Definition Definition and 1000 Threads

(a) State precisely the definition of: a function f is differentiable at a ∈ R. (b) Prove that, if f is differentiable at a, then f is continuous at a. You may assume that f '(a) = lim {f(x) - f(a)}/(x - a) x→a (c) Assume that a function f is differentiable at each x∈ R and...

Hello everybody, I am a new member and do not know what is the best way to asking question. in "Statical physics of particle , m.kardar" . how he derived eq1.6 ? he said that we can eliminate "C" from eq1.4, but how? http://libgen.org/book/index.php?md5=BC07A3BDB70689E0E612DED12C96CC78 the book...

Homework Statement For which value(s) of x is each expression not defined? Homework Equations x^2+3x-6 divided by x^2-x-12 The Attempt at a Solution The answer that was given turned out to be: x^2+3x-6 divided by (-4)(x+3) Restrictions: x ≠ 4,-3...

I've been reading Mermin's book on Quantum Computer Science, and in the section in which he discusses the construction of a QFT using 1-Qbit and 2-Qbit gates, he makes reference to some expressions involving linear operators that I'm not familiar with (at least if I've seen them before I've...

I tend to think of an operating system as an interface between the hardware and the application software. But, my lecturer says that that is a very misleading conception of an operating system. Why do you think he would say that?

Do {1,2,3} and {1,2} differ in, or by exactly one element? Or do those two things mean the same thing. Also, what about {1,2,3} and {1,2,4}.

So I have been told that I have a 240V-250V supply at 50Hz... What does 50Hz mean and please give analogy if possible, I am new to this stuff Thank you in advance

My textbook defines differential equation as Could someone explain what is meant by the part in parenthesis "dependent variables"? I don't see the difference.

I am learning how to solve 1st order linear ODEs using the integrating factor.However, I run into confusion at the definition of a linear ODE. According to a reliable source, a linear ODE must have the form: (dy/dt) + p(t)y= g(t) I don't understand what it means for an ODE to be...

\d{}{x}\frac{1}{\sqrt{x}} by the definition of the derivative. $$\lim_{{h}\to{0}}\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}=\lim_{{h}\to{0}}\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x^2+2xh}}=\lim_{{h}\to{0}}\frac{x-(x+h)}{h\sqrt{x^2+2xh}\left(\sqrt{x}+\sqrt{x+h}\right)}$$ Setting $h=0$...

Hello, I've been trying to pin-point the precise definition of charge, but I'm having difficulty. I asked my Professor at college what 'charge' is exactly, and he thought carefully for about a minute, and replied that he really has no idea. The best I can come up with is the following...

Hi, How can null geodesics be defined? Obviously the concept of parallel-transport, of the tangent to the curve, applies equally well to null curves as to time/space-like curves. Technically this is only the definition for an "auto-parallel", not for a "geodesic". For example in...

Anyone exposed to academic rigor would find the use of the term "biological evolution" to be rather vague. There is something about Darwin and common descent and then it starts to get somewhat vague. Is there a definitive definition of "biological evolution"? I offer this in the spirit...

I'm working through some problems from Stewart's Calulus, 6ed. and am having some difficulty with certain limit proofs. In particular, there is no definition provided for limits of the form: $$ \lim_{x \to - \infty} f(x) = L $$ One of the exercises is to come up with a formal definition...

Homework Statement What is the definition of p_Θ? Homework Equations I search all over the web using the keyword linear "momentum polar coordinates", "lagrangian in polar coordinates", "hamiltonian in polar coordinates", "momentum in 2D polar coordinates" for about half an hour. but i...

Homework Statement For standing wave, I have read that there are certain points on the wave that don't move at all, nodes. However, for a standing wave the shouldn't the resultant wave have no displacement because we now have two waves on the same rope with the same amplitude and frequency...

Homework Statement I've been given a question that makes use of 5^(n)*sin(pi*n!) The question merely asks if the sequence converges, and if so, to determine its limit. Am I right in assuming that this does converge, under the definition, but does so as n-> - infinity? So basically, I...

Homework Statement I'm referring to the question and solution for part (b) in the attached TheProblemAndSolution.jpg file. Homework Equations Definition of limit. The Attempt at a Solution Should the equation with the two things in brackets have absolute value bars instead of brackets...

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the general UMP-based definition of external and internal direct products ... ... On page 63, Knapp defines...

Hello, the tensor product definition of a two form is \alpha^{1} \wedge \beta^{1} := \alpha \otimes \beta - \beta \otimes \alpha \alpha \wedge \beta(v,w) = \beta(v)\alpha(w)-\beta(w)\alpha(v) But what is the definition in this sense for a three form?

Would anyone have a different explanation and maybe an example of radiation flux? Here is the Wikipedia definition "Radiation flux is a measure of the amount of radiation received by an object from a given source", which is quite clearly explained.

Definition/Summary Torque is the moment of a force about a point. Torque is a vector (strictly, a pseudovector, like angular momentum and any other moment of a vector), and is measured in units of Newton metres (N m or N.m). By comparison, energy is a scalar, and is measured in units of...

Definition/Summary Stress is force per area, and is a tensor. It is measured in pascals (Pa), with dimensions of mass per length per time squared (ML^{-1}T^{-2}). By comparison, load is force per length, and strain is a dimensionless ratio, stressed length per original length...

Definition/Summary "Rolling" means moving along a surface without sliding. The (instantaneous) point of contact is stationary relative to the surface. In other words: it is the instantaneous centre of rotation (if that surface is stationary). Friction at the point of contact of a...

Definition/Summary Inertia is the phenomenon that a force is required to cause change of velocity. The amount of inertial mass of an object is measured by measuring how much force it takes to accelerate it. The symbol for inertial mass is m. Equations Extended explanation The...

Definition/Summary Heat is the non-mechanical exchange of internal energy, U, between a system and its surroundings as a result of a difference in temperature. By contrast, work, W, is the mechanical exchange of energy as a result of force applied across a moving surface (such as the face...

Definition/Summary A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses. The number of elements of a group is...

i search all over the web but i cannot find the definition. Although I find all the literature saying that metal have valence band and conduction band overlapped, (and I cannot find a metal example with valence band and conduction band not being overlapped), I wonder if there is a...

I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between...

Hi everybody! Well, these are just two basic questions that are bothering me. 1.When it is said that, for example, a Cryocooler has 1W of cooling power at 4.2K, what does it really mean? To me, the action of "cooling" depends on the material that it's being cooled down. So, for different...

I am familiar with the fact that the number e can be defined several ways. One particularly interesting definition is the one based on limits, namely: e = limn \rightarrow ∞ (1 + \frac{1}{n})n My question is: wouldn't it be equally true to express e as the limit of the expression above as n...

I think i discovered a new way to define an integral, i don't know if it helps in any particular case, but its an idea worth posting i think. The idea is to define the height of the rectangles based on one single point of the function and then build up the next heights for the other rectangles...

Hello guys! I've been trying to get some intuition for differential forms. I know the formal theory and I know how useful they are. But then I came across the following paper: https://dl.dropboxusercontent.com/u/828035/Mathematics/forms.pdf It describes an intuition for forms that is very...

I am reading the book, "Introduction to Plane Algebraic Curves" by Ernst Kunz - which the author claims gives a basic introduction to the elements of algebraic geometry. The opening few paragraph of Kunz' text reads as follows:I am puzzled by Kunz statement: " \mathbb{A} (K) := K^2 denotes...

Homework Statement If f(x) = 3x+1 en assume δ > 0. Assume ε>0. Give a δ > 0 with the following property : |x-1|< δ => |f(x) - f(1)| < ε Homework Equations The Attempt at a Solution |f(x) - f(1)| < ε <=> |3x+1-(3*1+1)| < ε <=> |3x-3| < ε <=>...

I've been reading Kreyszig's functional analysis book, and I'm a little confused why he defines the spectral family of a self-adjoint operator the way he does. For an operator ##T## he defines ##T_{\lambda} = T - \lambda I##. Then he defines ##T_{\lambda}^+ =...

I'm watching V. Balakrishnan's video lectures on Classical Physics, and right now he's going through statistical mechanics. In that regards he's talking about Stirlings formula, and at one point, he wrote an integral definition of the factorial like the following n! =...

Please look a picture. I think the book made a mistake. The answer should be 89.9N/m. Why? Because they calculated the linear density μ wrongly. They should have done $$\frac{75*10^{-3}kg}{1.8m}$$

Hi, I was studying the impact of electronegativity in a row of the periodic table on the acidity of compounds. I came upon a problem of definition: what exactly is a compound? Does it have to be electrically neutral AT ALL TIMES?

Would it be wrong to describe Capacitance as being the charge stored per unit volt? I have found on the internet that the definition of a Farad is the charge needed to cause one unit volt. Which means that the definition of Capacitance should surely be the same because the internet definition...

hey pf! can you help me with this $$\lim_{h \to 0} \frac{f(x+3h^2) - f(x-h^2)}{2h^2}$$ i know the definition and have tried several substitutions, but no help. anyone have any ideas?

I am reading Paolo Aluffi's book Algebra: Chapter 0 which takes a (moderately) category theory oriented and infused approach to algebra. I am studying chapter 1: Set Theory and Categories and need help with formulating a definition of an epimorphism and with then proving it to be surjective...

Homework Statement Use formal definition of limits Find L = lim x→ c f(x). Then find a number δ > 0 such for all x f(x) = 3 - 2x c = 3 ε = 0.02 The Attempt at a Solution limx→3 3 -2x limx→3 3 - limx→3 2x 3 - 2(3) = -3 L = -3 I am not sure how to find delta

Hello, I have been self-learning Thermodynamics and I am having a bit of trouble with calculating the work in different circumstances. Along the lectures we have come up with three different equations for work 1) W = pΔV 2) W = nRTln(V2/V1) 3) W = CvΔT So my questions are: 1) which...

I am reading Steve Awodey's book, "Category Theory" (Second Edition). In Chapter 1 within a small section on monoids, Awodey defines Hom_{Sets} (X,X) as follows: " ... ... for any set X, the set of functions from X to X, written as Hom_{Sets} (X,X) is a monoid under the operation of...

Electric potential energy of a charge q at a point in an electric field due to any charge configuration as the work done by the external force in bringing the charge q from infinity to that point without any acceleration.. In this definition, why charge is moved without any acceleration?

(I am sorry, totally forgot about solving quadratic equations, close the topic please) Homework Statement Finding derivative of the equation like F(x) = 5x / (1+x^2) by definition (ƒ(a+h) - f (a))/h is easy (point 2;2), but I got really stuck in finding a way to solve it by basic...

I've seen it written two different ways: $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$ and $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x_0 + h, y_0) - f(x_0,y_0)}{h}$$ where the latter evaluates the function at...

Homework Statement ¿Why \Gamma^{k}_{i j} = (1/2) g^{k p} (g_{i p ,j}+ g_{j p ,i}- g_{i j , p}) can't be writed like \Gamma^{k}_{i j} = (1/2) g^{k p} (2 g_{i p ,j}+g_{i j , p}) if i can say that the metric is symmetric? Homework Equations That is the relevant equation The Attempt at a...

A lot of textbooks give the definition of an S-matrix element as: \langle \beta_{out}| \alpha_{in}\rangle = \langle \beta_{in}| S| \alpha_{in} \rangle=\langle \beta_{out}| S| \alpha_{out} \rangle=S_{\beta \alpha} and that S|\alpha_{out} \rangle =|\alpha_{in} \rangle I don't...