Definition Definition and 1000 Threads

I was reading Fundamentals of Inket Printing and it said the following: "The surface tension in a liquid causes a force to act in the plane of the free surface perpendicularly to a free edge in that surface." Can someone explain to me what this means? What's the direction of the force? I have...

Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X. The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...

So this article "Quantum leap for mass as science redefines the kilogramme" said that there is a new definition of the kilogram coming. But they neglected to mention what that new definition is exactly. All they said was that it's now based on Planck's Constant. So I worked my way backwards...

If a photon does the 4 things listed below, why do we say it doesn’t have mass? I guess I’m asking if there’s a clear definition of mass. follows geodesics in spacetime (classically: accelerated in a grav field), curves spacetime (classically: creates a gravitational field), has momentum /...

I' m trying to derive the work done by a torque from W = ∫ F ⋅ ds and I' ve looked up the internet, it said: W = ∫ F ⋅ ds ( since ds = dθ × r ) ---------------------------------------- ( Line 1 ) it can be written as W = ∫ F ⋅ dθ x r this is a vector triple product , thus can also...

Say that we are asked to prove, using the definition of limits, that the sequence ##\frac{4n^2+3}{n^2+n+2}## tends to ##4## as ##n## tends to infinity. The following is a screenshot of the solution I found in a YouTube video: (Note that in the definition above, "g" denotes the limit - in this...

Is it true that if ##f## is differentiable at ##a## that ##f'(a) = \lim_{h\to a}\frac{f(a+h) - f(a)}{h} = \lim_{h\to a}\frac{f(a-h) - f(a)}{-h}##. That is, can the sign of ##h## be flipped. I've seen this a few times and it seems a bit dubious.

this is probably a stupid question but for the fundamental domain for SL2(Z), we say the cusp is only at infinity. Compare say to hecke subgroups which are congruence subgroups where we say the equivalence classes are given by the points where the fundamental domain intercepts the real axis as...

I am reading the book "An Introduction to Differential Manifolds" (Springer) by Jacques Lafontaine ... I am currently focused on Chapter 1: Differential Calculus ... I need help in order to fully understand a remark by Lafontaine following his definition of differentials ... Lafontaine's...

Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity : $${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$ with ##v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}## and...

I apologize ahead of time for the simplicity of the question, but this has really been bothering me.Given the de Broglie relation, assuming natural units, where ##\hbar = 1##: \begin{equation} \begin{split} \vec{k} &= M \vec{v} \end{split} \end{equation}My question regards velocity and...

I'm learning APL and this is how a vector is defined https://tryapl.org: All data resides in arrays. An array is a rectangular collection of numbers, characters and arrays, arranged along zero or more axes. We can use more specific terms for some arrays, like a single number is a scalar, a list...

In the paper "Steady-state spin synchronization through the collective motion of trapped ions" it states the following: "Steady-state synchronization of atomic dipoles forms the foundation for ultra-stable optical lasers utilizing narrow-linewidth atoms coupled to a lossy cavity mode. The...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focuses on Chapter 3: Elements of Point Set Topology ... I need help regarding a remark of Apostol's made after Definition 3.2 and Theorem 3.3 ...Definition 3.2 and Theorem 3.3 read as follows: In a note at the...

I'm reading "The Geometry of Physics" by Frankel. Exercise 1.3(1) asks what would be wrong in defining ##||X||## in an ##M^n## by $$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to...

My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface? Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...

Hi, a basic question related to differential manifold definition. Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined...

Homework Statement Show that ##\displaystyle \lim_{n\to \infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n}) = \frac{1}{2}## Homework EquationsThe Attempt at a Solution We see that ##\displaystyle \sqrt{n}(\sqrt{n+1}-\sqrt{n}) - \frac{1}{2} = \frac{\sqrt{n}}{\sqrt{n+1}+\sqrt{n}} - \frac{1}{2} <...

Consider this Ito proces: $$dX_t = \mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ with W_t being a wiener process. My question: What is the diffusion coefficient of X? My motivation for asking: A lot if financial literature refer to "diffusion coefficient" and I haven't understood by googling it, because...

Homework Statement My textbook says the binding energy per nucleon is the energy needed to remove one nucleon from the nucleus, In a subsequent numerical which is as follows "The binding energy per nucleon for C12 is 7.68 MeV and that for C13 is 7.47 MeV. Calculate the energy required to remove...

I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.3 Ordered Sets ... I need some help in fully understanding some remarks by Searcoid following Definition 1.3.10 ...

In Micheal O. Searcoid's book "Elements of Abstract Analysis" we read the following: Help will be much appreciated ...Peter ========================================================================== It may help readers of the above post to hve access to Searcoid's text of Definition 1.3.4 ...

I do not want to get into a religious discussion. The question is an abstract one, not directed at any belief system. With that prologue: is it theoretically possible to give a rigorous and coherent definition of "supernatural" that would allow an observable event to be supernatural, independent...

The question is not to distinguish space from time, but in general, what distinguishes a spatial dimension from other types of dimensions? For example, Hilbert space has an infinite number of dimensions, but they are not spatial; string theories add extra spatial dimensions. Is there a...

prove the statement using the $\epsilon,\delta$ definition of a limit. $$\lim_{{x}\to{1}}\frac{2+4x}{3}=2$$ so then $$x_0=1\quad f(x)=\frac{2+4x}{3}\quad L=2$$ now $$0<|x-1|<\delta\quad\text {and}\quad\left|\frac{2+4x}{3}-2\right| <\epsilon$$ then...

Hi, I just had my first class of calculus where we mostly talked about sets. We defined the subset like this: Let A and B be sets. We say that A is a subset of B if ##x \in A \implies x \in B## But if we look at the truth table of the implication...

I am asked to caluculate the induced voltage between the two sides. To me it is confusing what is meant since the field is nonconservative. Is the question phrased in an unclear way or am I missing something about the actual definition of the voltage between two points.

I've searched around the internet and thought it would be a simple thing find and perhaps it is, but the more I dig the worse it becomes. According to online dictionary, 1 foot-candle is the amount of light from a 1 lumen light source on 1 sft of surface area at a distance of 1 foot. Sounded...

In the book "The Basics of Abstract Algebra" Bland defines a Euclidean Domain using two conditions as follows: In the book "Abstract Algebra"by Dummit and Foote we find that a Euclidean Domain is defined using only one of Bland's conditions ... as follows:What are the consequences of these...

I am confused by the definition of current density in Maxwell electrodynamics. Perhaps someone can help me out? As I understand it, the current density function can be written as $$ \vec{J} = \rho \vec{v}_S$$ where ρ is the charge density function and v_S is the continuous source charge...

Homework Statement 3 - 2 , 1 km wires ( so acting like infinite wires ) , both have same current , no direction specified , , separated by 1 meter distance and having between them a magnetic force of module 0.02 N . find the current i= Homework Equationsthe definition of 1 ampere has same...

Hi, my question is about the correct means of the expression "computer experiment" in a Monte Carlo simulation. What specifically is an "experiment" in a Monte Carlo simulation? Is the same of "Monte Carlo path" or is the whole process adopted to found a magnetization curve, for example? If the...

Many books sometimes for example define energy as quantity and sometimes as property. Also the definition of energy is the ability to do work or the meter of the ability to do work ? we define for example force as a quantity or as some quality and then we quantify this ?

The SI definition of a second was originally based on 1/86 400 of a day - but the Earth's rotation is a) unsteady and b) changing so it was changed to a defined number of cycles of an atomic clock. Was there a specific day for which the atomic clock was calibrated? Put another way, why were 9...

I was wondering how the rules work for observation in a quantum system. Particularly, about what happens if two separate entities try measuring at the same time. And also, what kinds of interactions are happening all the time that are considered measurements, for example in quantum...

i'm trying to review calculus and look a little deeper into proofs/derivations/etc. I'm doing this both for fun and to review before i go back to school. am i the only one who has difficulty understanding the "rigorous" definition of the limit? i found this web page...

Many sources give explanations of the Riemann tensor that involve parallel transporting a vector around a loop and finding its deviation when it returns. They then show that this same tensor can be derived by taking the commutator of second covariant derivatives. Is there a way to understand why...

Hi,t I am studying topology at the moment. I have seen that some authors define the neighborhood of a point using inclusion of an open set, while others define the term as open set that contains the point. In most of the theory I have seen so far, the latter is more convenient to use. Why is...

My class is difficult to teach, but I have a question that I think that I share the forum and if you give nice ideas it can be helpful. This is my last question of axioms and so on because I don't want to be as a mathematic cranck. So, this is my last question that deal with it. What I need to...

Homework Statement Homework EquationsThe Attempt at a Solution These assertion reasoning questions are little tricky . Even simple questions get wrong in a time bound objective test . In this question I think it should be option b) i.e Both the statements are correct but Statement R is not...

Homework Statement Hello I am solving a problem that requires me to find the max distance of a glider with spring attached to it. I am getting confused on Work total, in an example in my textbook W total is the W friction + W spring on glider = W total. My question is why did the textbook...

In my book, the definition of surjection is given as follows: Let A and B be sets and f:A->B. The function f is said to be onto if, for each b ϵB, there is at least one a ϵ A for which f(a)=b. In other words, f is onto if R(f)=B. A function which is onto is also called a surjection or a...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in $$\text{Mod}_R$$ ... ... I need some help in order to fully understand Definition 3.2.2 and Proposition 3.2.3 ... Definition 3.2.2 and Proposition 3.2.3 read as follows...

Hello, everyone! I am a lover of physics, specifically that of the quantum type. The very specific question I have is the exact scientific definition of "information" pertaining to string theory. Any insight is good insight!

hello i want to know if MCNP version 5 can define 8 sources by sdf card i need a help please

I'm modelling a scenario for the research that I'm working on, and I got the cells and surfaces all mapped out for the environment finally, but now I'm totally stuck on creating a source. I'd like a point source of Californium-252, but after hours of looking, I don't see any out of the 1000...

Homework Statement 1) Suppose ##t## is a subsequential limit for ##(s_n)##. Write the precise definition of the meaning of this statement. 2) Explain why there exists a strictly increasing sequence ##(n_k)^\infty_{k=1}## of natural numbers such that ##\lim s_{n_k}=t##. Homework EquationsThe...

HI .I'm trying to prove that, for a linear transformation, it is worth that: f(a\bar{x}+b\bar{y})=af(\bar{x})+bf(\bar{y}) for every real numbers a and b. Until now, I have proved by myself that f(\bar{x}+\bar{y})=f(\bar{x})+f(\bar{y}). and , using this result i proved that: f(a\bar{v}) =...

Right now, I am studying Advanced Calculus (proof based), and it is hard thinking through some of the definitions without first thinking about it concretely (meaning how to visualize it better geometrically, if that makes any sense?). I need help with this definition. Definition Let X be a...