Form Definition and 1000 Threads

Hi everyone, I am teaching myself Linear Algebra and I am confused with the terminology used in the subject. I am studying Linear Algebra based on Anton's. In the textbook, an augmented matrix in REF needs to have the first nonzero number in a given row to be 1. However, in other textbooks...

Assume M=xdy -ydx+dz ∈ Ω1(R^3). What's the restriction of M to the plane {z=2}? I think it's xdy-ydx. Is that right?

Homework Statement ##\frac{log_{2^{x^2+2x+1}-1}(log_{2x^2 + 2x + 3}(x^2 - 2x)}{log_{2^{x^2+2x+1}-1}(x^2 + 6x + 10)} \geqslant 0 ## The set of all real solutions to this inequality is of the form: ##(a) ## ##(a,b) \cup (b,c) ##, ##(b) ## ##(-\infty,a) \cup (c,\infty) ##, ##(c) ## ##(a,b) ## for...

What is the expected equation for total dark energy in universe as a function of size of the universe? ie size of universe=D Dark Energy f(D)= (D^n)*constant ; where n=-2,-1,-.5,0,.5,1,2 Dark Energy f(D)= D*constant or Dark Energy f(D)= (1/D)*constant or Dark Energy = constant or Dark Energy...

Suppose x ∈ Ω^(n−1)(Rn \{0}) is closed and the integral of x on S^(n-1) equals to 1. I am stuck on how to show there does not exist an n − 1 form y ∈ Ω(n−1)(R^n) with y|R^n\{0} = x.

Homework Statement Hello guys; I am currently dealing with a problem that I have faced before several times and I would like to know a consistent way on how to solve it. I think what I want to do is diagonalize a matrix but I'm not sure if that's exactly it. Basically I have two or three...

I know this has been asked before: "Why is there a negative in the Lagrangian: L = T - V" I have read the answers and am not happy with them so I tried to formulate my own justification and now ask if anyone could comment on it? First, I am not happy with those who say "Because it works and...

Good evening. Is there a way to take a decimal approximation and see if there is a relatively simple expression? I'm guessing there might be software for this, but I'm not sure I'm even asking the appropriate question. If it matters, the number I'm after is...

Homework Statement I have the following complex numbers : -3,18 +4,19i I must put it in polar form. Homework Equations r=(a^2+b^2)^(1/2) cos x = a/r sin x = b/r The Attempt at a Solution I was able to find with cos x = a/r that the x = 127,20 But when I do it with sin x = b/r I obtain like...

Tried simplifying it of course, but didn't get far. Here's tbe problem: ''Express 3sin(3x)-4cos(3x) in the form Rcos(3x+\alpha),\alpha\ge0;R>0. Hence, find the smallest possible value of x for which 3sin(3x)-4cos(3x)=4.'' Bit confusing for me, especially the last part. How do you solve this, lads?

Homework Statement Show that all ##n \times n## unitary matrices ##U## leave invariant the quadratic form ##|x_{1}|^{2} + |x_{2}|^{2} + \cdots + |x_{n}|^{2}##, that is, that if ##x'=Ux##, then ##|x'|^{2}=|x|^{2}##. Homework Equations The Attempt at a Solution ##|x'|^{2} = (x')^{\dagger}(x')...

Homework Statement Show that the set of all ##n \times n## orthogonal matrices forms a group. Homework Equations The Attempt at a Solution For two orthogonal matrices ##O_{1}## and ##O_{2}##, ##x'^{2} = x'^{T}x' = (O_{1}O_{2}x)^{T}(O_{1}O_{2}x) = x^{T}O_{2}^{T}O_{1}^{T}O_{1}O_{2}x =...

Ehecatl posted very helpful content on this ,.. Just wondering if anyone can describe the actual reaction that takes place?

Homework Statement Show that all ##n \times n## (real) orthogonal matrices ##O## leave invariant the quadratic form ##x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}##, that is, that if ##x'=Ox##, then ##x'^{2}=x^{2}##. Homework Equations The Attempt at a Solution ##x'^{2} = (x')^{T}(x') =...

Homework Statement Write the expression i2i in the form a + bi Homework Equations Honestly we haven't treated such subjects during the classes, but I've made some researches and found the Euler identity might help me. The Attempt at a Solution By using the Euler identity, I found that i =...

I think my question is more appropriate here than in the computation section. My question is: (In the context of inverse fast-fourier transforms and fast-fourier transforms) Knowing ifft(fft(x)=x might be trivial as it is almost a definition; associating it with a domain ##t## is perfectly...

I read about how MRI works briefly, by flipping the water molecules using a magnetic field to the correct state then send the radio wave to these atoms and have it bounces back to be received by receiver coils and apply Fourier Transform to figure out the imaging. My question is, how does...

what is the closed analytical form of exp(ξG) ? could you help me !

Hello everyone I would like to ask a question that seems simple but can't find the (detailed) proof/ how its derived. Simply we know that Vrms = Vpk /[Sqrt (2)]. But how is that derived? Thank you in advance Fawzi

In another forum someone states that "cacao powder" cannot be considered as a "solid state" since "it cannot sustain shear stresses". Has this statement any basis? -- lightarrow

Homework Statement Homework EquationsThe Attempt at a Solution I chose choice no.2&3 but only choice no.1 is correct. I've understand why no.2 is wrong, but why is no.3 wrong? I thought N exists as N2, which is highly inert?

Hello! (Wave) A linear programming problem is in canonical form if it's of the following form: $$\pm \max (c_1 x_1+ \dots + c_n x_n) , c_1, \dots, c_n \in \mathbb{R} \\ Ax=b, A \in F^{m \times n}, x=\begin{bmatrix} x_1\\ \dots\\ \dots \\ x_n \end{bmatrix}, b=\begin{bmatrix} b_1\\ \dots\\...

Is it possible for a molecule like water to exist in BEC form?

This is actually a pretty simple thing, but the ref(A) that I compute on paper is different from the ref(A) that my TI-89 gives me. Compute ref(A) where A = \begin{bmatrix} 1 & 2\\ 3 & 8 \end{bmatrix} \\ \begin{bmatrix}1 & 2\\ 3 & 8\end{bmatrix} \ r_2 \rightarrow r_2 - 3 \times r_1 \\ \\...

I have come across the following paragraph from my book "The gravitational force is always attractive .Whereas the electric force is attractive or repulsive acording to whether q0 is negative or positive.Each of the above forces is conservative,so a potential energy is associated with each of...

I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that ## \nabla\cdot E=\frac{1}{\epsilon 0}\rho##, but when I calculate it directly I get that ## \nabla\cdot E=0## (at least for ## r\neq0##). 2) Now ##...

How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...

I am not entirely sure how to convert the conservation of mass and momentum equations into the Lagrangian form using the mass coordinate h. The one dimensional Euler equations given by, \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0...

What is the basic matrix form for a uniform (unit) sphere centered at the origin? Given a vector that specifies the radii (1,1,1) == (r1,r2,r3), I would like the matrix that implies no rotation (is it [[1,0,0],[0,1,0],[0,0,1]]?) and covers the rest of the necessary parameters. I am testing...

I understand that monoatomic ions such as Sodium and Magnesium form to fill there outer shells, but why do polyatomics form? I know that CO2 is a pretty stable compound, so why does carbonate even form? Likewise what leads to the formation of ammonium?

How can I find the parametric vector form of a cartesian equation under a specific condition? Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1...

Homework Statement Assume the number https://www.physicsforums.com/tel:1111111 Is in one's complement form, what is its decimal values? Homework Equations 2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0 1 in first sign bit indicates negative 0 in first sign bit indicates positive The Attempt...

Homework Statement I am doing an experiment where I have to test the validity of the equation: (m1-m2)g=(m1+m2+I/R^2)a. The lab instructions say to linearize this equation. What would be the "m" and "b" values? Homework Equations y=mx+b (m1-m2)g=(m1+m2+I/R^2)a The Attempt at a Solution I...

We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis. The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or...

I wrote x² - (a + b)x + (ab) in the wolfram and polynomial discriminant was: a² - 2ab + b². Factoring: (a-b)² --- So, I wrote x³ - (a+b+c) x² + (bc+ca+ab) x - (abc) and the polynomial discrimant given was: Factoring: (b-c)² (c-a)² (a-b)² --- Now, I wrote x² - 2Ax + B² and the polynomial...

Homework Statement Q7.[/B] a) Express x^(n-1)⋅³√(y^2/2x^5) in its simplest, rationalised surd form. b) Given that the solution to part a) is 5, and that y can be expressed as 1/x^(6n+5)/4), determine the value of x. Again, express your answer in rationalised surd form. *Note, this is a...

How can you tell when a compound will form a covalent bond or a coordinate bond? I know that a coordinate bond is a special type of covalent bond and if during covalent bonding, if the elements taking part do not obtain a noble gas configuration, they for coordinate bonds. But take for example...

Hey! :o Let $R$ be any integral domain of characteristic zero. We consider the Pell equation $$X^2-(T^2-1)Y^2=1\tag 1$$ over $R[T]$. Let $U$ be an element in the algebraic closure of $R[T]$ satisfying $$U^2=T^2-1\tag 2$$ Define two sequences $X_n, Y_n, n=0, 1, 2, \dots $, of polynomials in...

I have x1 + 3x2 - x3 = b1 x1 - x2 + 3x3 = b1 -2x1 - 5x2 - x3 = b1 So using an augmented matrix I get this [1 3 -1 | 1] [1 -1 3 | 1] [-2 -5 1 | 1] [1 3 -1 | 1] [0 -4 4 | 0] R2 - R1 = R2 [0 1 -1 | 3] R3 + 2R1 = R3 [1 3 -1 | 1] [0 1 -1 | 3] Swap R2 with R3 [0 -4 4 | 0] [1 0 2 | -8] R1 - 3R2 =...

Homework Statement Hi all, I have this quiz on MasteringPhysics, but I can't seem to get the right answer.[/B] Consider two positively charged particles, one of charge q0 (particle 0) fixed at the origin, and another of charge q1 (particle 1) fixed on the y-axis at (0,d1,0). What is the net...

"Most vegetables, annuals and grasses prefer their nitrogen in the nitrate form and as such do better in alkaline inclined soils dominated by bacteria. Most trees, shrubs and perennials prefer their nitrogen in the ammonium form and as such do better in acid inclined soils dominated by fungi."

Homework Statement Express $$4sin\theta-3cos\theta$$ in the form $$rsin(\theta-\alpha)$$ Hence find the maximum and minimum values of $$\frac{7}{4sin\theta-3cos\theta+2}$$ State the greatest and least values. Homework EquationsThe Attempt at a Solution Okay so putting it in the...

Hi there, I was reading up on Holonomic constraints and came across this equation on the Wikipedia page: The page says it is a differential form. Can anyone explain the notation for me or provide a link or two to documents or pages which explain this notation? Thank you very much, Geoff

The EFE are tensor equations in 4-dimensional spacetime and by virtue of their tensorial form indepedence from the choice of coordinate system is guaranteed, and the same goes for the metric tensor solutions. When looking for assumptions that help simplify the process of solving the EFE to find...

Hello, Please, someone, explain what the // in the hint below stands for: "Show that the numbers of the form ±m√2/n for m, n ∈ N are dense." Hint: "To find a number in (x, y), find a rational in (x//√2, y//√2). Conclude from this that the set of all (irrational) numbers of the form ±m√2/n is...

I am no Einstein but I would like to expand my knowledge and share it. When two deuterium atoms fuse together they become a helium nuclei. Now deuterium nuclei contains 1 proton and 1 neutron. When the deuterium nuclei fuses to form a helium nuclei. Helium nuclei contains 2 protons, and 2...

Hi All, I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199). Most appreciated if someone could enlighten me this. Many thanks in advance. Peter Yu

Hey, all. I have a question concerning the treatment and use of vectors when solving problems (or in general, really). I know that vectors have both magnitude and direction, while scalars only have magnitude. However, in solving problems and looking at how others have solved them, I've noticed...

To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...

I have been studying the Maxwell equations recently (namely the integral forms of them). Of course I had to study line integrals before that. Well, I went to a hyperphysics page to look up the equations: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html I noticed that the...