Definition Definition and 1000 Threads

How far is the definition of time that time is change in space, correct?

This is a very sound stupid question, but I'll go ahead anyway. Current is defined as the displacement of charge through a cross-section of a conductor per unit time. Okay. But how does relate to its math-definition, current = change of charge with respect to time, I=dQ/dt? I mean, for every...

definition of "mathematical object" ...And "mathematical existence." Do these phrases have accepted definitions? Back when Dedekind was rejecting Cantor's transfinite ideas, could there have been a definition Cantor would refer to and say definitively "my (infinite) sets have mathematical...

Homework Statement I'm trying to understand how this equation was came together, mainly, I hardly understand why velocity is squared, and why there's the constant of '1/2'. Homework Equations In classical mechanics, the kinetic energy of a point object (an object so small that its mass...

I was reading this page: http://en.wikipedia.org/wiki/Tensor which said the definition of a tensor was a relation between two vectors. I then went down to the examples section and it had some sort of (n,m) notation. I had some theories on what they meant but none of them made sense. What do n...

I wish to know what exactly magnetic declination and inclination means. I used to feel that angle of declination was constant throughout the Earth equalling≈11.3°.But, it apparently isn't. Then, why is so much significance attached to 11.3° if it is not universally constant?

Homework Statement What is the definition for first law of thermodynamic? I was confused and which one is correct? (a) The CHANGE in internal energy of a closed system is equal to the heat that enters a system and the workdone on the system OR (b)...

Graded poset on wiki: Graded poset - Wikipedia, the free encyclopedia Wikipedia defines a 'graded Poset' as a poset $P$ such that there exists a function $\rho:P\to \mathbb N$ such that $x< y\Rightarrow \rho(x)< \rho(y)$ and $\rho(b)=\rho(a)+1$ whenever $b$ covers $a$. Then if you go to the...

I have seen two definitions with oposite signs (for one of the pressure terms in the formula) all over the web and books. I suspect it is related to the chosen metric signature, but I found no references to that. General Relativity An Introduction for Physicists from M. P. HOBSON...

Hi, I have a question, because I am confused with the definition of surface tension. In my book it is defined as follows: "Surface tension is the energy required to increase the surface area of a liquid by a unit amount" What do they mean with increasing the surface area, how do you do...

I'm a bit confused about how my book defines convergence. Definition: A sequence {an} convergences to l if for every ε > 0 there is a natural number N such that, for all natural numbers n, if n > N, then l a,-l l < ε note, l a,-l l = the absolute value Maybe someone could give me an...

It seems weird that such a relatively complex concept is simply given as a definition in most textbooks and then dismissed for further explanation other than using it intact or as a basis for further proofs.

How is a dimension defined in quantum mechanics?

Hello, I just whant to know what mathematical rule alows me to do this? I mean i think it is u-substitution, but i am not sure how it is done here? It is weird to me as it seems that ##dt## just cancel out and limits are changed... $$ \int\limits_{0}^{t} \frac{dv}{dt} \cdot mv \gamma(v)\, dt...

I don't quite get the significance of the delta limit definition, if n>N and |sn−s|< ϵ , why does the limit converges does this simply means that there exist a number ε such that if n is great enough it will be greater than s by ε? But this doesn't make sense, because s is the value...

I was fiddling around with the definition of limits at infinity and believe I have found a statement that is equivalent to the definition. So the question is this: are the following two statements equivalent? (1) \lim_{x\rightarrow\infty}f\left(x\right)=L (2) \exists c>0\exists...

Homework Statement "We say a set T is a subset of a set S if every element of T also belongs to S( i.e T consists of some of the elements of S). We write T ⊆ S if T is a subset of S and T ⊄ S if not. For example, if S = {1, {2}, cat}, then {cat} ⊆ S, {{2}} ⊆ S, 2 ⊄ S. As another example, the...

I am confused about the counting of degrees of freedom. Yes, I know that it is the number of vectors which are free to vary. But that definition gives way to different interpretations: (1) the number of data points minus the number of independent variables. This seems to be the basis of the...

So, it seems that in a real-valued setting, the limit and the derivative of a real-valued function is defined only if the domain is an open subset of Euclidean space. I'm a little confused as to why this is the case, and why we can't just define a limit and derivative on any subset of Euclidean...

definition of an integral question... if f(x) ≥ 0, how can you use the definition of an integral to prove that ∫(a,b)f(x)dx ≥ 0? This seems like it is an easy question, and seems like one of those things that seems obvious but hard to explain, and the only definition of an integral I've been...

How do we mathematically define a inertial frame in GR? Is it only a basis in some tangentspace or does it have to be induced by a coordinatechart? :/

Homework Statement Using the definition of derivative find f'(x) for f(x) = x - sqrt(x) Homework Equations None. The Attempt at a Solution lim h --> 0 : ((x + h) - sqrt(x + h) - x + sqrt(x))/h 1 - (sqrt(x + h) - sqrt(x))/h Multiply by conjugate.. 1 - h/(h*(sqrt(x) +...

One volt is when: The difference in (electrical potential energy per unit charge (q)) between two places equals one. Where electrical potential energy equals EPE at distance R from charge Q = (1/4piEpsilonNought) * Q/R Is this Correct? Thanks!

I don't like the way Wiki describes entanglement. Here is my own definition. Tell me if it is in essence correct. If I have left out an important detail please let me know objects are entangled if and only if by changing the property of one object one instantaneously changes the property of...

Why can $H = \left(\ell - \frac{b}{\sin\theta}\right)$ where $\ell$ is the length of the rope. (everything is frictionless.) http://img690.imageshack.us/img690/579/pulley.png

I've seen in some probability theory books that the classical definition of probability is a probability measure, it seems fairly trivial but what is the proof for this? Wikipedia gives a very brief one using cardinality of sets. Is there any other way?

Homework Statement Is the set $$ \{cos(x), cos(2x)\} $$ linearly independent?Homework Equations Definition: Linear Independence A set of functions is linearly dependent on a ≤ x ≤ b if there exists constants not all zero such that a linear combination of the functions in the set are equal to...

In my class notes my professor defined a true vector as a vector which does not depend on origin placement. Once he defined it he went on with an example of how a vectors magnitude is conserved in two different coordinate systems. So my question is what is the definition of a true vector? Is...

Hello, I am working through some proofs from the following document: Function Definitions Under Calculation of Big - Oh, some theorems are provided that classify the growth rates of functions in relation to one depending on what the limit is as the input approaches infinity. One proof is...

In general terms a manifold can be defined simply as a topological space locally resembling Euclidean space with the resemblance meaning homeomorphic to Euclidean space, plus a couple of point set axioms that avoid certain "patological" manifolds and that some authors reserve for the definition...

Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x2 And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and...

Homework Statement Can you guys explain to me what the following mean. We are working on probability and unions, and these came up on the homework and need to know what these mean in order to solve the problem. Thanks P(AB) P(AB)c Where c is the compliment. Also i want to...

Homework Statement Find the derivative of f(x) = x using the definition of a derivative. (when Δx → 0) Homework Equations (x + Δx) - x -------------- Δx The Attempt at a Solution I know the answer is 1. I graphed the function of f(x)=x and confirmed this, however the...

Hi All, Usually the Fourier transform is defined as the one in the Wiki page here (http://en.wikipedia.org/wiki/Fourier_transform), see the definition. My question is can I define Fourier transform as \intf(x)e^{2\pi ix \varsigma}dx instead, i.e., with the minus sign removed, as the...

So I was looking through Wald when I noticed his definition of the stress-energy for an arbitrary matter field: T_{ab}=-\frac{\alpha_M}{8\pi} \frac{1}{ \sqrt{-g}} \frac{\delta S_M}{\delta g^{ab}} where S_M is the action for the particular type of matter field being considered, and \alpha_M...

I've been poking around, learning a little about homology theory. I had a question about the boundary operator. Namely, how it's defined. There's two definitions I've seen floating around. The first is at: http://en.wikipedia.org/wiki/Simplicial_homology The second, at...

I used the linear equation method to solve a D.E. and got y=3/4 at the end. I'm asked to find the interval of definition but I don't know how to do that when Y is just a constant :/

You sometimes see statements regarding so-many-minutes after the big bang. Or 10^-23 seconds after the big bang. But this exactly is the event this is measuring from? How is it defined? I presume it has some definition from within the framework of general relativity.

Attached is a scan of how Balanis define plane wave with circular polarization with Ey having a phase of +∏/2 respect to Ex component of the E field. I don't quite agree with the book. The second attachment is my derivation. The definition of CW or CCW is with respect to direction of...

Hi, I am currently reading about differential forms in "Introduction to Smooth Manifolds" by J. M. Lee, and I was wondering exactly how you define the wedge product on the exterior algebra \Lambda^*(V) = \oplus_{k=0}^n\Lambda^k(V) of a vector space V. I understand how the wedge product is...

Hi, I have this report to do on "Charge injection in heterostructures". I have been searching and reading but I still have some trouble with the basics, i.e. defining the concept. As far as I understood a heterostructure is a junction between two or more different semiconductors and the...

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.