Definition Definition and 1000 Threads

Homework Statement Suppose that Pr(X = 0) = Pr(X = 1), Pr(X = k + 1) = (1/k)Pr(X = k), k = 1,2,3,··· Find Pr(0). Homework Equations The Attempt at a Solution Ok I started with k = 1 and went to k = 5. The pattern I noticed is For k=n we have p(X=n+1) =...

Hello! The definition of Line Integral can be this: \int_s\vec{f}\cdot d\vec{r}=\int_s(f_1dx+f_2dy+f_3dz) And the definition of Surface Integral can be this: \int\int_S(f_1dydz+f_2dzdx+f_3dxdy) However, in actually: \\dx=dy\wedge dz \\dy=dz\wedge dx \\dz=dx\wedge dy What do the...

Conformal transformations as far as I knew are defined as g_{mn}\rightarrow g'_{mn}=\Omega g_{mn}. Now I come across a new definition, such that a smooth mapping \phi:U\rightarrow V is called a conformal transformation if there exist a smooth function \Omega:U\rightarrow R_{+} such that...

Is there a rigorous integral definition of the exterior derivative analogous to the way the gradient, divergence & curl in vector analysis can be defined in integral form? Furthermore can it be formulated before stating & proving Stokes theorem? Finally, a beautiful classical argument for...

Hi all, first math post here. I was just wondering- after having read from quite a few textbooks that intuitively, a set is a collection of objects- if there's a rigorous definition of the concept of set. It's just out of curiosity- I mean, is a rigorous definition even necessary? I guess I'm...

So there is something I don't understand in the definition of limit that is applied to some problem I have some intuition for like the rigorous limit definition but I don't have full understanding when applied to some problems. Use definition 2 to prove lim as z → i of z^2 = -1 The...

Once I said "there is no such thing as time" in my high school physics class and got laughed at. I still don't understand how time can be anything "natural" though and I was wondering if someone could help me understand it. I get the concept of an arrow of time, and how our minds organise...

Given this definition of two homeomorphic spaces, Definition 1.7.2. Two topological spaces X and Y are said to be homeomorphic if there are continuous map f : X → Y and g : Y → X such that f ° g = IY and g ° f = IX. Suppose I know f and g are both continuous. Would it be safe to assume...

On page 56 (see attachment) Dummit and Foote define the notation Z_n as follows: "Notation: For each n \in \mathbb{Z}^+ , let Z_n be the cyclic group of order n (written multiplicatively). " (my emphasis) But this notation is surely a bit counter-intuitive since Z_n is an additive...

One of the definitions of a subbasis ##\mathcal{S}## of a set ##X## is that it covers ##X##. Then the collection of all unions of finite intersections of elements of ##\mathcal{S}## make up a topology ##\mathcal{T}## on ##X##. That means the collection of all finite intersections of elements of...

What does the definition" the energy is not continuous" mean? Title is the whole question

Suppose I define closed sets and open sets in a single definition environment. Now I want to put two labels in the environment. \label{Open Set} and \label{Closed Set}. I have tried this and this doesn't cause any trouble and doesn't give any warning. But are there any hidden pitfalls in doing...

I am currently reading Munkres' book on topology, in it he defines an open sets as follows: "If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T." Firstly, are the open sets a property of the set X or the topological...

Homework Statement If ε = 10, give a value of δ that satisfies |δ - a| where a = 2 and 0 < δ ≤ 1 and also guarantees that|f(x) − 1/4|< ε where f(x) = 1/(x^2) Homework Equations N/A The Attempt at a Solution My problem is the solution. The solution is δ = min(1,8) = 1. I am...

Homework Statement I know what a limit is and I understand the idea behind it, but I am misunderstanding something in the formal definition of a limit. DEFINITION Let f(x) be defined on an open interval about x0, except possibly at x itself. We say that the limit of f(x) as x approaches...

Can someone give me the correct definition of time dilation(or explain it in such a way that it can be used to tackle any problem)?? What i believe now is "a moving clock ticks more slowly than a clock at rest"..but according to me this is inadequate because there can be two situations...

ϵ-N definition of limit Use the ϵ-N definition of limit to prove that lim[(2n+1)/(5n-2)] = 2/5 as n goes to infinity. The way I do it is Let ∊ > 0 be given. Notice N ∈ natural number (N) which satisfies {fill this box later}< N. It follows that if n>=N, then n > {fill this box later}, so for...

Homework Statement Let f(x) = 2x^{3} + 3x\log{x}, prove f \in O(x^{3}) using the Big-O Definition. Homework Equations Big-O definition: f(x) \in O(g(x)) if |f(x)| \leq C|g(x)| and x \geq k where C and k are both positive integers. The Attempt at a Solution I basically set C=4 and k=4...

Application on the limit definition of "e" Hi, I have known that: (i) (1+\frac{a}{n})^n=((1+\frac{a}{n})^\frac{n}{a})^a\to e^a (ii) (1-\frac{1}{n})^n=(\frac{n-1}{n})^n=(\frac{1}{\frac{n}{n-1}})^{(n-1)+1}=(\frac{1}{1+\frac{1}{n-1}})^{(n-1)}\cdot (\frac{1}{1+\frac{1}{n-1}}) \to \frac{1}{e}\cdot...

The definition of convergence is given by : ## \forall \epsilon > 0, \exists N \in \mathbb{R} ## such that ## |x_n - l | < \epsilon ## ## \forall n \in \mathbb{N} ## with ## n > N ## negate this statement and prove that the sequence ## x_n = (-1)^nn ## is divergent using only the negation of...

what is fermi energy?

Hello everyone, This may seem like a weird question, but today I had a discussion with my teacher in which I won't give up so easily. There was a test question that went like this: "A + 2 H+ + 2 e- ---> B Use this half-reaction to define whether A is a reducing agent or an oxidizing agent."...

Homework Statement Give a recursive definition for the set of bit strings \{ 0^{r} 1^{s} 0^{r} \| r, s \in N \}. Note the number of 0’s must be equal, but the number of 1’s may be different from the number of 0’s. Homework Equations N/A The Attempt at a Solution I believe this is: Basis...

Homework Statement If KVL was not true, would our definition of Voltage still make sense? Clearly explain and prove your answer Homework Equations KVL definition Voltage definition The Attempt at a Solution I said no, because if KVL were not true, then definition of voltage would...

For every interval [ f(a)-e, (fa)+e ] there exists an interval [ f(a-d), f(a+d) ] such that [ f(a)-e, (fa)+e ] includes [ f(a-d), f(a+d) ] is this definition equivalent to the epsilon-delta definition?

please i am new to math. I don't know exact meanings of epsilon-delta definition. i don't comprehend it. Would Anybody help me. thanks in advance

Homework Statement Let f(x) =\begin{cases} 0 & \text{ if } x\leq 0 \\ e^\left ( -1/x \right ) & \text{ if } x> 0 \end{cases} Compute f'(x) for x < 0 and x > 0. Homework Equations f'(x) = \lim \ \ \ \ \ \ \displaystyle{\frac{e^{(-1/(x+h)} - e^{-1/x}}{h}} \\ \ \ \ \ \ \ \ \ \...

Homework Statement http://i.minus.com/jbicgHafqNzcvn.png Homework Equations The limit definition of a derivative: [f(x+h)-f(x)]/h as h approaches zero is f'(x) The Attempt at a Solution I'm just not understanding the wording of the question. The limit given in the question is...

Homework Statement http://i.minus.com/jbzvT5rTWybpEZ.png Homework Equations If a function is differentiable, the function is continuous. The contrapositive is also true. If a function is not continuous, then it is not differentiable. A function is differentiable when the limit definition...

Definition of "hypersurface orthogonal" Hi all! I'm not sure if the thread belongs more to General Relativity or Differential Geometry, but I guess the border is labile. I've come across the term "hypersurface orthogonal" many times, but I still haven't found a clear definition. Apparently...

naive (intuitive) definition of "set" I happened upon a book by a Joseph Landin, once head of the math department at University of Chicago and subsequently Ohio State University, in which he gives this as a definition of a set and states this property: Shortly thereafter, he writes...

Homework Statement What exactly is op-amp saturation? Homework Equations V_{out}=A_{v}V_{in} The Attempt at a Solution I am doing a lab where we use an LM339A quad voltage comparator and a voltage divider network to create a voltmeter. A single red LED is connected to each of...

Homework Statement let a_{i}=x^{i} and b_{i}=1\div i ! and c_{i}=(-1)^{i} and suppose that i takes all interger values from 0 to ∞. calculate a_{i}b_{i} and calculate a_{i}c_{i} Homework Equations i know that in suffix notation a_{i}b_{i} is the same as the dot product as when you have to...

What is the definition of entropy of a thermodynamic irreversible process? In the case of reversible process from initial state 'a' to final state 'b' ,one may define entropy by 1) Constructing infinitely many reservoirs having temperatures corresponding to the temperature at every...

Homework Statement Consider the function g(x) = 1/x. b. Find a number δ′ > 0 different from the number δ you found in the previous question such that all the values x ∈ (2 − δ′, 2 + δ′) satisfy that g(x) is within 0.1 of 1/2. Note, I found I value δ>0 in an earlier question in which the...

Homework Statement Evaluate derivative of (sin sq. root x) w.r.t x? Homework Equations Limit Δx--> 0 (sin√(x+Δx) - sin(√x)) / Δx The Attempt at a Solution i couldn't operate it from here... Δy = (2cos((√x+Δx) + (√x)) . sin((√x+Δx) - (√x)) / Δx...?

what is the defination of complex no?

In another thread, DH and I have been discussing the definition of an ideal gas. DH, who appears to be a physicist, seems to use a definition different from that which we engineers use. I am soliciting responses from both physicists and engineers as to their understanding of the term "ideal...

Hello MHB. Can someone please check if these definitions are correct. Definition. Let $U$ be a subset of the real numbers. A function $f:U\to\mathbb R$ is said to be a real analytic function if $f$ has a Taylor series about each point $x\in U$ that converges to the function $f$ in an open...

Homework Statement Prove using ε/δ definition, lim x tends to -1 (x^3+2x^2) = 1 Homework Equations The Attempt at a Solution I have done to the step where δ(δ^2-δ-1) ≤ δ ≤ ε so i choose ε=min(2,ε) Not sure whether I am correct or not

Here is the question: I have posted a link there to this topic so the OP may see my work.

Homework Statement lim (x - 4) = 2 as x -> 6 eps > 0 delta > 0 No eps given Homework Equations The Attempt at a Solution Not sure how to apply equation.. Usually I would do I assumed you would do.. 6 - eps < x - 4 - 2 < 6 + eps 12 - eps < x < 12 + eps So, it would...

Some define the extrinsic curvature tensor as $$K_{\mu \nu} = h^{\ \ \ \sigma}_\nu h^{\ \ \ \lambda}_\nu \nabla_\sigma n_\lambda.$$ From the expression it seems like the index of the covariant derivative in can be any spacetime index. However, does it makes sense to ask what the...

Here is my question: Solve the initial value problem \(y\prime=\dfrac{3x^2}{3y^2-4},~y(1)=0\) and determine the interval in which the solution if valid. Hint: To find the interval of definition, look for points where the integral curve has a vertical tangent. My work so far...

In the definition of an integral what does the astrix (*) mean above the x_i? I got confused in class today because the prof used an astrix but just to mean the equation we had been talking about. Also, I sometimes see the top of the sigma being n-1 instead of n. I guess it doesn't really...

\According to Newton's second law, force is defined as rate of change of momentum or force is defined as product of mass into acceleration, I both of these definitions are same and right. My questions are, (1) how can i prove this two definitions are same?

I am slightly confused by the definition of average power if the power function $p(t)$ is sinusoidal. Why is it that only one period is considered? I mean I know that it simplifies calculations but if we assume that the period of $p(t)$ is $T$ and I compute the average power over...

A red a book in classical mechanics , the author says A particle is an idealised body that occupies only a single point of space and has no internal structure clarify me these terms Idealised body and Internal structure

Homework Statement A vector is a geometrical object which doesn't depend on the basis we use to represent it, only its components will change. We can express this by \vec{A}=ƩA_i \hat{ε_i} = Ʃ\tilde{A_i} \vec{ε_i}, where it has been emphasized that the basis ε is not necessarily orthonormal...

Hi, can somebody tell me the definition of : and,or Thank you