Definition Definition and 1000 Threads

Homework Statement Use the ε-δ definition of limits to prove that limx→2 x2 = 4. The Attempt at a Solution |x2 - 4| < ε 0 < |x - 2| < δ |x - 2| |x + 2| < ε And that's where I get stuck, can I divide both sides by |x + 2| to yield |x - 2| < ε/|x + 2| = δ In which case, where do I go from...

Hello all I have this function with 2 variables: \[f(x,y)=\frac{\sqrt{x}+\sqrt{9-x^{2}+y^{2}}}{ln(\frac{1}{4}x^{2}+\frac{1}{16}y^{2}-1)}\] and I wish to plot the definition region (where the function is defined). I did the calculation manually, and asked MAPLE to plot my inequalities, and I...

Just wondering if there's a precise definition of what it means to be an elementary particle. I had assumed it was related to not being able to convert it into multiple "smaller" things, but then a photon is called elementary when it can be converted into smaller energy positrons and electrons.

We know that we define a function from the set A to the set B ,denoted by f: A=>B iff: 1) f is a subset of AxB 2) For every a belonging to A ,there exists a unique b belonging to B,such that (a,b) belongs to f In trying now to negate the above definition i got stuck ,particularly in negating...

Hi there, I have a question about the definition of a charge of a free electron. Let's suppose that QED is the true theory of the interactions of charged particles. Presumably the charge on an (effectively) free electron, then, is the charge on an electron in which the electromagnetic...

I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...

From my research I am showing that if an input signal becomes too slow (ie: a low slew rate) then the noise can cause multiple state changes. But I am being told that if the slew rate is low then it will get rid of unwanted noise. I read my results from this: (I lost the link but I did copy...

Hi! According to this http://en.wikipedia.org/wiki/Electrical_mobility, the definition of electrical mobility ##\mu## is: ##\vec{v} = \mu \vec{E}##. But since electrical mobility is always positive, this means that the velocity is always parallel to the E-field regardless of charge. How can...

Hi I'm new to limits and calculus in general. Our professor told us there existed some rigorous proof for a limit, but it was "beyond the scope of the course". All we needed to know about a limit was that (1)$$\lim_{x\to a} f(x)$$ is true iff when x approaches a from both directions p(x)...

I want to ask a strange question. When the gun has 10% accuracy, what does 10% mean? I mean it doesn't really tell me anything. How large is the area the bullets spread if the gun has 10% accuracy? I have already attempted to find articles in google, and I still didn't get the answer about the...

Hi there, I have a question about the rest mass of an electron. As we all know, the charge of an electron is a function of the energy at which the system is probed. When defining the charge, we typically use as our reference scale the charge measured in Thompson scattering at the orders of...

I'm trying to wrap my head around the epsilon-delta definition. "Let ##f## be a function defined on an interval that contains ##a##, except possibly at ##a##. We say that: $$\lim_{x →a} f(x) = L$$ If for every number ##\epsilon > 0## there is some number ##\delta > 0## such that: ##|f(x) - L| <...

In textbooks, a tensor is usually defined in terms of its transformation properties. But this definition actually seems vague when it comes to checking a set of quantities to see whether they form a tensor or not. Imagine I have four functions and want to see whether they form a 2d 2nd rank...

Hi, I have a very simple question. What is meant by "angular spread"? I'm not too sure what is meant by that. I have tried looking on this forum as well as on google have not found any sort of definition or description. For example, the spread of light when diffracting? Or the spread of light...

Hello, I found the equation for torque to be t = r x F, where r and F are vectors. I have several questions about this; is r a true position vector, or is it the distance from the axis of rotation to the mass? is r the initial position of the mass with respect to the axis of rotation? My...

Von neumann and bell pointed out that basically the non isomorphic fact that the spectrum $$\sigma(A+B)!=\sigma(A)+\sigma(B)$$ leads to contradictions. If we but replace the sum by $$A\otimes 1+1\otimes B$$ then the above inequality becomes an equality. This would make things much easier. We...

Hey, i am trying to teach accuracy and precision. I understand precision to mean two things. How repeatable measurements are (variation), and how specific measurements are (sensitivity). I am having a bigger problem defining accuracy. Most places i look define accuracy as how close...

Homework Statement The website says this: "It is Linear when the variable (and its derivatives) has no exponent or other function put on it. So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is). More formally a Linear Differential Equation is in the form: dy/dx +...

I have a question I'm a little embarrassed to be asking: what is meant in condensed matter when someone describes a system with "open boundary conditions," say in one-dimension for simplicity? I am comfortable with the statement of fixed (Dirichlet) or free (von Neumann) boundary conditions, as...

I have little problem in understanding emf in a circuit. There are three types of emf mentioned in textbooks and the first two are very similar: 1. The emf provided by a battery or other stationary power source. The emf is actually the voltage difference provided to the circuit. 2. The emf...

The lecturer in my dynamics class defined Newton's First Law to be 'There exists at least one inertial frame with respect to which mass m moves in a straight line with a constant velocity. In this frame no net force acts on m.' This has confused me; I thought inertial frames could not...

Homework Statement It is not exactly a homework question, but why does the definition of a limit use strict inequalities as follows: if 0 < |x - a| < δ, then |f(x) - l| < ε rather than weak inequalities, for example if 0 < |x - a| < δ, then |f(x) - l| ≤ ε Could the addition of the equality...

I'm looking at 2 sources. One has it defined as ##T^{c}_{ab}=-\Gamma^{c}_{ab}+\Gamma^{c}_{ba}## And the other has ##T^{c}_{ab}=\Gamma^{c}_{ab}-\Gamma^{c}_{ba}## ##T## the torsion tensor and ##\Gamma^{c}_{ab}## the connection. Or is it more that different texts use different conventions? Thanks.

Derive an exact formula (solve the recurrence definition) for the following recursive sequence: $s_n = 2_{s_n-1} - s_{n-2}$ where $n \ge 2$, and $s_0 = 4$, $s_1 = 1$. So I saw someone solving this by making it a differential equation or something? How would you do that? should I do let...

In some books the radial distribution function is defined as 4pi*r^2*R(r)^2 and in others as r^2*R(r)^2. Thus, a factor of 4pi is differing. Which expression is correct and why?

Homework Statement The definition of gravitational potential at a point in my textbook is "the work done per kg to move a small test mass from infinity to that point" I am having difficulty grasping this concept, how is work done bringing an object closer to earth?? shouldn't work be done...

According to my thermo textbook the density of states should really be called the density of orbitals because "it refers to the solutions of a one particle problem and not to the states of the N particle system". This makes perfect sense to me but now I'm confused about references to the density...

In 1-D if I have an infinite potential at x<0 so the wavefunction is zero for x<0 but for x>0 the potential is zero so the wavefunction oscillates to infinity is that a bound state ? I presume this isn't bound as it can't be normalized but most definitions state that bound means the wavefunction...

I have a general question. For example I have some relation A=\frac{iB+j}{k}C where i,j,k are integers. How to obtain minimal triplet and what is definition of minimal triplet. Let for example we have case ##A=\frac{1}{10}##, ##B=\frac{1}{2}##, ##C=\frac{2}{10}##. Find minimal triplet ##i,j,k##...

1. "An object is in "free fall" when the only force acting upon it is gravity". 2. Is gravity in this case singular or plural? Is the acting gravity the resultant force of all bodies in the universe? 3. In theory, my own bodys gravitational force is acting on the object and thus it's not...

Hi all, I have the definition of a linear transformation in terms of a transformation matrix. So the mapping is a function $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$, where $f(\textbf{x})=A\textbf{x}$ and $A$ is a $n\times m$ matrix. I'm looking for a similar definition for a transformation that...

Hello, I got interested in the concept of rapidity and would like to know a bit more about it. Unfortunately hyperbolic trigonometric functions are not taught in school, at least not where I'm living, so despite the fact that they preserve many characteristics of ordinary trigonometric...

Homework Statement My textbook says that the slope of the tangent line at a point can be expressed as a limit of secant lines: m = \underset{x \rightarrow a}{\lim} \, \frac{f(x) - f(a)}{x - a} \, . If x > a and we approach a from the right, why do we have to insist that this limit exists...

Please help me to define correctly, in the language of mathematics, the configuration of sets shown on the picture. Homework Statement I'd like to define the following rules: U is a set with infinite members. L is a list or set of properties. Every property (Ls1, Ls2 ... ) have a value (...

A lot of the less maths-y definitions of entropy talk about disorder and how disordered a system is. I'm given to understand that entropy is a measure of energy over temperate. Could someone clear up these misconceptions? I don't understand why 'disorder' is used. Isn't that subjective? Second...

Wikipedia states the following definition of a closed, thermodynamical system: "In a closed system, no mass may be transferred in or out of the system boundaries. The system always contains the same amount of matter, but heat and work can be exchanged across the boundary of the system."...

The two tensor definitions I'm (newly) familiar with, by transformation rules, and as a map from a tensor product space to the reals, don't tell me what a tensor does, and to the best of my knowledge they don't make it apparent. So, I'm looking for an operational definition, and suggesting the...

I'm coming from a physics background, but find pure mathematics extremely interesting, so have decided to try and gain a more fundamental understanding of the subject. I've recently been reading up on relations and how one can define them as sets of ordered pairs. I am particularly interested in...

In classical physics E=Fd and F=ma so E=mad. a=d/t2 so E=md2/t2 Measurements of d and t will get complicated by Lorentz transformation, so, is E=Fd still a correct definition of energy, or is it a Newtonian approximation which is not accurate at relativistic velocities?

If A is an additive subgroup of a ring R, A is said to be an ideal if Ra is contained in A for all a in A; that is, if every multiple of an element of A is again in A. Is it true that A is an ideal of R if Ar is contained in A for all r in R? To me it seems like they are equivalent statements...

Below is a test problem I recently had in Probability class. I missed points on this problem (event B) because I counted 0 as a multiple of 3. But...0 is a multiple of 3 right? I approached my professor with this concern and he told me that 0 is definately not a multiple of 3... If I am...

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's definition of the natural logarithm function (page 234 -235) The relevant section of Stoll reads as follows: In this section we read: " ... ... To prove (a), consider the function L(ax), x \gt 0. By...

M = intersection. Textbook: "The following are equivalent for subgroups G1, G2, ... ,GN of a group. 1) (G1*G2*...*G(K-1)) M GK = {1} for each k=2,3,...,n 2) If g1*g2*...*gn = 1, where each gi is an element of Gi, then gi = 1 for each i." If these conditions are met then the subgroups are...

I was wondering, if we take a "group" G (so multiplication is defined among the elements) it forms a group if it has the following properties: Closure Contains the identity element Contains the inverse elements follows associativity. I was wondering if associativity is not a must though... like...

Hello Big minds, In the book of Arfken [Math Meth for Physicists] p 134 he defined contravariant tensor...my question is about a_ij he defined them first as cosines of an angle of basis then he suddenly replaced them by differential notation...why is that? cosines are not mention in this...

1. Are uncountable unions of sigma algebras on a set X still a sigma algebra on X? 2. Are uncountable intersections of sigma algebras on a set X still a sigma algebra on X? (I think this statement is required to show the existence of sigma algebra generated by a set) 3. If 2 is true, can we...

Hi! (Smile) I am looking at the proof of the theorem of Cantor- Schröder-Bernstein, that states the following: Let $A,B$ be sets. If $A$ is equinumerous with a subset of $B$ and $B$ is equinumerous with a subset of $A$ then $A, B$ are equinumerous. Or equivalently, if $f: A \overset{1-1}{B}$...

Why do we need infinitesimally small charge in definition of electric field? Since the test charge cannot exert force on itself, F on test charge will not change whatever the value of test charge q is. So, F/q will be same for any value of test charge. Then why do we need this limit of...

I would be grateful for some direction on this. I wish to implement the following - Given a deterministic signal (the feedback signal of a closed-loop stable system) I would like to plot the power spectral density. The definition I am working with is this: Implementation (MATLAB): %...

I have tried to search on the internet for a clear and concise definition for the mathematical term "pointwise," but I cannot find one that is comprehensible. The context of needing an answer to this question is this: "operations on real functions in a vector space are defined pointwise, such...