Notation Definition and 1000 Threads

So to write 'in section 2' I would say 'in \S 2' for 'in subsection 4.2' would I then say 'in \S\S 4.2'? or just 'in \S 4.2'? Thanks

lim (e^x-sin(x)-cos(x))/(e^(x^2)-e^(x^3)), x->0 = lim (1+x+o(x)-x+o(x)+1/2(x^2)-1+o(x^2))/((x^2)+1+o(x^2)-1-(x^3)+o(x^3)), x->0 = lim 1/2(x^2)/(x^2+o(x^2)), x->0 = 1/2 is it correct?

Homework Statement The electron in a hydrogen atom is described by the following superposition of two states: |\psi> = \frac{1}{\sqrt{2}}(|n=2,l=1,m=0,s_z= +1/2> + |n=2,l=0,m=0,s_z= +1/2>) (b) Let J = L + S be the total angular momentum. Express state \psi in basis |n, l, J, J_z>...

Good day! I am reading the paper of Marc Yor (www.jstor.org/stable/1427477). equation (1.a) seems unfamiliar to me since the $ds$ comes first before the exponential part; $$ \int_0^t ds \exp(aB_s + bs). $$ Can you please help me clarify if there is a difference with the above notation as...

Homework Statement My Calc II final is tomorrow, and although we never learned it, it's on the review. So I have a few examples. Some I can figure out, some I cant. Examples: f(x)=sinh(x), f(x)=ln(x+1) with x0=0, f(x)=sin(x) with x0=0, f(x)=1/(x-1) with x0=4 The only one of those that I was...

How do I evaluate the sum of this sigma notation problem? 20 ∑ k k=10 Normally, I would think to use the theorem for the sum of the first n integers: n ∑ k = n(n+1)/2 k=1 I don't know how to do this, however, since the lower limit is k=10, not k=1. My professor wrote this note on the board...

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...

Hello, Can anybody tell me the meaning of f:N^n -->N

Hi, I noticed in mathematics the O symbol is used in the following way: A term T is in O(x^p), if lim x->0 T/x^p=c, for a constant c. While in computer science the O symbol is used is this way: A term T is in O(x^p), if lim x->∞ T/x^p is a constant. What gives, these two notations...

Hello, I am reading the paper "Covariant Action for a D=11 Five-Brane with the Chiral Field". I don't understand the (\partial a)^2 in the dual field strength \widetilde{F}_{mn}\equiv\frac{1}{\sqrt {(\partial a)^2}}F^*_{mnl}\partial^la(x). Is it ##\partial_ra\partial^ra##? Regards.

Suppose I have a function of x and t such that f(x,t) \leq x^{1/2} \sqrt{1+t^2}. How should I express this in big Oh notation in terms of both x AND t?

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...

Homework Statement Homework Equations Work = F* delta x The Attempt at a Solution My thought process since the force is given in vector notations multiply the pair of i components together same applies with j and k components. However this process actually strangely does not...

Hi there. First time PF poster. I have a question about interpreting IR spectro graphs. I get the general idea of interpreting the sharpness and strength of the line but when I need to ID the pulse down I have to choose between \delta and \upsilon (or mabe its just v) version of the...

My prof uses this all over his notes, and I'm still not 100% sure what he means by it: C[T]B or B[T]B From what I can gather, it has something to do with a transformation matrix, but where the B and C come into play, I have no idea.

Alright... So I'm in an 'introductory' Q.M class in college right now, it's the only one that this two-year college has, so I don't have an upper division Q.M Profs to talk to about this, and since my prof is equally confused, I turn to the internet. Okay, so everyone knows that <ψ|Aψ> = <a>...

In probability, P(A) is the probability of event A occurring. What is O(A), in the same context? I've tried looking this up, but I can't find the definition/explanation of this notation anywhere. For an example, see the definition of "Bayes' rule".

Homework Statement Imagine you have two vectors |a> and |b> such that: |c> = |a> + |b> Now imagine you want the dot product: <c|a> Is that the same as: <c|a> = [ <a|*+<b|* ] |a> = <a*|a> + <b*|a> where * represents the complex conjugate of the vector? Homework Equations...

Homework Statement Simplify the following, where A and B are arbitrary vector fields: f(x) = ∇\bullet[A \times (∇ \times B)] - (∇ \times A)\bullet(∇ \times B) + (A \bullet ∇)(∇ \bullet B) I know that the correct solution is A \bullet ∇2B, according to my professor. However, I can't...

Is there notation to denote the i-th row of a Matrix or j-th column?

Hi! I've got a problem with understanding notation in this lecture: http://www.youtube.com/watch?v=FZDy_Dccv4s&feature=BFa&list=PLF4D952FA51A49E66 For example, at 00:44:13, what does all lowercase deltas stand for? He writes: δA=∫(∂L/∂q)δq + (∂L/(∂q dot))δ(q dot) Why lowecase delta? What...

Homework Statement Determine the complexity of the following code: for (i = 0; i < 2*n; i += 2) { for (j=n; j > i; j--) { a++; } } The Attempt at a Solution Well.. The first for block is O( n ) because i is incremented by 2 each loop up to 2n. The second block...

Hi, I have been going round in circles trying to figure it out so I figured it would be quickest and easiest to just ask: What is the definition of: \int f(x) d^4 x I first thought it was: =\int \int \int \int f(x) dx^1 dx^2 dx^3 dx^4 but now I am starting to think it is...

f: ℝ^n→ℝ\qquad g: ℝ^n→ℝ^m

While taking notes in class, I was trying to write the moment of inertia tensor in Einstein notation as Iij instead of a 3x3 matrix, but when we diagonalized the matrix with diagonal elements Ii, I was confused on how to write it. Ii doesn't work because that means that treats it as a vector...

Hi, I've been wondering this since I started learning integration. I get that ∫ is basically an elongated S for "sum", because that is what it is basically doing. But then Ʃ does the same thing as well. If I'm understanding the difference, it is that Ʃ increments by finite measures, whereas ∫...

Hello, I have read several different sources on this very topic, and the one thing that confused a little was defining it using sigma notation. Could some please explain to be what it means?

Homework Statement I see in chapters on perturbation theory and time-dependent variation, the Hamiltonian (usually expressed as H) is now printed as H(t). This is still the same Hamiltonian, correct? I assume this notation simply helps to signify that the total energy varies with time? If...

Homework Statement consider the position vector expressed in terms of its cartesian components, r=xiei. Let w=wjej be a fixed vector whose components wj are constants that do not depend on the xi, so that δwj/δxi = 0 Homework Equations I am trying to evaluate ∇((wXr)^2) The...

Hi I have a question regarding notation of matrices. I am trying to conserve space in my report, so instead of writing my matrix fully like this \left( {\begin{array}{*{20}c} 1 & 2 \\ 0 & { - 5} \\ \end{array} } \right) my plan is to write it as (1,0 ; 2 -5)^T. Is this notation...

hello, after some struggle,i finally understand these notations, but there is a problem. i'm studying with "principles of quantum mechanics" [R. Shankar]. in the beginning of the chapter where he discusses 'bra-kets' he mentioned the axioms of inner product. one of the axioms he mentioned...

Homework Statement Prove that: \sum_{n=1}^{14} 10n = \sum_{n=1}^{7} (20n+70) Homework Equations properties of sigma notation The Attempt at a Solution I know several properties of sigma notation but none that I know can be used to prove this. I don't know how to change 10 n to 20...

Determining the commutation relation of operators -- Einstein summation notation Homework Statement Determine the commutator [L_i, C_j] . Homework Equations L_i = \epsilon_{ijk}r_j p_k C_i = \epsilon_{ijk}A_j B_k [L_i, A_j] = i \hbar \epsilon_{ijk} A_k [L_i, B_j] = i \hbar...

I had taken a multi-variable calculus course and since have misplaced my notes. I recall the prof inventing his own notation because somewhere partial derivatives using Leibniz notation don't show the correct path. I think it was something like if you had a function f(x,y)=z and y depended on x...

How would you simplify this expression: <a|<b|a>|a> where ψ = |a>|b> and I'm finding ψ*ψ.

Hi all! If you are given an operator such that A|1> = √(1/3) |1> +√(2/3) |2>, how do we interpret it? I do know that 1/3 and 2/3 are probabilities but is this operator application on state one suggesting that this state in state 1 and 2 with probabilities 1/3 and /3 respectively? Thank you!

Hi If C is an operator such that C|1> = |1> and C|2>=|2>, then C^3 |1>= |1>|1>|1> =|1> ^ 3 ? If yes, then what does this C^3 represent? :confused:

Homework Statement How many elements of each cycle-type are there in S5? The Attempt at a Solution One way of working this out would be to write out each permutation and see how many 2-cycles, 3-cycles ,4-cycles and 5-cycles there are but given that there are 5! permutations this would...

In reviewing some basic GR (just to keep my old brain sharp), i was looking at the Einstein notation cinvention and was a bit confused. I see how you do the dot product of say: ei.ej = δij (i.e. 1 or 0) But then the book I'm reading talks about ei.ej or ei.ej. Isn't that just the...

Homework Statement Prove that (AxB) is perpendicular to A *We know that it is in the definition but this requires an actual proof. This is what I did on the exam because it was quicker than writing out the vectors and crossing and dotting them. Homework Equations X dot Y = 0 when...

I have some trouble understanding the attached section of my book. Basically I can't see why the marked equations are equivalent - that is the first two are contained in the last one. I can follow the derivation but when I do an example for myself where I just have two variables (q,p) being...

The attached is a section of the derivation of canonical transformation from the symplectic formulation. I tend to get very confused by the subscripts i and j. For me they both run from 1 to 2n and can be used interchangeably. But of course that is not the case since equation (9.53) on the...

(r×∇).(r×∇)=r.∇×(r×∇) now in index notation it is written as, =xi∂jxi∂j-xi∂jxj∂i but when I tried to prove it ,it just came out twice.can anyone tell how it is correct(given is the correct form).i really mean that i was getting four terms which gave twice of above after reshuffling...

Homework Statement ∫ ∂k(gixiεjklxl dV Can anyone make sense of this? I know I'll need to apply the chain rule when taking the derivative, but I'm not quite sure how to proceed. Also, this is part of a larger problem where g is a gravity vector existing purely in the -z direction, but I...

I've attached the .pdf from which I have questions. After it says "Take the square of the numerator" halfway down the page, there is an equation that lists vector components with hats and arrows at the same time, and the arrow on other vectors. Can someone help explain the difference to me...

Homework Statement In an intro Economics course, the teacher gave us some questions regarding function notation. I learned calculus to a decent level, but I have problems with notation. Homework Equations Question 1. Give an explicit expression for the following set: { x : x ∈ Z and x < 2 }...

Homework Statement Verify that (n^2 + 3n -3)/n^3 = 0 + O(2/n) Homework Equations The Attempt at a Solution I really don't have an attempt. I understand Big O notation, but I don't know how to verify this.

Hello everyone, Recently I started to use index notation, but still the division is not clear for me. I'll mention just some simple examples that I'm not sure about: Does a =\frac{1}{b_i} mean that a = \sum_{i=1}^{3}\frac{1}{b_i} or a = 1 / \sum_{i = 1}^{3}b_i ? Similarly, does a_i...

Homework Statement Xuv is a 4x4 tensor and Vu is a vector. Vu = (-1, 2, 0, -2) (i.e. it is a 1x4 vector). Find the quantities Vu Vu and Vu Xuv Homework Equations Given above The Attempt at a Solution I'm having trouble finding Vu. Initially I thought that it should be the...

Homework Statement http://imgur.com/gTapO Homework Equations The Attempt at a Solution The first one is easy, just use the fact that δi = δ/δxi and it reduces to the sum from with i=1,2,3 of δxi/δxi = 1 + 1 + 1 I tried to do a similar thing with the second one, also using...