Vector Definition and 1000 Threads

Hi, This feels like such a stupid question, but it's bugging me. Two displacements can be represented with two vectors. Let's say their magnitudes are expressed in metres. The scalar (dot) product of the two vectors results in a value with the units of square metres, which must be an area. Can...

let ##f : R^3 → R## the function ##f(x,y,z)=(\frac {x^3} {3} +y^2 z)## let ##\gamma## :[0,## \pi ##] ##\rightarrow## ##R^3## the curve ##\gamma (t)##(cos t, t cos t, t + sin t) oriented in the direction of increasing t. The work along ##\gamma## of the vector field F=##\nabla f## is: what i...

My idea is to evaluate it using gauss theorem/divergence theorem. so the divergence would be ## divF = (\cos (2x)2+2y+2-2z ( y+\cos (2x)+3) ) ## is it correct? In this way i'ma able to compute a triple integral on the volume given by the domain ## D = \left\{ (x, y, z) ∈ R^3 : x^2 + y^2 +...

Hello! I am reading about spin-orbit coupling in Griffiths book, and at a point he shows an image (section 6.4.1) of the vectors L and S coupled together to give J (figure 6.10) and he says that L and S precess rapidly around J. I am not totally sure I understand this. I know that in the...

Summary:: Suppose that [x, y] = e^{-3t} [-2, -1] is a solution to the system $x' = Ax$, where A is a matrix with constant entries. Which of the following must be true? a. -3 is an eigenvalue of A. b. [4, 2] is an eigenvector of A. c. The trajectory of this solution in the phase plane with axes...

I've attached a .txt file of my script for those who want to take a look at it Here's a picture of my vector field at time t = 0 I'm very concerned about this picture because from my understanding the Poynting vector is supposed to point outwards and not loop back around, this looks nothing...

is it correct if i use Gauss divergence theorem, computing the divergence of the vector filed, that is : div F =2z then parametrising with cylindrical coordinates ##x=rcos\alpha## ##y=rsin\alpha## z=t 1≤r≤2 0≤##\theta##≤2π 0≤t≤4 ##\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{4} 2tr \, dt \, dr...

Given ##F (x, y, z) = (0, z, y)## and the surface ## \Sigma = (x,y,z)∈R^3 : x=2 y^2 z^2, 0≤y≤2, 0≤z≤1## i have parametrised as follows ##\begin{cases} x=2u^2v^2\\ y=u\\ z=v\\ \end{cases}## now I find the normal vector in the following way ##\begin{vmatrix} i & j & k \\ \frac {\partial x}...

I need to prove this using the given equations. $$\vec{N}(t) = \frac{\vec{a}_{v\perp}}{|\vec{a}_{v\perp}|}$$ Here is the entirety of my work up to this point. So far I've wanted to use what I have to find something that is perpendicular to the velocity vector and maybe show that with the dot...

For a specific wave vector, ##k##, the group of wave vector is defined as all the space group operations that leave ##k## invariant or turn it into ##k+K_m## where ##K_m## is a reciprocal vector. How the translation parts of the space group, ##\tau##, can act on wave vector? Better to say, the...

I'm struggling to get the hang of killing vectors. I ran across a statement that said energy in special relativity with respect to a time translation Killing field ##\xi^{a}## is: $$E = -P_a\xi^{a}$$ What exactly does that mean? Can someone clarify to me?

Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...

I understand that the Dual Space is composed of elements that linearly map the elements of the Vector Space onto Real numbers If my preamble shows that I have understood correctly the basic premise, I have one or two questions that I am trying to work through. So: 1: Is there a one to one...

Hello, I start by applying the integral for the vector potential ##\vec{A}## using cylindrical coordinates. I define ##r## as the distance to the ##z##-axis. This gives me the following integral,$$\vec{A} = \frac{\mu_0}{4\pi} \sigma_0 v 2 \pi \hat{x} \int_0^{\sqrt{(ct)^2-z^2}}...

I am so confused.If F and d are both vector quantity.How come W is a scalar quantity?

I am trying to understand “divergence” by considering a one-dimensional example of the vector y defined by: . the parabola: y = -1 + x^2 The direction of the vector y will either be to the right ( R) when y is positive or to the Left (L). The gradient = dy/dx = Divergence = Div y = 2 x x...

I don't know which part gets wrong

vector=(abc) 1. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) \\ 0& sin(\theta) & cos(\theta) \end{pmatrix}$$ The rotation part is correct. 2. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0& 0 & 0 \end{pmatrix}$$ is wrong apparently how do I do the mirroring? step 3 i can do...

In studying gyroscopic progression, the angular momentum vector is added to the torque vector. As intuitively these two vectors seem to be qualitatively quite different, how do we know that both vectors are in the same vector field and that they can be manipulated using the rules of vector...

As a simple example, the probability of measuring the position between x and x + dx is |\psi(x)|^{2} dx since |\psi(x)|^{2} is the probability density. So summing |\psi(x)|^{2} dx between any two points within the boundaries yields the required probability. The integral I'm confused about is...

question1 : if you draw a small circle around the north pole (it should be the same at every points because of the symmetry of the sphere),then it is approximately a flat space ,then we can translate the vector on sphere just like what we have done in flat space(which translate the vector...

I am having problem with part (b) finding the vector potential. More specifically when writing out the volume integral, $$A = \frac{\mu_0}{4\pi r}\frac{dq}{dt}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{?}\frac{1}{4\pi r'^2} r'^2sin\theta dr'd\theta d\phi$$ How do I integrate ##r'##? The solution...

Here is his paper. I don't see what the big deal about it is. https://arxiv.org/pdf/1911.00892.pdf

So I'm trying to figure out the integral of phi hat with respect to phi in cylindrical coordinates. My assumption was that the unit vector would just pass through my integral... is that correct? (I reached this point in life without ever thinking about how vectors go through integrals, and...

I'm currently working out quantities that include the vector and axialvector currents ##j^\mu_B(x)=\bar{\psi}(x)\Gamma^\mu_{B,0}\psi(x)## where B stands for V (vector) or A (axialvector). The gamma in the middle is a product of gamma matrices and the psi's are dirac spinors. Therefore on the...

I was thinking about this while solving an electrostatics problem. If we have a vector ##\vec V## such that ##\oint \vec V \cdot d\vec A = 0## for any enclosed area, does it imply ##\vec V = \vec 0##?

Before writing out each component I'm going to simplify ##\vec{I}## to the best of my abilities $$\vec{I} = \int \left(\hat{r}\cdot\vec{r'}\right) \vec{r'} \rho\left( \vec{r'} \right)\, d^3r'$$ $$\vec{I} = \hat{r} \cdot \int \vec{r'} \left( x' , y', z' \right) \rho\left( \vec{r'} \right)\...

Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged. Hello! I am struggling with understanding the meaning of "each member is a unit vector": I can see that N would represent the number of samples, and the pointy bracket represents an...

I'm new to classical mechanics. I've done enough work with vectors to get the basics. But, I'm having trouble understanding the notation on this MIT presentation I found on circular motion: http://web.mit.edu/8.01t/www/materials/Presentations/Presentation_W04D1.pdf On slide 23, for example, I...

Hello Everyone, A small dilemma: is force, which is a vector, a free vector, since it can be slid along its along of application, thus changing its point of application (principle of transmissibility) or a bound vector, since the point of application of the force is crucial for the effect the...

If I wanted to write ##\hat{r}##in terms of ##\hat{x}##and ##\hat{y}##, is it ##\frac{\hat{x} + \hat{y}}{\sqrt{2}}## ?

Hi everyone, While finding the solution for one of my exercises, I found the following answer. I'm seriously questioning if the equations provided in that answer are reversed. According to my understanding, if two vectors ##\vec{S}## and ##\vec{T}## are parallel (same direction) the magnitude...

Matt and Hugh play with a tennis ball and a brick. Then they do some working out to derive the formula for the centripetal force (a = v^2/r) by differentiati...

I spent a good amount of time thinking about it and in the end I gave up and asked to a friend of mine. He said it's a 1-line-proof: just "integrate by parts" and that's it. I'm not sure you can do that, so instead I tried using the identity: to express the first term on the right-hand side...

If a "stand" on the ball, I would feel a centrifugal force, which would be pulling me out of the circle. But in the equation of centrifugal force we have ##\vec r##, which is the vector that goes from the centre of the non inertial frame to the body in motion. But if I'm on the ball, my system...

The sketch above shows the situation of the problem. Clearly, as the rotation is taking place in the ##y-z## plane, the x-components of the two vectors remain unchanged : ##A_x = B_x##. Let the projection of the vector ##\vec B## on to the y-z plane be vector ##(\vec B)_{yz} = B_y \hat y + B_z...

Homework Statement: Mike the Mailman takes his oath seriously: "Neither snow, nor rain, nor heat, nor gloom of night stays these courageous couriers from the swift completion of their appointed rounds". Even though a blizzard is raging outside, he goes out to deliver the mail. He makes four...

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As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations. However, what I noticed in Source #2 was that, when functions are represented as vectors, the...

I am assuming the set ##V## will have elements like the ones shown below. ## v_{1} = (200, 700, 2) ## ## v_{2} = (250, 800, 3) ## ... 1. What will be the vector space in this situation? 2. Would a subspace mean a subset of V with three or more bathrooms?

If I'm using the basis vectors |u> and |r> for two polarisation states which are orthogonal in state space, I've seen the representation of a general state oriented at angle theta to the horizontal written as $$\lvert\theta\rangle = \cos(\theta) \lvert r \rangle + \sin(\theta) \lvert u...

Homework Statement: The homework problem is included below, but I am looking at the derivatives of vectors. Homework Equations: I have the properties of derivatives below, but not sure they help me here...

Starting with LHS: êi εijk Aj (∇xA)k êi εijk εlmk Aj (d/dxl) Am (δil δjm - δim δjl) Aj (d/dxl) Am êi δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...

Lorentz gauge: ∇⋅A = -μ0ε0∂V/∂t Gauss's law: -∇2V + μ0ε0∂2V/∂t2 = ρ/ε0 Ampere-Maxwell equation: -∇2A + μ0ε0∂2A/∂t2 = μ0J I started with the hint, E = -∇V - ∂A/∂t and set V = 0, and ended up with E0 ei(kz-ωt) x_hat = - ∂A/∂t mult. both sides by ∂t then integrate to get A = -i(E0/ω)ei(kz-ωt)...

I have tried to solve it and I would like a confirmation, correction or if something else is suggested... :) Helicoidal movement

If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...

Suppose I have a vector of matrices: \mathbf{v}=(A_{1},\cdots,A_{n}) How would I vectorise this in MATLAB? This question comes from a requirement to compute a Greens function for the spherical heat equation. I can easily compute a single function for a single position in space, but can I do...

I'm currently watching lecture videos on QFT by David Tong. He is going over lorentz invariance and classical field theory. In his lecture notes he has, $$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##. He mentions he uses active...

The only thing tripping me up here is that the answer needs to be in vector form. If the question was asking for the scalar form, then I would just find the distance between the charges (plot the charges according to their vector coordinates, then use pythagorean theorem to find the distance...