Definition Definition and 1000 Threads

Hello! (Smile) I am given this exercise: $$f(x)=\left\{\begin{matrix} \frac{e^x-1}{x} &, x \neq 0 \\ 1& ,x=0 \end{matrix}\right. , x \in [0,1]$$ Show that $f$ is integrable in $[0,1]$,knowing that if $f:[a,b] \to \mathbb{R}$, $f$ continuous,then $f$ is integrable in $[a,b]$. So,I have to...

Hello I am working on deriving the expression relating the equilibrium constant K to the change in Gibbs energy. This part seems to be followed okay, but here I am not following why the change in Gibbs energy of reaction is defined this way. I can see why K is defined in a way because...

Einstein has a thought experiment with two trains which he uses to prove linear motion without acceleration is inertial. Inertial means there is no physical test which will prove which train is moving and which is stationary, no coordinate system is preferred and that coordinate system are...

I guess I have several definitions of df flying at me, and I am having trouble getting a continuous definition. So in basic Calculus, we are taught df = f'(x)dx, and it's taught as sort of a linear approximation of the change of f for small values dx, whch makes sense with the definition of the...

I found in the wiki a definition for the max of 2 numbers: https://en.wikipedia.org/wiki/Ramp_function But is definition is only for 2 numbers, how would be the definition for 3 numbers? Also, which is the definition of minimum function?

Are set operations on a set ##X## defined as operations on ##2^X##? In other words a binary operation on ##X## is an operation ##\omega:2^X\times{}2^X\rightarrow{}2^x##? Surely the basic set operations could be defined that way, but then some weird non-standard operation like...

Can anyone give me a good definition of Euler's number and its significance. I see it everywhere, it's prolific in science and engineering.

Homework Statement A separable differential equation is a first-order differential equation that can be algebraically manipulated to look like: a. f(x)dx +f(y)dx b. f(y)dy = g(x)dx c. f(x)dx = f(y)dy d. g(y)dx = f(x)dx e. both f(y)dy=g(x)dx and f(x)dx = f(y)dy Homework Equations...

Could someone give me a word definition of Ampere's Law as I have googled some definitions and they all seem a bit confusing. Thanks :)

Accidentally I wrote in the wolfram f(x) = f(1/x) the the wolfram give me the solution for this equation (f(x) = Abs(log(x))). Hummmm, nice! Thus I thought: given the definition of derivative, ##f'(x) = \frac{f(x+dx)-f(x)}{dx}##, is possible to isolate f(x) in this equation? If yes, how? I...

what is phase constant and how is possible to go about figuring it out in an unscaled graph that has no values associated with it.

It's not a homework problem, just a problem that suddenly popped out of my mind. Homework Statement So, my question is : how to calculate, or how to define 'average inclination'? Suppose I am given an equation y=f(x) that resembles the shape of a section of a hill, and I want to...

On page 113 Munkres (Topology: Second Edition) defines a J-tuple as follows: I was somewhat perplexed when I tried to completely understand the function \ x \ : \ J \to X . I tried to write down some specific and concrete examples but still could not see exactly how the function...

Hi :) I'm reading a didactic paper and the author defined the initial state ket as |\Phi_{in}> = {\int}dq\phi_{in}(q)|q> where q is a coordinate and \phi_{in}(q) = <q|\Phi_{in}> = exp\left[\frac{-q^{2}}{4\Delta^{2}}\right] I don't know if I'm missing something but isn't this definition a...

A question about an anhydride that has nothing to to with acetic acid! How novel! I am not entirely sure I understand what an anhydride is. I found a reference to disulfuric acid, H2S2O7, and thought "Ah! Sulfuric Anhydride", but when I went to my other references, specifically the oxoacid...

Hi, While reading Sean Carroll's book, I came across the following statement: Okay so this has me confused. Perhaps I am nitpicking, but isn't f a scalar function, i.e. a 0-form? So shouldn't he really be saying "why shouldn't df be considered the one-form..."? If f is a scalar, then df (as...

Homework Statement lim x→2 (x2+1) = 5 Homework Equations 0 < |x-a| < delta |f(x) - L| < ε The Attempt at a Solution 0 < |x-2| < delta |x2-4| < ε |(x-2) (x+2)| < ε |x-2| |x+2| < ε.....and I am stuck here, any help

Hi all, I'm reading through Zwiebach's String Theory text on my own and am thinking about one of his very elementary exercises on "units." We are asked to define temperature in Kelvin with reference to the fundamental units of mass, length, and time. My thought is the following: We take...

Definition of "Pointer" I am in the process of writing a review-type paper for my intermediate quantum mechanics course. I have chosen to do my paper on the topic of measurement, with a focus on weak measurement. In all of the papers that I am reading, the term "pointer" is thrown around, but I...

Energy -- need a proper definition please Please could someone give me a proper definition of energy. My teacher said in class today that "there is no such thing as different types of energy". He said that although we learn this in introductory physics, a physicist would not agree there are...

Homework Statement My textbook gives me this definition of a connected set. http://media.newschoolers.com/uploads/images/17/00/69/80/76/698076.png I have been working through my practice problems and I got to one that asked me to sketch the set given by|z+2-i|=2and note whether it is...

Hello! In my studying of work I've always been told that work done (by what? ) = Force (component along its path) times the distance it moves. Mathematically, W=F.s ... My question here is what is the geometrical presentation of saying Force x Distance ... In other words, when we multiply...

In the book Galois Theory by Emil Artin (2nd Ed 1965 of a work copyrighted 1942), he says By contrast the modern definition of a field is that it is a commutative ring in which each nonzero element has a multiplicative identity. What developments caused the change in the definition with...

In wolframpage there is follows definition for shape operator in a given point by vector v: I think that this equation means: S(\vec{v})=-\frac{d\hat{n}}{d \vec{v}} correct, or not? If yes, of according with the matrix calculus...

Can someone tell me if there is a definition for energy. I don't mean a type of energy such as kinetic or potential etc. We talk about empty space having energy but what is it? Thanks

Hi, I have a question about the definition of the poisson process. Check out the definition here: Would you say that one can prove point (2) from point (3)? The reason I have some discomfort about this is that something seems to be hidden in the poisson distribution to make it all work? For...

Prove Lim x^2=9. With the epsilon/delta definition of a limit. x->3 My work so far. For every ε>0 there is a δ>0 such that if 0<|x-3|<δ , Then |x^2-9|<ε so, |(x-3)(x+3)|<ε |x-3|* |x+3|∠ε what do I do from here? My book is not very clear (Stewart Calculus 7ed)...

If the abs(a+b) = abs(a) + abs(b), so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ? √[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z) It's no make sense. What there is of wrong with those definions?

So I think I've just proven a preposition, where ##0## is divisible by every integer. I prove it from the accepted result that ##a \cdot 0 = 0## for every ##a \in \mathbb{Z}##. From then, we can just multiply the result by the inverse of ##a##, to show that the statement holds for ##0##. That is...

After a recent conversation, I was trying to determine if there is an online authoritative definition for the age of the universe. The Wiki article states "the International Astronomical Union presently use 'age of the universe' to mean the duration of the Lambda-CDM expansion", but the...

So when we have an open cover of a set X means we have a collection of sets \{ E_\alpha\}_{\alpha \in I} such that X \subset \bigcup_{\alpha \in I} E_\alpha . My question comes from measure theory, on the question of finite \sigma -measures, The definition I'm readying says \mu is \sigma...

page 71 he appears to define the affine connection in terms of derivatives on the locally inertial coordinates with respect to the laboratory coordinates and then the very next page claims that all you need is the affine connection and metric tensor to determine the locally inertial...

Find the limit L. Then use the epsilon-delta definition to prove that the limit is L. $\sqrt(x)$ as x approaches 9 I figure out the first part of the question. the Answer is three. Yet I have some difficulty to answer the second part of the question.Thank you Cbarker11

What is the precise definition of the open set? The definition I have been using until now has been that an open set is a set such that all of its points have some neighborhood that's contained in the set. The definition of neighborhood as far as I know is a collection of all the points within...

Definition of a "Feeble" acid? I'm currently trying to re-familiarize myself with chemistry, after a very long absence of any study in this field. At the moment I'm working my way through the text "Fundamentals of Chemistry" by "David E Goldberg" and I notice the author introduces the concept...

Hi, In Spice, the capacitance between two points x, y is defined as follows: Cxy = - dQx / dVy for x != y Cxx = dQx/dVx Could anyone help me explain why it is defined like that? I know that C = dQ/dV but the formula above seems strange to me. Is my understanding below correct? Cxy =...

Hi all, I'm wondering whether an expression which is used to describe a function in a certain domain is a different function for the same expression with a differing domain. For example: expression; x^2. f(x) = x^2 for domain {1 < x < 10} f(x) = x^2 for domain {10 < x < 11} Are these two...

Is correct to define Fourier series like: f(t)=\sum_{k=0}^{\infty}a_k \cos \left (\frac{2 \pi k t}{T} \right ) + b_k \sin \left (\frac{2 \pi k t}{T} \right ) Where ak and bk: a_k=\frac{1}{T} \int_{-T}^{+T} f(t) \cos \left (\frac{2 \pi k t}{T} \right ) dt b_k=\frac{1}{T}...

In the wikipedia page a distinction is made in the definition of the SR 4-acceleration between the case with rectilinear vs. curvilinear coordinates, the latter requiring the use of Christoffel symbols of the coordinates wrt the Minkowski space and therefore an additional term. Finally it...

The usual definition of an n-dimensional topological manifold M is a topological space which is 'locally Euclidean', by which we mean that: (1) every point in M is contained in an open set which is homeomorphic to ##\mathbb{R}^n##. (2) M is second countable. (3) M is an Hausdorff space...

The "definition" of synchronized clocks This is a purely terminological question. Consider the following setup: we have two inertial observers A and B at rest with respect to one another each equipped with a clock (clock A and clock B respectively), a means of exchanging light signals between...

The circle seems to be of great importance in topology where it forms the basis for many other surfaces (the cylinder ##\mathbb{R}\times S^1##, torus ##S^1 \times S^1## etc.). But how does one define the circle in point set topology? Is it any set homeomorphic to the set ##\left\{(x,y) \in...

I know Newton described it as 'the quantity of motion' but are there any other more descriptive or better definitions? Thanks in advance!

Ok so I roughly know what rotational motion is (e.g. the spinning of a DVD) but I wish to know the textbook definition for it. I tried Googling it but to no avail. The first link on Google gives this: http://en.wikipedia.org/wiki/Rotation_around_a_fixed_axis But as mentioned above, that...

in class we learned the definition of temperature to be \frac{1}{T}=\frac{∂S}{∂U} i don't understand why it's U as opposed to Q. afterall, Q is the only form of energy that contributes to temperature isn't it? If i take a bathtub of water and i swirl my arm in it, i just gave it some...

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...