Form Definition and 1000 Threads

Hello, Today I started to think about why graphs, of the same equation, look different on the Cartesian plane vs. the polar grid. I have this visualization where every point on the cartesian plane gets mapped to a point on the polar grid through a transformation of the grids themselves...

Hey! 😊 Let $v\in \mathbb{R}^2$ be a vector and let $X\subseteq \mathbb{R}^2$ be a subset, then we define the subset of $\mathbb{R}^2$ : $$v+X:=\{v+x\mid x\in X\}$$ Let \begin{equation*}L:=\mathbb{R}\begin{pmatrix}1 \\ 2\end{pmatrix}=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\mid 2x-y=0\right...

Hi, I was wondering by what other methods iron can form in stars? Iron can form during silicon burning and i assume it can also form via the beta- decay of neutron rich isotopes around the iron peak? Are these athe only two processes that relate to the cosmogenesis of iron?

If it is known that such an equation exists, I would much appreciate seeing a link to a reference. If it is known that such an equation does not exist, I would much appreciate seeing a reference with an explanation of implications regarding the compatibility of general relativity with a...

Hey! 😊 Calculate the Cholesky decomposition of the matrix, the only non-vanishing elements are the diagonals $1,2,3, \lambda$ and all under and upper secondary diagonal elements are $1$. For which $\lambda$ is the matrix singular? Could you please explain the form of the Matrix? Does the...

I would like to investigate a function that sends ##f(x)## to ##f(x) - \frac{1}{c}f(x^{c})##, or a function ##g## such that ##g(f(x)) = f(x) - \frac{1}{c}f(x^{c}).## Since symmetries produced by groups are used in physics, I thought there might be someone here who could help explain what the...

$\tiny{1.5.12}$ Describe all solutions of $Ax=0$ in parametric vector form, where $A$ is row equivalent to the given matrix. RREF $A=\left[\begin{array}{rrrrrr} 1&5&2&-6&9& 0\\ 0&0&1&-7&4&-8\\ 0& 0& 0& 0& 0&1\\ 0& 0& 0& 0& 0&0 \end{array}\right] \sim \left[\begin{array}{rrrrrr} 1&5&0&8&1&0\\...

What is your personal practice in dealing with expressions of fullness that end with "full"? (I'm just curious what other people do, I'm not likely to change my own habits.) I don't know if USA English has a rule about writing expressions like "cup full" (two words) vs "cupful" (one word sans...

Describe all solutions of $Ax=b$ in parametric vector form, where $A$ is row equivalent to the given matrix. $A=\left[\begin{array}{rrrrr} 1&-3&-8&5\\ 0&1&2&-4 \end{array}\right]$ RREF $\begin{bmatrix}1&0&-2&-7\\ 0&1&2&-4\end{bmatrix}$ general equation $\begin{array}{rrrrr} x_1& &-2x_3&-7x_4...

Write the solution set of the given homogeneous systems in parametric vector form. $\begin{array}{rrrr} -2x_1& +2x_2& +4x_3& =0\\ -4x_1& -4x_2& -8x_3& =0\\ &-3x_2& -3x_3& =0 \end{array}\implies \left[\begin{array}{rrrr} x_1\\x_2\\x_3 \end{array}\right] =\left[\begin{array}{rrrr}...

Standing waves in a string fixed at one end is formed by incoming and reflected waves. If reflected waves are 180° out of phase with incoming wave, how could they combine to give an oscillating wave? Shouldn't it be completely destructive interference all the time across the whole length of string?

I don't really know how to begin. I've done alkylations by having two of the same compounds react with each other e.g. two aldehydes but never started out with dimethyl malonate. I was thinking I need 1,4 dibromobutane to form the cyclopentane ring but apart from that I'm clueless

Hello, I am trying to design a mechanism that moves a piece towards the center of a disk by rotating a second disk as described in the picture below. There is a stationary disk which restricts the motion of the piece, so it can move vertically only. There is also a second disk that it is located...

In a writing class, my professor taught us that "any" should be followed by plural form of nouns, so as "no". For example, we should say, "Are there any students in the classroom?", and "There are no students in the classroom.". And it is incorrect to say something like "Is there any student in...

I drew the red and green tangent lines and I found that the angles in blue are equal to theta 1. Also , as the BCD triangle is equilateral, theta 2 = 30. With this I can calculate the side of this equilateral triangle as a function of the radius R of the circumference. After that, I can't go on...

I was trying to find this form of the Taylor series online: $$\vec f(\vec x+\vec a) = \sum_{n=0}^{\infty}\frac{1}{n!}(\vec a \cdot \nabla)^n\vec f(\vec x)$$ But I can’t find it anywhere. Can someone confirm it’s validity and/or provide any links which mention it? It seems quite powerful to be...

... And if it won't form spontaneously with reversed flow, then will a vortex at least sustain itself once we nudge it in one direction?

I recently came across a paper (referenced below) containing the statement that:"The differential form notation is much more concise and elegant than the tensor notation, but both contain the same information.", and the paper left me with a desire to understand the notation of differential...

Summary:: Hello, my question asks if the complex exponential equation 4e^(-j) is equal to 4 ∠-180°. I tried to use polar/rectangular conversions: a+bj=c∠θ with c=(√a^2 +b^2) and θ=tan^(-1)[b/a] 4e^(-j)=4 ∠-180° c=4, 4=(√a^2 +b^2) solving for a : a=(√16-b^2) θ=tan^(-1)[b/a]= -1 b/(√16-b^2)=...

Having read a few Wiki articles on and around the subject, I am now somewhat aware of the means by which these ##H^-## anions acquire the extra electrons "donated" by e.g. ionized alkali atoms in stellar atmospheres, and the means by which they provide opacity. What I don't understand is how it...

Problem: See Attachment. Parts (a) & (b) are clear, but my confusion arises in (c)-- I feel like there is a much simpler form. While technically my answer is correct, there must be something I'm missing. I parameterized the curve C=(t, e^2t, e^2t) and got c'(t)=(1,2e^2t,2e^2t), which should...

I have an long equation I managed to simplify to that form, but now I haven't the foggiest on how to solve for x, any help appreciated

In the framework of tight binding approximation, does the wavefunction for atom A (or B) has two spinorial components(2 components) in "real space"? If so how does this spinorial component propagate in the graphene?

I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12). And in Topology, Geometry and Physics by Michio...

I have asked this question before but badly and just caused confusion. So I thought to ask it again but without muddled presentation. If ##M## is a closed surface embedded in ##R^3## that has strictly positive Gauss curvature then its second fundamental form is positive definite and so is a...

I managed to expand a general expression from the alternatives that would leave me to the answer, that is: I will receive the alternatives like above, so i find the equation: C = -sina, P = cosa So reducing B: R: Reducing D: R: Is this right?

So the standard explanation for star formation says we have a disk of gas collapsing into itself until a certain pressure/temperature is reached at which point the star "ignites" and pushes away the rest of the material in the disk. My thinking is, surely this pressure/temperature needed for...

i have been stuck this problem for a few days now. could somebody help me please?

I'm reading a text on special relativity (Core Principles of Special and General Relativity), in which we start with the equation for composition of velocities in non-standard configuration. Frame ##S'## velocity w.r.t. ##S## is ##\vec v##, and the velocity of some particle in ##S'## is ##\vec...

At first I thought it was C. A few sources agree with me. The book says D is correct. Here was my reasoning for C: With both salts having a formula of MX3, their [OH-]'s can be compared according to their Ksp values. Tl(OH)3 has a smaller Ksp, so it has a smaller [OH-] at saturation than...

Please help me solve this, Given digits 2,2,3,3,3,4,4,4,4, how many distinct 4 digit numbers greater than 3000 can be formed? Thank you

Hi, K₁cos(θt+φ)=K₁cos(θt)cos(φ)-K₁sin(θt)sin(φ)=K₁K₂cos(θt)-K₁K₃sin(θt) Let's assume φ=30° , K₁=5 5cos(θt+30°) = 5cos(θt)cos(30°)-5sin(θt)sin(30°) = (5)0.866cos(θt)-(5)0.5sin(θt) = 4.33cos(θt)-2.5sin(θt) If only the final result, 4.33cos(θt)-2.5sin(θt), is given, how do I find the original...

Hi friends How exactly do we change the general equation of a line in space( given two intersecting planes) into the canonical form Thanks

1. From the series, I got that ##a_n=\frac{(-1)^{n+1} n^2}{n^2+1}##. I think my answer is correct. But if I try to find the limit as n approaching infinity, I got indeterminate form (isn't it?) : ##\frac{-1^\infty}{1+0}## 2. What is the limit as n approaching zero? Thanks, all!

##ds^2=g_{μν}(x)dx^μdx^ν= -(r^6/l^6)[1-(Ml^2/r^2)]dt^2+{1/[(r^2/l^2)-M]}dr^2+r^2dΦ^2## Does anyone know how to solve this?

In theory, does an algebraic expression exist for the ground state of the Klein Gordon equation with \phi^4 interactions in the same way an algebraic expression exists for the simple harmonic oscillator ground state wavefunction in Q.M.? Is it just that it hasn't been found yet or is it...

program Lab5A implicit none ! This program introduces Fortran string handling capabilities character*26 upper, lower, name, cap character str*2, one*1 integer from, to, i, m lower = "abcdefghijklmnopqrstuvwxyz" upper = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"...

I've been deeply disturbed by recent brain-related experiments that involve driving around insects, rats, and probably other animals like remote-controlled toys. No doubt this will lead somehow to some benefit for humanity. To cut straight to the point, do you feel (because this is a moral...

The steps are as follows (apologies for my awful formatting, I'm not sure which Chemistry latex package there is!), $$(CH_3)_2CO \xrightarrow{HCN} (CH_3)_2C(OH)CN \xrightarrow{H_2SO_4} (CH_3)_2C(OH)COOH \xrightarrow{PCl_5} (CH_3)_2C(OH)COCl \xrightarrow{NH_3} (CH_3)_2C(OH)CONH_2$$ As far as I...

Hi, I want to use cobertura for coverage. I followed the following link...

Hey there, I was looking at the Higgs sector of the standard model, particularly its coupling to the fermions: ##\mathscr{L}_{ yukawa }=-\sum _{ a,b=1 }^{ 3 }{ \left( { Y }_{ ab }^{ u }{ \bar { Q } }_{ a }{ \hat { \varepsilon } }_{ 2 }{ H }^{ \dagger }{ u }_{ b }+{ Y }_{ ab }^{ d }{ \bar { Q }...

I am not sure if this is the right forum to post this question. The title says it all: are there examples of Lie groups that cannot be represented as matrix groups? Thanks in advance.

Hey! :o Let $\displaystyle{a:=\begin{pmatrix}2 & 1 & 0 & 5 \\ 1 & 0 & 1 & 1 \\ 4 & 1 &2 & 7\end{pmatrix}\in \mathbb{R}^{3\times4}}$ and $\displaystyle{b_1:=\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} , \ b_2:=\begin{pmatrix}-2 \\ 1 \\ 0\end{pmatrix} \in \mathbb{R}^3}$. I applied the Gauss...

Summary:: Pretty sure they have something to do with path integrals, or what not. But obviously it's hard to *search* for this stuff. Basically, I'm looking for a textbook, any textbook--physics, mathematics, etc.--that deals with integrals that look something like this (mistakes are mine): S...

how would the year 2011 appear in standard form

In this derivation,i am not sure why the second derivative of the vector ## S_j '' ## is equal to ## R^{u_j}{}_{xyz} s^y_j v^z y^x## could anyone explain this bit to me thank you it seems ## S_j '' ## is just the "ordinary derivative" part but it is not actually equal to ## R^{u_j}{}_{xyz}...

in the language of general relativity,we know that we can write $$\nabla_{V}W $$ in this form such that: $$\nabla_{V}W = = w^i d ( V^j e_j)/du^i = w^j e^i (V^j e_j ) = W( V)$$ where $$w^i * d/ (du^i) =W$$ will act on the vector V where $$W = w^i d( ) /du^i $$ and W is a vector as a...

I've looked for a while and can't find an answer to this, hence the post. Ultimately this is a question about how the first pieces of "classical" matter formed from quantum matter. My study of self-organizing systems shows that you need a hierarchical build-up of structures to allow a complex...

hello, please can you help me in this problem

I'm reading through Thomson's "Modern Particle Physics", and I've gotten stuck at a point in the derivation of the form factor for electron scattering in a static potential due to an extended charge distribution. It's just a mathematical "trick" i don't quite get. He goes from $$\int\int...