Notation Definition and 1000 Threads

Hi All, I'm extremely confused by what's going on in section 10.1.3, pg 377 of the 2nd edition of Nakahara, in regards to his notation for lie algebra-valued one forms. We let \{U_i\} be an open covering of a smooth manifold M and let \sigma_i be a local section of M into the...

Notation question - R^m ---> r^n I've come across this notation a lot lately. I'm not sure what it really means. R^{m} \rightarrow R^{n} I can't find the place in my book where it explains it.

When doing calculations on AC circuits, what is in your opinion the best notation? Here they mostly use complex notation. For example: V(t) = Vmaxeiωt. In US textbooks (at the non-EE level), they only use ordinary notation. Example: V(t) = Vmaxcos(ωt). I have a feeling that complex notation is...

Hello, I was studying the theorem of smoothness/compactness in Fourier theory and at the very last step of the proof one gets the result that \omega F(\omega)\to 0 when x\to \infty. The author of the book writes this result in little-o notation as: \omega F(\omega) = o(|\omega|^{-1}) which I...

When one writes: \int^{t}_{t_{0}} f(s) ds Do they generally mean "the antiderivative of f(t), and ignore the arbitrary constant/pick t naught so that it is 0?"

Homework Statement This problem is about the 1-dimensional harmonic oscillator. The normalized energy levels are labeled |n>, n=0,1,2... Two operators are given by \widehat{A} = \alpha (\widehat{a}++\widehat{a}-) \widehat{B} = i\beta (\widehat{a}+2+\widehat{a}-2) where \alpha and...

Hey everyone, I'm struggling with the summation notation for matrices and vector operations, multiplication in particular. Please refer to the image below where I've typed it all out in Word, its too cumbersome here and I want my meaning to be clear...

I’m taking an electronics course and in the book it’s talking about a period in AC electricity and it asking me to find the frequency for the period and also the time for the frequency. The problem is this: Calculate the period for the two frequencies of 1 MHz and 2 MHz. For 1 MHz I...

Greetings, I would like to find the commutator \left[Lx^2,Ly^2\right] and prove that \left[Lx^2,Ly^2\right]=\left[Ly^2,Lz^2\right]=\left[Lz^2,Lx^2\right] I infer from the cyclic appearance of the indices that using the index notation would be much more compact and insightful to solve the...

Homework Statement 1. Given that |ψ> = eiπ/5|a> + eiπ/4|b>, express <ψ| as a linear combination of <a| and <b|. 2. What properties characterise the bra <a| that is associated with the ket |a>? Homework Equations The Attempt at a Solution 1. <ψ| = e-iπ/5<a| + e-iπ/4<b| 2. a. The bra <a|...

Surface charge density is denoted with a lower case sigma: σ Though sometimes I also see it denoted as σ with a subscript 0. What is the difference in these notations? Thank you.

Hello, I'm in an introductory course about quantum computing. My math experience is fairly solid, but not very familiar with Dirac (bra-ket) notation. Just would like to clarify one thing: In a single cubit space, we have |0 \rangle , and | 1 \rangle . I understand that these form an...

Just to clarify in the dirac equation (i\gamma^{\mu}\partial_{\mu} -m)\psi=0 Is it equal to (-i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i} -m)\psi=0 in (-,++++) notation?

Hi everyone, We just started learning partial derivatives and I understand the fx notation, but I'm confused when I'm asked for the value of fxy. Does this mean multiply the two derivatives together? For example: What is fxy when f(x,y) = (x+2y)ln(xy) Thanks!

Hey, I'm studying for a physics degree and have a general curiosity about vector calculus. Having learned about surface and line integrals for scalar functions in multivariable calculus I've been having some issues translating them into vector calculus. Though conceptually I haven't had much...

is there an algebraic meaning to expressing the derivative of a function as (d^2)y/(dx)^2 in the liebniz way

Good day! I was trying to make sense of the notation $P(X \in dx),$ where $X$ is a continuous random variable. Some also write this one as $P(X \in [x, x+dx])$ to represent the probability that the random variable $X$ takes on values in the interval $[x, x+dx].$ I have seen similar notation a...

Homework Statement Use exponential notation to write (√3-i)(1+i√3) in the form a+ib. Homework Equations The Attempt at a Solution Let (√3-i) = z1 r1=|z1|=√((√3)2-i2) =√(3-1) =√2 Therefore, √3=√2cosθ and -1=√2sinθ cosθ = √3/√2 and sinθ = -1/√2 θ= arcsin(-1/√2) θ= -∏/4...

Homework Statement sorry about the lack of LaTex but I don't know how to do bra-ket notation in tex vectors |1> and |2> are a complete set of normalized basis vectors. the hamiltonian is defined as |1><1|-|2><2|+|1><2|+|2><1| find the eigenvalues and eigenvectors in ters of |1> and |2>Homework...

Hey! How to transform the equation \bigtriangleup\vec E=\operatorname{div}(\operatorname{grad}(\vec E))=\epsilon_0\cdot\mu_0\cdot\frac{\partial^2\vec E}{\partial t^2} in Einstein Notation? Thank you all for your help!

I'm continuing through Dirac's book, The Principles of Quantum Mechanics. You can view this as a google book in the link below. http://books.google.com.au/books?id=...page&q&f=false On page 28-29 he proves this Theorem: If ξ is a real linear operator and ξm|P> = 0 (1) for a...

Homework Statement A force F=12NÎ-10NĴ acts on an object. How much work does this force do as the object moves from the origin to the point R=13mÎ+11mĴ? Homework Equations Work is the product of Force and Displacement (W=F*s) The Attempt at a Solution Things I know, or think I know: The object...

I see summation examples in my textbooks often give something where there is an ai next to the summation itself. It also shows the index of notation to be i=m at the bottom an m at the top with the whole thing equal to am+am+1+...+an-1+an What mathematical representation are they trying to...

Homework Statement Add the following 3 displacement vectors and give the answer in ijk notation and in magnitude and direction format. A = 3i - 3j m B = i - 4j m C = -2i + 5j m. Then I have to do a scetch of the resultant vector...which i'll call D. Homework Equations The...

Notation confusion; ## |\pi N; I, I_3 \rangle ## states In my book it says for the ##\pi N ## state: ##|\pi N; \frac{3}{2},\frac{3}{2}\rangle =|\pi ;1,1\rangle | N; \frac{1}{2},\frac{1}{2}\rangle## firstly, does this mean: ##|\pi N; \frac{3}{2},\frac{3}{2}\rangle =|\pi ;1,1\rangle...

Homework Statement Questions 11 and 12 specifically.. Homework Equations The Attempt at a Solution 11(a) 11(b) 12(a) 12(b) 12(c) 12(d) (I did the last part of 12(d) by normal vector methods and got 2a instead...which is the correct answer)

Homework Statement Given two functions f and g with derivatives in some interval containing 0, where g is positive. Assume also f(x) = o(g(x)) as x → 0. Prove or disprove each of the following statements: a)∫f(t) dt = o(∫g(t)dt) as x → 0 (Both integrals goes from 0 to x) b)derivative of...

Homework Statement (\sin x) (\cos x)^{n - 1} + (\sin x) (\cos x)^{\frac{n - 1}{2}} + (\sin x) (\cos x)^{\frac{\frac{n - 1}{2}}{2}} + (\sin x) (\cos x)^{\frac{\frac{\frac{n - 1}{2}}{2}}{2}} ... n is an odd number and the series ends when (n-1)/2^k = 1, and the last term ends up being sin(x)...

Homework Statement Show logical notation for being prime given N=(P1, P2,...Pn) +1Homework Equations The Attempt at a Solution I came up with the following, but I am not sure if it makes sense (I used trial division) (\existsx=((Pn+1)/((M>1)\wedge(M\leq√(Pn+1))))\inN) => (Pn+1 \neg=Prime...

So I've just started working with the index/einstein notation for matrices and vectors the other day. I've been doing a few exercises from a booklet I have, but I am still a bit confused. I am pretty sure my confusion is rather stupid though, so I apologize in advance. Homework Statement So...

Problem: Show that if $P(A_i)=1$ for all $i \ge 1$ then $P(\bigcap_{i=1}^{\infty}A_i)=1$. What is strange about this question is the first part, $P(A_i)=1$ for all $i \ge 1$. If I'm understanding this correctly that's saying that $P(A_1)=1$, $P(A_2)=1$...$P(A_n)=1$, for $n \ge i \ge 1$. This is...

Homework Statement Let f(x) = x6 + x3 + 1 in Z_2[x]. Show that f(x) is irreducible. Let E = Z_2[x]/(f(x)), and let αdenote the image of x in the quotient field. Show that E* = <α>. Homework Equations I have solved the first part, but what does E* mean? I have seen the asterix in...

given the vector in the first equation below, does that necessarily imply the third equation, as shown? {{u}_{a}}{{e}^{a}}={{x}_{a}}{{e}^{a}} {{u}_{a}}{{e}^{l}}g_{l}^{a}={{x}_{a}}{{e}^{l}}g_{l}^{a} {{u}_{a}}{{e}^{l}}={{x}_{a}}{{e}^{l}}

Use indicial notation to show that: $$ \mathcal{A}_{mi}\varepsilon_{mjk} + \mathcal{A}_{mj}\varepsilon_{imk} + \mathcal{A}_{mk}\varepsilon_{ijm} = \mathcal{A}_{mm}\varepsilon_{ijk} $$ I'm probably missing an easier way, but my approach is to rearrange and expand on the terms: $$...

Homework Statement See attachment: what is the interval notation of the continuous lineHomework Equations no equationThe Attempt at a Solution (-2,-1)U(-1,inf) ? not sure then it goes to 1 then to -inf? There are two continuous lines (-2,-1)U(-1,1)U(1,inf)? No... i feel like an idiot on this...

Homework Statement I have found the complexity of an algorithm as the expression below. How can I find the complexity in big O notation for such expression? Or proved that it's bounded by n^3 or n^4 ? Thank you! Homework Equations \sum_{j=3}^{n} \left[(j-1)[2(j-2)-1] +...

Trying to show that $\mathbf{a}\times\mathbf{b}\cdot\mathbf{a} = 0$. \begin{alignat}{3} \mathbf{a}\times\mathbf{b}\cdot\mathbf{a} & = & \varepsilon_{ijk}a_jb_k\hat{\mathbf{e}}_i\cdot(a_1\hat{\mathbf{e}}_i + a_2\hat{\mathbf{e}}_j+a_3\hat{\mathbf{e}}_k)\\ & = &...

Using indicial notation, I am trying to show that $\mathbf{v}\cdot\mathbf{v} = a^2b^2\sin^2\theta$ where $ \mathbf{v} = \mathbf{a}\times\mathbf{b}$ and $\mathbf{v}_i\hat{\mathbf{e}}_i = a_j \hat{\mathbf{e}}_j\times b_j\hat{\mathbf{e}}_k = \varepsilon_{ijk}a_jb_k\hat{\mathbf{e}}_i$. So...

Hi, I've been wondering about this forever and I finally decided to ask on the forums. In relativistic index notation (with c= \hbar =1) with the minkowski metric g\mu\nu=diag(1,-1,-1,-1), the 4-vector x^{\mu}=(t,x,y,z)=(x^0,\vec{x}), and with the del operator defined as \partial_{\mu}\equiv...

Hi, I didn't put this into homework since is only a question about notation: In a problem, given a Lagrangian and a transformation (x,y) -> (x',y'), where these x' and y' depend on λ, in particular like e^{\lambda}. The problem asks for the derivative \frac{\delta L}{\delta \lambda}. What...

I am using an old textbook as a reference, and the equation given for the electrical permittivity of free space ε0 is given in terms of: (Fm-1) where this measurement notation represents farads per meter. Why the superscript of -1 in the notation?

Homework Statement I don't have a particular problem to solve, but I need help with understanding the notation I have found in an academic paper. I have a system of linear equations which each take the form: pi xi r1 + pi yi r2 - pi u + qi xi r4 + qi yi r5 - qi v + xi x + yi y -...

I've always been curious why points in polar coordinates are defined as (r,θ) when all equations (including parametric equations formed from them) are defined as r=f(θ). Considering that point in cartesian coordinates are defined as (x,y) where y=f(x). Also a,b=(r,θ) ∫1/2[f(θ)]2 further...

I think I get summation notation when when there are more numbers than variables 6 Ʃ i/6 <---I can figure that out. i=1 But I'm confused on how to find what this equals: N Ʃ (i^3)/(N^3) = ? i=1 How do you add something N times? ...I could deal with a number like 6, but I'm...

It has been a while since I've had to figure out summation notation. Would you please look through my solutions, and tell me if they're correct? Thank you so much! :) 1a. 6 Ʃ 1/6 = ? i=1 1/6 + 1/6 + 1/6 + 1/6 + 1/6 +1/6 = 6/6 = 1 What makes me doubt my answer is that it seems like...

Sorry for disregarding the template; I'm not really working out a homework problem as much as just trying to follow the reasoning in the text. I'm working through the first chapter of Quantum Mechanics, McIntyre, and I'm a little bit confused by the following. The text introduces bra-ket...

The summation convention for Tensor Notation says, that we can omit the summation signs and simply understand a summation over any index that appears twice. So consider a 3X3 matrix A whose elements are denoted by aij, where i and j are indices running from 1 to 3. Now consider the...

I was wondering if the O notation definition could be exchanged with the Ω notation and o could be exchanged with the ω notation. I ask this because of this: 2n² O(n²) means that 2n² <= c*n² which it is true for c=3 and n>=1 for example Instead, it would be like this: c*2n² <= n² which...

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?