'm confused to the core right now :confused:. Unnecessary background to explain the why of my question: (You don't need to read it as it makes no difference to my question for which I ask for help) I ask this question because a Wikipedia...

Homework Statement In problems 11-14 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. Question 13: y'' + y = tanx ; y = -cos(x)ln(sec(x) + tan(x)) Anyways, next problem... In...

In another thread Peter Donis mentioned there may be a way to define energy properly GR. I always thought it highly problematical because you don't have time transnational symmetry so Nother can be applied. John Baez wrote an interesting article about it...

Homework Statement What is the definition of resistance? a. Resistance is the potential difference per unit current b. Resistance is the gradient of the graph of potential difference against current c. Resistance is the voltage required for a current of 1 A d. Resistance is defined by the...

Hi. Thanks for letting me ask this question. It is a stupid one. I'm a newbie at relativity theory, and I'm reading On the electrodynamics of moving bodies, By Einstein, A. He associates time with space, and tells us that you can measure the "time" of an event if you are in the coordinates of...

Is there a definition of "random(ness)"? Is it defined?

From Newton's 2nd law F = ma and a = F/m(acceleration and mass are inversely related when force is constant) But in w = F.d , F =w/d(but d and F are not inversly related just as above) I think there's something wrong in my question, please point it to me or please answer it.

Hi guys, I have been reading some of the literature recently concerning the Kardar-Parisi-Zhang equation and the words "universality" and "KPZ universality class" keep appearing. I already did a cursory wikipedia search on the subject, but it did not make much sense to me. Can you please...

In general relativity we demand that the physical law can be stated as a form which does not depend on the choose of particular coordinate system, So the vector field is defined as a changing object following a regular pattern under the transformation of coordinates. For example, we can define...

Hi people! A may be philosophical question, in fact my doubt comes from epistemology class: Newton´s second law is a definition of force based on 2 previously defined things: momentum and time or is a relation found by Newton between 3 previously defined things: force; momentum and time? Thanks.

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... I need help in fully understanding Browder's comments on Definition 8.9 ... ... Definition...

I require your help to list all phenomena described as interferences in physics ( as teached nowadays ) with their citations in scholar documents if they are not well known by non-specialists. I am open to adjacent domains like information theory and mathematics. There are already light...

Hi, so I have just a small question about cyclic groups. Say I am trying to show that a group is cyclic. If I find that there is more than one element in that group that generates the whole group, is that fine? Essentially what I am asking is that can a cyclic group have more than one generator...

Homework Statement Prove that \lim \frac{n+100}{n^{2}+1} = 0 Homework Equations (x_{n}) converges to L if \forall \hspace{0.2cm} \epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N\in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall n\geq N \hspace{0.2cm} , |x_{n}-L|<...

Hello, I would like to create few functions in loop by Table (not ordinary change a parameter, but different numerical solution of problem). The names should differ by "number"of the step, let's say "i", in the name. I.e. I would like to get set of function f"i"={f1=Interpolation[data]...

Dear Members, I was going through some video lecture (Quantum Mechanics) when I encountered a definition of momentum as shown in the attached picture. I do not understand how iota and ħ is ignored ? There are some negligible terms after plus sign. What are those ? In short how they have...

Dear All I would like to understand a paper for Xiao-Gang Wen " A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model" . Can anyone help me by suggesting another reference that is a little bit easier. Thank you

My teacher told me to find the definition of matrix which is in function form, but haven't seen it. The definition of matrix that I know is a rectangular arrangement of mn numbers, in m rows and n columns and enclosed within a bracket, but it is not right which my teacher wants. I want to know...

I am reading "An introduction to manifolds" by Tu. He starts off in Chapter 1 by introducing some definitions on ##\mathbb{R}^n## that will carry across to general manifolds. In Chapter 1, 2.2, he defines germs of functions as a certain equivalence class of smooth functions ##C^\infty_p##. I...

Can we say that the First law of Newton defines the force whereas the Second law gives the magnitude of force?

Homework Statement Homework Equations Derivative as a limit: $$y'=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ The Attempt at a Solution $$f'(x)=\lim{\Delta x\to 0}\frac{f(x)f(\Delta x)-(1+xg(x))}{\Delta x}=\bigstar$$ $$\left\{ \begin{array}{l} f(\Delta x)=1+\Delta x...

My textbook states that an alternative definition of the change in potential energy is the work required of an external force to move an object without acceleration between two points. I am confused on why it says acceleration. Wouldn't that mean that the acceleration is 0 and therefore the...

Hi everyone, apologies if this question has been asked already. My search didn't give any results. Can anyone tell me the definition of \sigma_8 in terms of the power spectrum? A reference where I could find it is perfect too! Thanks a lot

What is the definition of prompt in Matlab. Here is an example sentence. "commands are entered at the prompt with looks like two successive greater than signs" Are prompt and command window the same things in Matlab or all computer terminology? If so, is the prompt the environment where we...

Suppose if we have a cube: The volume of the cube is the product of the length, width and the height. All this time, I've been looking at it as: To get the volume, multiply the area of the cross section of the cube by how many "layers" it has. To elaborate with the diagram given, one can see...

Homework Statement Does the delta-epsilon limit definition in reverse work for describing limits in monotonic functions? By reversed, one means for lim (x -> a) f(x) = L if for each δ there corresponds ε such that 0 < | x-a | < δ whenever | f(x) - L | < ε. Homework EquationsThe Attempt at...

Consider the Gaussian position measurement operators $$\hat{A}_y = \int_{-\infty}^{\infty}ae^{\frac{-(x-y)^2}{2c^2}}|x \rangle \langle x|dx$$ where ##|x \rangle## are position eigenstates. I can show that this satisfies the required property of measurement operators...

What is definition of heat? And why it's unit is kgm^2/s^2? I couldn't be able to link the unit to the definition.

Is it possible to learn to prove limits by the formal definition without doing a course of real analysis? I'm not talking about just following the model that the Calculus books give, what I want is to understand the why of all the steps in formally proving the limit, to understand the why to use...

Is there a precise definition for the statement that two differently worded probability problems are "equivalent"? One technique of (purportedly) solving a controversial probability problem is to propose an "equivalent" problem whose solution is not controversial. (e.g. The Sleeping Beauty...

I am reading Dummit and Foote, Chapter 14 - Galois Theory. I am currently studying Section 14.2 : The Fundamental Theorem of Galois Theory ... ... I need some help with Corollary 10 of Section 14.2 ... ... and the definition of \text{Aut}(K/F) ... ... Corollary 10 reads as follows: Now...

I am reading Dummit and Foote, Chapter 14 - Galois Theory. I am currently studying Section 14.2 : The Fundamental Theorem of Galois Theory ... ... I need some help with Corollary 10 of Section 14.2 ... ... and the definition of ##\text{Aut}(K/F)## ... ... Corollary 10 reads as follows: Now...

Forgive me for asking a rather silly question, but I have thinking about the following definition of the extrinsic curvature ##\mathcal{K}_{ij}## of a sub-manifold (say, a boundary ##\partial M## of a manifold ##M##): $$\mathcal{K}_{ij} \equiv \frac{1}{2}\mathcal{L}_{n}h_{ij} =...

The definition of activity is: $$\mu _{i}=\mu _{i}^{0}+RT\cdot \ln a_{i}$$ where μi is the chemical potential of i in current state and μi0 is the chemical potential of i in standard state. The current and standard state have the same temperature or can their temperature differ? If their...

I can say that Electric Flux is Electric field passing per unit area perpendicular to electric field lines. But the formula came up as Φ = ∫ E⋅dA. Well, take the case of pressure! That is force acting per unit area perpendicular to direction of force. But it's formula came up as P = F/A. Now my...

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows:https://www.physicsforums.com/attachments/6636... and Bland's second example is as...

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows: ... and Bland's second example is as follows: I am uncomfortable with, and do...

Is absolute zero nothing? Can a quanta exist in nothing? There could be no quantum fluctuations at absolute zero.

Hello! I was wondering how does a mathematical statement come to be an axiom? I understand that an axiom can't be proven using other mathematical statements. But how does one know that a statement can or can not be proven? For example, why isn't Riemann Hypothesis considered an axiom? I also...

Hi,I was just wondering if someone could provide clarity on this matter: that if a straight-line is initially defined as "a shape that forms the shortest distance between two points" and conceptualising that shape [that forms the shortest distance between two points] as one that, at an...

Probability of an event happening = (Number of ways the event can happen)/(Total number of outcomes) Please, explain the above definition How is the above definition applied to the following question. A coin is tossed 100 times. How many heads will pop up? Solution: Let P = probability...