Definition Definition and 1000 Threads

Hi, I'm trying to determine ##\vec{\bigtriangledown }\times \vec{a}## , where ##\vec{a}=\vec{\omega }\times \vec{r}##, being ##\vec{\omega }## a constant vector, and ##\vec{r}## the position vector, using this definition: ##\vec{curl}(\vec{a})=\lim_{V\rightarrow 0}\frac{1}{V}\oint...

It is my understanding that while instantaneous speed is the magnitude of instantaneous velocity, average speed is , in general, not the magnitude of average velocity, since average speed is total distance traveled divided by change in time. Why is the mathematical definition of average speed as...

Hello, I was just wondering, we have what could be called the indefinite derivative in the form of d/dx x^2=2x & evaluating at a particular x to get the definite derivative at that x. But with derivation, we can algebraically manipulate the limit definition of a derivative to actually evaluate...

Homework Statement In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...

Does the discriminant have any specific definition? I've come across two discriminant definitions that don't seem to be very relevant to each other, and I was wondering if there is any particular property that discriminants convey. In finding roots and using the quadratic equation, I've seen ##...

Homework Statement Suppose that an amount function ## a(t) ## is differentiable and satisfies the property ## a(s + t) = a(s) + a(t) − a(0) ## for all non-negative real numbers ## s ## and ## t ##. (a) Using the definition of derivative as a limit of a difference quotient, show that ## a'(t) =...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the definition of a noetherian module - in particular I need help with the nature of an ascending chain of submodule ...

I know it follows a geodesic in space-time, but is it not being able to bend space-time itself, part of the definition? When I google i just find 'follows a geodesic in space-time'. I thought that it also should not bend space-time itself - but this isn't included in the definitions I've...

Homework Statement [/B] I'm supposed to find the derivative of 2^x using the definition of a derivative. I am really confused as to how I can factor out the h. Homework Equations y=2^x The Attempt at a Solution limit as h->0 in all of these, I don't want to write it out because it's going to...

How will we define gravity in terms of General Relativity.? Technical and scientifically approved definition are appreciated

what is definition of closed packing?a close packing plane is a plane that the atoms cannot be packed any closer?

Hello! I hope I posted this in the right section. I have tried to google the definitions of what follows but I didn't get what I expected. I would like to ask you then: What is the definition of a self-dual field and a self-dual topologically massive vector gauge field? Thank you very much...

"I've looked this up and came up to similar questions but haven't seen it been explained very clear yet. Excuse me if such questions were already posted here. There are two different kinds of simultaneity it seems to me:Let's take the event of two lightning bolts striking earth. Observer A is...

Homework Statement Here's the problem: Mod note: the last line is extraneous, and unrelated to this problem. Homework Equations Definition of e (1-1/n)^n, etc. The Attempt at a Solution [/B] My question is this: a) When do you know to use e in a situation like this? Is it when you have...

Why is the electric flux defined as (electric field) x (area), Φ = EA? How do you come up with that equation? Is it because the electric flux is proportional to the charge and surface area?

Homework Statement A subgroup ##H## of a group ##G## is normal iff ##\forall g \in G## and ##\forall h \in H##, ##ghg^{-1} \in H##. Homework EquationsThe Attempt at a Solution I am having trouble seeing how this is not saying the same thing as the definition of a normal subgroup. Could...

Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...

Homework Statement My question is, to gain more knowledge on, is in Physics, what is the terminology for D-offset in the Parallel axis theorem? Homework Equations I= Icm + Md^2 The Attempt at a Solution From my understanding, the offset is the distance away from the axis of rotation.

A=limn→∞Rn=limn→∞[f(x1)Δx+f(x2)Δx+...+f(xn)Δx] Consider the function f(x)=4√x, 1≤x≤16. Using the above definition, determine which of the following expressions represents the area under the graph of f as a limit. I knew the correct answer was \sum \frac{15}{n} (4√x+\frac{15i}{n}) I figured...

I tried to get the potential from the potential energy and I get a positive sign for potential...! I cannot find what I did wrong. - Gravitational force is conservative so its work W is symmetrical to the change in potential energy U. - Potential is the work done by gravitational force, per unit...

Hi. I am a bit confused on the definition of multivariable functions. Say you have ##f(x) = x^2 + x## and ##g(x,a) = x^2 + a## where ##a=x##. Is ##g(x,a)## then a mathematically legal multivariable function? Because if you take ## \frac{\partial f(x)}{\partial x}=2x +1## you'll get a different...

Homework Statement 2*Lim (as k approaches infinity) of (| (k/(k+1))^k |) The answer to this limit is 2/e I know there is a definition of e used, but I am unclear what to do/how to do it. If someone has a link I can look at or could point me in the right direction I would be thankful.

I am facing problems while comparing the results of solving a problem individually using both the concept of Binomial Distribution of Probabilities and the Classical Definition of Probability. Let me formulate the problem first: "The probability that a pen manufactured by a company will be...

Homework Statement Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement. (a) lim x-> a [f(x) + g(x)] = infinity (b) lim x-> a [f(x)g(x)] = infinity if c > 0 (c) lim x-> a [f(x)g(x)] = negative infinity if c < 0 Homework Equations...

For a 2x2 matrix, does the general definition hold? If so, how exactly is the minor ## M_{i,j} ## computed in this case? If A is a 2x2 matrix, is det(A) only defined as ad - bc?

this is a rather stupid question regarding preliminaries for the definition of boundaries the question is whether every closed n-1 dim. closed submanifold C of an arbitrary n-dim. manifold defines a volume V; i.e. whether \partial V = C can be turned around such that V is defined as the...

Homework Statement Show that ##\lim_{x \to a} f(x) = L## if and only if ##\lim_{x \to 0} f(x+a) = L## Homework Equations - The Attempt at a Solution For the forward direction (ie ##1 \Rightarrow 2##), I tried to first assume that 1. holds true (ie ##\forall \epsilon>0, \exists \delta>0...

Do we say something is an EM wave only if the EM field is oscillating at a constant frequency? What exactly is the definition of an EM wave? If an electron moves in a direction and then stops moving, is an EM wave produced by that electron?

Hello, I really need your help dear friends, I had this question in an interview : What is the definition of future according to quantum mechanics? And how can we create and improve our future?? they asked me to write a three-page texte. I must respond them tomorrow.

In my maths textbook, it says that work done can be defined as Force x Distance moved in direction of force, AND change in kinetic energy. I feel both these definitions can be contradictory Example: A box moves at a constant velocity along a rough horizontal plane. It has a driving force of 5N...

Based on the following problem from http://math.uchicago.edu/~vipul/teaching-0910/151/applyingformaldefinitionoflimit.pdf: f(x) = \begin{cases} x^2 &, \text{ if }x\text{ is rational} \\ x &, \text{ if } x\text{ is irrational} \end{cases} is shown to have the following limit: \lim_{x\to 1}f(x)...

what is Stanton Number? and its use/application in convection heat transfer coefficient?

Here is the definition of the limit of f(x) is equal to L as x approaches a: "For every positive real number ϵ > 0 there exists a positive real number δ > 0 so that whenever 0 < |x − a| < δ, we have |f(x) − L| < ϵ." But what is the difference if I use this definition? "For every positive real...

Hi, Suppose you want to prove $|x - a||x + a| < \epsilon$ You know $|x - a| < (2|a| + 1)$ You need to prove $|x + a| < \frac{\epsilon}{2|a| + 1}$ So that $|x - a||x + a| < \epsilon$ Why does Michael Spivak do this: He says you have to prove --> $|x + a| < min(1, \frac{\epsilon}{2|a| +...

For a vector : ##\nabla_a V^b=\partial _a V^b+T^b_{ac}V^c## I am trying to derive for a covector: ##\nabla_a w_b=\partial _a w_b+T^c_{ab}w_c## I am told to use the Leibniz Rule and the definition that for a scalar ##f## : ##\nabla_a f =\partial_a f ## to do so My thoughts: Define ##w_b##...

Homework Statement The title. I am confused with the definition as my textbook does not have a good definition for this. P.S I have my exams tomorrow so I can't ask my teacher now.(Mid night) Homework Equations $$V=\frac{E}{Q}$$The Attempt at a Solution E.M.F is the energy given to each...

hello we define sin/cos/tan as the ratio of sides of a right triangle, which we have proved (? not sure, or we assume by a theorem or something?) that are constant for a specific angle this makes some sense but what about sin/cos/tan for degrees like 0, 90, >90 ? what about negative degrees...

Suppose we have a random service time ##T## with residual service time ##R## observed at some point along the way. What is the correct way to call ##T## (in 1-2 words, without having to introduce ##R##) if: 1) For any observation time, ##\mathbb{E}R\leq\mathbb{E}T##? 2) For any observation...

Hi! (Smile) According to my notes: Let $p \in \mathbb{P}$. The set of the integer p-adic numbers is defined as: $$\mathbb{Z}_p:= \{( \overline{x_n})_{n \in \mathbb{N}_0} \in \Pi _{n=0}^{\infty} \frac{\mathbb{Z}}{p^{n+1} \mathbb{Z}} | x_{n+1} \equiv x_n \pmod {p^{n+1}}\}$$ Could you explain...

Homework Statement Calculate the value of the limit and justify your answer with the ε-δ definition of the limit. lim (x->1) x2 Homework Equations My professor gave us the hint that we have to take δ as 0<δ≤ k0 so that δ(ε)=min{k0,ε/ (k0+2)} I'm guessing that k0 is meant to be any number...

Four definitions: 1) Define M_n( \mathbb{R} ) as the set of all n x n matrices over \mathbb{R}. 2) Define O(n) = \{ A \in M_n ( \mathbb{R} ) | A A^T = I \} 3) Define GL(n, \mathbb{R} ) = \{ A \in M_n( \mathbb{R} ) | \det A \neq 0 \} 4) Define SL(n, \mathbb{R} ) = \{ A \in GL(n, \mathbb{R} ) |...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings, on Page 66 we find a definition of right Artinian rings ... The relevant text in Cohn's book is as...

Homework Statement The lim(z->i) of [z^2+(1+i)z+2] using the epsilon-delta proof. Homework Equations z=x+iy Triangle Inequality: |z+w|<or=|z|+|w| The Attempt at a Solution For every epsilon>0, there exists a delta>0 such that |(z^2+(1+i)z+2)-(i)|<epsilon whenever 0<|z-i|<delta I'm not sure how...

Hello, i'm having some trouble understanding the definition of an asymptote, or rather the conditions that must be met in order for a line to be one. I have; "Let f : A \longrightarrow B be a function and A \subset R, B \subset R. A straight line is called an asymptote if one of the following...

If a function is continuous (nothing else specified), is it defined over R? Continuity means a function's value being the same as the limit for that point IIRC, but I don't know if it being continuous (over R presumably) means that it is also defined over R, or just that it's continuous wherever...

I am reading "An Introduction to Mechanics" by Kleppner and Kolenkow (2014). On page 241 is the definition of the angular momentum: "Here is the formal definition of the angular momentum $$\vec{L}$$ of a particle that has momentum $$\vec{p}$$ and is at position $$\vec{r}$$ with respect to a...

I'm confused by a set of problems my teacher created versus a set of problems in the textbook. My textbook states that "A vector space V over a field F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so that for each pair of...

(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...