Definition Definition and 1000 Threads

Integral through a path in 2D (or ND) What's the usual "definition"? [Bold letters are vectors. eg: r] We have a scalar function f(r) and a path g(x)=y. I see two ways to reason: (1) The little infinitesimals are summed with the change of x and on the change of y separately. (2) The little...

What does it really mean in physical term? Does it mean no friction? No loss of mechanical energy? Thanks!

I saw the equation here http://en.wikipedia.org/wiki/Magnetic_moment#Current_loop_definition for the definition of the magnetic moment for a non-planar loop. Can someone tell me if there's a name for this equation m= \frac { I }{ 2 } \int { \overrightarrow { r } } \times d\overrightarrow { r }...

I'm not a mathematician of any sort so excuse me if my question is stupid. I just realized that I could not define the set of whole numbers without referring back to them or to the operation of addition, which then itself can't be defined. How would you define whole numbers?

definition of electric charge as "rationalized charge" Hi All, I wonder about the meaning of the term "rationalized" when saying "rationalized electron charge." Does this mean that the charge is given in natural units? Thank you very much! Best

Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?

Limit definition and "infinitely often" If we have a sequence of real numbers x_{n} converging to x, that means \forall \epsilon > 0, \exists N such that |x_n - x| < \epsilon, \forall n \geq N. So, can we say P (|x_n - x| < \epsilon \ i.o.) = 1 because for n \geq N, |x_n - x| < \epsilon...

Hello everyone, I have some conceptual issues with aforementioned definitions. How is exactly multiplicative inverse defined? Say, for a rational, nonzero number a/b, its reciprocal is b/a. Is there a certain operation that transforms a/b to b/a? Also, the notation for multiplicative...

Why is change in potential energy is defined as PE1 - PE0 = -W I mean I could see it for example for gravity if we took PE0 to be zero at ground and we integerated -mgy(y^) we get -mg(y0 - y1) -> -mgh,but is their a proof somewhere where it shows it will be always negative work ? Thank you.

I could not figure out definition of arc voltage and arc current. Is that true arc voltage if the potential differences between dual electrode and the current is flow through one of these electrode? any help would be appreciated.

why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how...

I have just finished reading stuart clark's book 'The Universe' and i find myself pondering the question of space its possible infinite size,shape, and its relation to our universe. a) if space did not exist before our universe's expansion. What are we expanding into and what are we pushing...

What does the "N" mean in a Cauchy sequence definition? Hi everyone, I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question. I believe I have an intuitive understanding of what a Cauchy...

Okay so in a HDM scenario, I have seen it described that the neutrinos were relativistic at freeze out. (If I could find it I would reference it.) Is this a contradictory statement? The condition for relativistic travel is E>>m but just before freezeout, the neutrino has energy equal to...

Hi For some strange reason I just can't see why this is true?? Can anyone help me explain why the left side can be written as the right side? I added a picture.

There is any space ? In physics space has any definition ?

The geometric and physical properties of derivatives and integrals to an integer order are easy to describe, but fractional calculus is obviously present in modern mathematics and physics. That being said, are there a generalizations of the definitions derivatives and integrals that include...

Homework Statement I've been asked to write what this is explicitly - γμ∂μ Homework Equations (γμ∂μ-im)ψ = 0 - Dirac's equation The Attempt at a Solution I understand that γμ is a matrix but depending on what μ is they're all completely different :S How can it be written...

The total work is the same in all adiabatic processes between any two equilibrium states having the same kinetic and potential energy. That is another way to describe first law of thermodynamics , and define internal energy. My question is what does "the same kinetic and potential energy "...

Good day, In my book, the following definition for flow velocity is given: So summarized, the flow velocity at a point in space is the velocity of an infinitesimally small fluid element as it sweeps through that point. But now my question; how is the velocity of an infinitesimally small...

Common extensive quatities such as mass, charge, volume can be defined for general systems. I can imagine that we can measure and define them without any problem in case of any kind of complex system as well. However, I do not know the general definition of the entropy, only the thermodynamic...

Sorry to spam my problems all over this forum but series have me struggling somewhat. Last problem on my homework is the sequence an defined recursively by: a1=1 and an+1= \(\frac{a_n}{2}\) + \(\frac{1}{a_n}\) First part was the only part i know how to do. it was to find an for n=1 through 5...

Homework Statement It can be shown that lim n→∞(1 + 1/n)^n = e. Use this limit to evaluate the limit below. lim x→0+ (1 + x)^(1/x) Homework Equations The Attempt at a Solution So i guess what i need to do is try to get that limit in the form of the limit definition for e...

Prove that the following definition cannot be satisfied if Π can encrypt arbitrary-length messages and the adversary is not restricted to outputting equal-length messages in experiment PrivKeavA,∏. A prive-key encryption scheme ∏=(Gen, Enc, Dec) has indistinguishable encryptions in the...

Are inflection points critical points? and what about at the value that f(x) undefined? Is that critical point too?

What is it in a simple definition, and how does it differ from relativistic quantum mechanics?

Homework Statement http://store2.up-00.com/Sep12/JB498124.jpg 2. The attempt at a solution No attempts because i can't understand how to solve it

i was trying to formalize the definition of the supremum in the real Nos (supremum is the least upper bound that a non empty set of the real Nos bounded from above has ) but the least upper part got me stuck. Can anybody help?

Wavelength of a sinusoidal wave is defined as the spatial period of the wave, which can be measured between any two points on the wave where the shape repeats. But if my wave is defined as a function of time (like those of a simple harmonic oscillator), how can you say the distance along the...

Which of the following two definitions is correct: 1) ##\forall x\forall y[ |x|=y\Longleftrightarrow( x\geq 0\Longrightarrow x=y)\wedge(x<0\Longrightarrow x=-y)]## 2) ##\forall x\forall y[ |x|=y\Longleftrightarrow( x\geq 0\Longrightarrow x=y)\vee(x<0\Longrightarrow x=-y)]## I think the...

Hi, I was just having a little trouble of understanding what it... is saying, well first I'll state what my book says the definition is: A function T:D* \subseteq R2 → R2 is called one-to-one if for each (u,v) and (u',v') in D*, T(u,v)=T(u',v') implies that u = u' and v = v' A function...

What must actually be specified in order for a function to be fully defined / or in what combinations if not all 3 need to be specified? I.e - from knowing the function you can determine the co-domain - e.g - if it is specified that real functions are going in, and for something simple like 2x...

Homework Statement I am curious if all modules contain 0. Homework Equations A left R-module M over the ring R consists of an abelian group (M, +) and an operation R × M → M such that certain properties hold... The Attempt at a Solution The definition of a module says that it is an...

Sometimes the dielectric function is defined as the connection between the total electric field in a material and the external field, \mathbf{E}(\mathbf{r},\omega) = \int \epsilon^{-1}(\mathbf{r},\mathbf{r'},\omega) \mathbf{E}_{\text{ext}}(\mathbf{r'},\omega) d \mathbf{r'}, and sometimes...

Homework Statement Prove lim x--> -1 1/(sqrt((x^2)+1) using epsilon, delta definition of a limit Homework Equations The Attempt at a Solution I know that the limit =(sqrt(2))/2 And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if...

According to my book, a vector space V is a set endowed with two properties: -closure under addition -closure under scalar multiplication and these two properties satisfy eight axioms, one of which is: "for all f in V there exists -f in V such that f+(-f)=0" But then isn't this axiom...

Homework Statement use the formal definition to show that lim as t goes to infinity of (1-2t-3t^2)/(3+4t+5t^2) = -3/5 Homework Equations given epsilon > 0, we want to find N such that if x>N then absolute value of ((1-2t-3t^2)/(3+4t+5t^2) + 3/5) < epsilon The Attempt at a Solution...

Homework Statement I'm reading through Taylor's advanced calculus and came across this question in section 7.2 : http://gyazo.com/6b0c5a2e4e605ff77bf6584eb3295948 Homework Equations The definition of the partial of f with respect to some variable at some point (a,b), let's say the...

My question is rather simple but it puzzles me for a long time actually. If we have a look at differential as physicists usually do we came up with a simple definition of "infinitesimal variable change". And this idea then preserves elsewhere like in the definition of entropy: \mathrm{d} S =...

Homework Statement This is a problem from Introduction to Analysis by Arthur P. Mattuck,chapter 20,problem 20-1. <a href="http://www.flickr.com/photos/86024731@N04/8090259684/" title="arctangent by gnu is not unix, on Flickr"><img...

Homework Statement Let f be a continuous function on ℝ. Suppose that \mathop {\lim }\limits_{x \to - \infty } f(x) = 0 and \mathop {\lim }\limits_{x \to \infty } f(x) = 0. Prove that there exists a number M > 0 such that \left| {f(x)} \right| \le M for all x \in ℝ. Homework Equations...

Homework Statement Find derivative of f(a) for f(t)=(2t+1)/(t+3) using the definition of a derivative Homework Equations f '(a)=lim as x goes to a of (f(x)-f(a))/(x-a) The Attempt at a Solution f '(a)=lim as x goes to a of (f(x)-f(a))/(x-a) f '(a)=lim as t goes to a of...

hello , I don't understand the meaning of the next definition , so , I hope that you can make it easy to understand it for me definition if G is an arbitrary group and ∅≠S⊆G , then ,the symbol (s) will represent the set (S) = ∩ { H∖S ⊆ H : H is a subgroup of G } can you give me some...

Homework Statement I want to show that \lim_{x \rightarrow 0}\frac{1}{x} does not exist by negating epsilon-delta definition of limit. Homework Equations The Attempt at a Solution We say limit exists when: \forall \epsilon > 0, \exists \delta > 0 : \forall x(0< \left| x\right| < \delta...

Hi everyone! I wanted to know why in the circuit analysis a generator doesn't represent an electrical branch? And the second question is if two resistors are in series on a wire, does it represents only a branch(the series of the resistance) or two branches ?

Homework Statement I'm wondering why we can't use less than or equal to for the formal definition of the limit of a function: Homework Equations lim x→y f(x)=L iff For all ε>0 exists δ>0 such that abs(x-y)<δ implies abs(f(x) - L)<ε Why not: lim x→y f(x)=L iff For all ε>0 exists...

Hello, my book defines the complement of a set like so: {x ∈ U | x /∈ A} To me, it seems like the definition should be (x| x \in U \wedge x \notin A) Which is more proper?

Good day! Im currently reading the book of Steven R. Lay's "Analysis with an Introduction to Proof, 3rd ed.". According to his book, if a subset S of ℝ contains all of its boundary then it is closed. But i find this wrong since if we consider S={xεQ;0≤x≤2}, then it can be shown that S...

Hello Everybody, I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator: \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, where N is the number of systems in the ensemble and the...

If one uses the limit definition of a derivative (lim of (f(x)-f(a)) / (x-a)) as x approaches a) on a function and you get a value (ie. it is not undefined) does that mean the derivative of the function at that point exists? In other words, even if the limit definition of the derivative works...