$$ {\Lambda}^{i}_{j} $$
When indices are written on top of one another I am confused wich is the inner index and which is the lower one when we lower the upper index.
Suppose i have a term like this one (repeated indices are being summed)
$$x = \psi^a C_{ab} \psi^b$$
Such that ##C_{ab} = - C_{ba}##, and ##\{\psi^a,\psi^b\}=0##. How do i evaluate the derivative of this term with respect to ##\psi_r##?
I mean, my attempt g oes to here
$$\frac{\partial...
The actual problem that i was looking at with my students was supposed to be
##x^\frac{2}{3} - x^\frac{-2}{3}-6=0##(which is easy to solve using quadratic equations) of which i wanted them to solve, ...then i realized then that i had erronously posted
##x^\frac{2}{3} - x^\frac{-3}{2}-6=0## on...
In《Introducing Einstein's Relativity Ed 2》on page 106"lowering the first index with the metric,then it is easy to establish,for example by using geodesic coordinates..."
In 《A First Course in General Relativity - 2nd Edition》on page 159 "If we lower the index a,we get(in the locally flat...
I‘m reading the chapter 4 《Perfect fluids in special relativity》of《A First Course in General Relativity》.In the process of deriving conservation of energy-momentum,it said:##\frac {\partial T^0{^0}} {\partial t}=-\frac {\partial T^0{^x}}{\partial x}-\frac {\partial T^0{^y}}{\partial y}-\frac...
Hi everyone,
I am a new member and would like to ask a naive simple (my guess) question.
I am reading Weinberg’s Gravitation and Cosmology. On page 59, Eq. 2.12.10 therein reads
$$
\begin{aligned}
\left[\sigma_{\alpha \beta}\right]_{\gamma \delta}{}^{\varepsilon \zeta}
&=\eta_{\alpha \gamma}...
My approach;
$$(a-a^{-1})(a^{\frac {4}{3}} + a^{\frac {-2}{3}})
=(a-\frac {1}{a})(a^{\frac {4}{3}} +\frac {1}{a^{\frac {2}{3}}})
=(\frac{a^2-1}{a})(\frac{a^2+1}{a^{\frac {2}{3}}})
=\frac{a^4+a^2-a^2-1}{a^{\frac {5}{3}}}
=\frac{a^4-1}{a^{\frac {5}{3}}}$$
now at this point i multiplied both...
I am reading《Relativity - An Introduction to Special and General Relativity》
my question:
##1=-\eta_{44}L^{n{'}}{_4}L_{n{'}}{^4}=-\eta^{n'}{^{m'}}L_{n{'}}{_4}L_{m'}{_4}##
##\eta## is Minkowski Metric，##L## is Lorentz transformation matrix...
1.Since ##-\eta_{44}##=1,what's the usage of it...
I'm still confused about the notation used for operations involving tensors.
Consider the following simple example:
$$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$
Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get...
I have just met linearized gravity where we decompose the metric into a flat Minkowski plus a small perturbation$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\ \ \left|h_{\mu\nu}\ll1\right|$$from which we 'immediately' obtain $$g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$$I don't obtain that. In my rule book...
In dynamic programming:
1. what is the definition of the space of subproblems? does it have a mathematical definition?
2. why is it necessary to have an arbitrary index for the subproblem to vary?
To elaborate on question 2, I've taken the following paragraph from chapter 15.3 in...
Property (a) simply states that a second rank tensor that vanishes in one frame vanishes in all frames related by rotations.
I am supposed to prove: ##T_{i_1 i_2} - T_{i_2 i_1} = 0 \implies T_{i_1 i_2}' - T_{i_2 i_1}' = 0##
Here's my solution. Consider,
$$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1...
The ##I_i## are the intensity of the rays, in other words energy per surface units per radians by seconds.
The d##\Omega## are the solid angles
The equation p75 isis what I don't understand. I suppose that each side represent the energy going and out of the surface dS but I don't understand...
Hello,
I am clear on 1D and 2D Numpy arrays, how to create them and address them).
1D array: single list
2D array: list containing multiple lists as elements
3D array: list containing lists which contain lists as elements
Array elements can be address using indices as a[], a[][], a[][][]...
Hi everyone
Could someone please help me with a yr 10 maths problem? It's for my niece. I've done 2nd yr uni maths and can't seem to solve it.
Write 2^(-m) x 3^(-m) x 6^(2m) x 3^(2m) x 2^(2m) as a power of 6.
I've attached my attempt in the file. I get stuck at the point where I need...
Hi,
I have this question about the variation of wavelength and frequency as light travels to an environment with a different index.
As we have learned in class, celerity can change as light enters a different environment, however frequency and wavelenght are independent and remain constant...
I have an equation $$
\chi_\nu\nabla_\mu\chi_\sigma+\chi_\sigma\nabla_\nu\chi_\mu+\chi_\mu\nabla_\sigma\chi_\nu=0
$$so we also have$$
g_{\nu\rho}g_{\mu\tau}g_{\sigma\lambda}\left(\chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho\right)=0
$$Does...
I know what Carroll refers to as 'conservation of indices' is just a trick to help you remember the pattern for transforming upper and lower components, but nonetheless I don't understand what he means in this example:
E.g. on the LHS the free index ##\nu'## is a lower index, and on the RHS...
I am able to simplify/evaluate the above equations correctly, however I end up with an incorrect sign for each answer (i.e positive when it should be negative) and I can't see where the error is. I feel I am clearly missing something but having checked my working including with a calculator for...
Can someone help me understand why what I wrote is correct? That is: If I have a sequence with double indices and if the summation of the elements modules of this sequence converges (less than infinite) than it does not matter how I make this sum (second line) they are going to be always the...
is it true that##\frac{\partial f}{\partial x_\mu}=\frac{\partial f}{\partial x^\mu}*g_{\mu \mu}##?
how to lower some of indices if same indice is many times in same tensor or multipication? for example:
##T^{i_1 i_1 i_2}## to ##T_{i_1 i_1 i_2}##
##a^{\alpha}*b^{\alpha}*c^{\alpha}## to...
Hi.
I would like to check that my understanding is correct. For ##f(x)=x^{1/n}## where n is an integer. If n is odd then f(x) is an odd function while if n is even then f(x) is neither odd or even as it involves the square root function which is only defined for non-negative x.
For ## f(x) =...
I want to calculate transition amplitudes in QCD for processes like ##q(k)q^\prime(p)\rightarrow q(k^\prime)q^\prime(p^\prime)##, where ##q,q^\prime## are quarks. However, I am unsure what to do with the colour indices of the quark spinors upon squaring the matrix element. For the sake of...
I have a technical problem.
1. Accordingly to historical E.B. Christoffel’s work (I think year 1869), (Christoffel’s) symbols are symmetric in the two (today writing) lower indices.
2. These symbols have been introduced when studying the preservation of differential forms of degree two. The...
To analyze the LHS of this equation, I used (k-1) , k and (K+1) to get
## \frac {(-1)^{k-1} } { (k-1)} \ . \frac {(-1)^k} { (k)} \ . \frac {(-1)^{k+1} } { (k+1)} \ ##
Nothing cancels out in these terms and the sign of each term is the opposite of the previous term.
I calculated...
Helping my daughter with her math and hit this one and not sure how to advise. All help welcome(x-2y10)3 / (x-4yz4)-5
This one throws me off because I don't know how to deal with the z, as only on the right side of the divide
Let ##Q_ik## be a symetric tensor, so that ##Q_ik= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j## (here k is also a sub, couldn't do it better with LaTeX).
How do we derive such a tensor, with respect to time? And what could such a tensor mean in a physical sense? It really looks like the...
When dealing with any tensor quantity, when making a coordinate transformation, we should put a bar (or whatever symbol) over the functions or over the indices? For exemple, should the metric coefficients ##g_{\mu \nu}## be written in another coord sys as ##\bar g_{\mu \nu}## or as ##g_{\bar \mu...
I found notation in the form
##\vec{e}_{\alpha} \cdot \vec{x}=x^{\alpha}##
and also
## \langle \vec{e}^{* \alpha}, \vec{x} \rangle =x^{\alpha} ##.
If I understand this ##\vec{e}_{\alpha}## is unit vector in direct space, and ##\vec{e}^{* \alpha}## is unit vector in dual space, and the bracket...
I'm self-studying field theory and trying to solidify my understanding of index manipulations. So I've been told that there is a general rule: " If the index is lowered on the 'denominator' then it's a raised index". My question is whether this is just a rule or something that can make sense...
Hello, I just want to clarify some things with a simple exercise: I have the equation ## \frac{\partial^2 f}{\partial A^\mu \,\partial A^\nu} = 0## and I want to integrate it once assuming that ## f=f(A^1,A^2,...,A^n)=f(A^\rho) ##.
I think the solution should be ## \frac{\partial f}{\partial...
Homework Statement
both filled to height h in the vessel of length 2h. one has refractive index root 2 and the bottom fluid has refractive index n. find the apparent depth of vessel.
Homework Equations
n = real depth/apparent depth
The Attempt at a Solution
what these guys have done ...
I am reading Leonard Susskind's Theoretical Minimum book on Quantum Mechanics. Excercise 7.4 is as follows:
Calculate the density matrix for ##|\Psi\rangle = \alpha|u\rangle + \beta|d\rangle##.
Answer:
$$ \psi(u) = \alpha, \quad \psi^*(u) = \alpha^* \\
\psi(d) = \beta, \quad \psi^*(d) =...
Homework Statement
Homework Equations
find intercept ==> reciprocal ==> reduction ==> (hkl) miller indices ==> i ==> (hkil) miller indices
The Attempt at a Solution
by using axis a1 a2 z,
intercept: a = 0, b = inf, c=inf
reciprocal: a= inf,b=0,c=0
(hkl)=(inf 0 0)
i=-(h+k) = no conclusion
I...
Homework Statement
Using the E-L equations to get the EoM from the action.
Homework Equations
I am using E-L equations in the form:
## \frac{\partial}{\partial_u} \frac{\partial L}{\partial_u \phi}-\frac{\partial L}{\partial \phi} ##
where ##L ## is the Lagrangian
The Attempt at a...
Homework Statement
We've been told there's this operation called 'contraction' where if you have a superscript and a subscript that are the same they cancel. I don't understand how that works, partly in the sense that we haven't got round to what the superscripts and subscripts actually mean...
Homework Statement
Write out this covariant derivative in terms of partial derivatives and Christoffel symbols:
##\nabla_{\mu} S^{\nu}_{\nu \rho}##
Homework EquationsThe Attempt at a Solution
I think you can contract that so it reads
##\nabla_{\mu} S_{\rho}##, in which case the solution...
Hey,
I thought I understood Wick contractions but a formula in Zee's "Quantum Field Theory in a Nutshell" disproved me:
In the section on Feynman Diagrams it is tried to evaluate the "four-point Green's function" in (I.7.10) by the integral $$
\int_{-\infty}^\infty \left ( \prod_m \mathrm{d}...
Hi everyone, to find the draw the direction for a given miller index say, [1234] we first convert this miller index consisting of 4 indices into one containing 3 indices.
To do so, we have a set of formulae prescribed in almost every book. Sadly I haven't been able to come across a single book...
Is there any way to show sub or sup indices in a text file such as this Ab? Text files are very easy to use. They are fast, can be opened fast and created easily.
Thank you.
Hi everyone,
I'm currently studying Griffith's Intro to Elementary Particles and in chapter 7 about QED, there's one part of an operation on tensors I don't follow in applying Feynman's rules to electron-muon scattering :
## \gamma^\mu g_{\mu\nu} \gamma^\nu = \gamma^\mu \gamma_\mu##
My...
Homework Statement
A parallel beam of light containing two wavelengths, λ1= 400 nm and λ2= 650 nm, enters the silicate flint glass (at a angle of 41 degrees, relative to normal) of an equilateral triangle (60 degrees at each angle) prism. At what angle, relative to the normal, does each beam...
Compare this with the definition of the inverse transformation Λ-1:
Λ-1Λ = I or (Λ−1)ανΛνβ = δαβ,...(1.33)
where I is the 4×4 indentity matrix. The indexes of Λ−1 are superscript for the first and subscript for the second as before, and the matrix product is formed as usual by summing over...
Hi there. I am working with a numerical quadrature in some scheme to solve a set of equations. At this point I am working in two dimensions. The thing is that I have some function ##\psi_m(x,y,\Omega_m)## with ##\Omega_m=(\Omega_{x,i},\Omega_{y,j})## with ##\displaystyle...