What is Vector: Definition and 1000 Discussions

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...

Since coordinate transformations should be one-to-one and therefore invertible, wouldn’t there be no restriction on pushforwarding or pullbacking whatever fields we feel like (within the context of coordinate transformations)?

I'm stuck on a few Vector homework problems. I don't quite understand how to write vectors A+B and A-B for questions 1b and 2b. I tried starting with calculating the magnitude for vector A+B on question 1b and then followed by finding theta, but I'm not sure if that's what I'm supposed to do...

I'm stumbling on something rather basic here, will explain with an example. (Pardon the LaTeX problems, trying to fix..) Suppose I have a plane, and in the plane I put the familiar (x,y) Cartesian coordinate system, and the metric is the usual Euclidean metric with ds^2 = dx ^2 + dy^2 . Now...

Trying to solve it

I learned in a vector calculus class that the operation of vectors is not defined. The professor mentioned it had to do with topology. How does the operation of vector subtraction relate to topology and how does topological properties prevent vector subtraction from being defined?

1. I consider this problem algebraically, ##c\cdot \vec{u}+(1-c)\cdot \vec{v}=c(1,2)+(1-c)(2,1)=(c,2c)+(2-2c,1-c)=(2-c,1+c)##; since the constraint I know is ##c\geq 0##, I can conclude the expected vectors##(x,y)## must have ##x\leq2, y\geq 1##. 2. Similarly, I get...

Given that the Set of 1-Forms is a Vector Space distinct from, but complimentary to, the Linear Vector Space of Vectors. And given that there is an Isomorphism between the linear space of vectors and the dual vector space of 1-forms, does it make mathematical sense to combine a vector space and...

--##ker(T^2)=ker(T)## if ##T(V)=T^2(V)##-- Suppose that ##T^2(V)=T(V)##. So ##T:T(V)\mapsto T^2(V)=T(V)##. Hence, ##T## is one-to-one and so ##ker(T)=\{0\}##. Suppose that ##T^2(w)=0## for some ##w\in ker(T^2)##. Then ##T^2(w)=T(T(w))=0## which implies that ##T(w)\in ker(T)## and so ##T(w)=0##...

Okay, so the answer is quite easy if you draw a diagram and notice that cosine law solves everything rapidly. But at first, I tried doing some vector algebra and apply properties to see if I could get to something. This is what I could develop. Consider ##|\vec u|##=12, then $$\langle \vec...

Problem Statement: Why are vector mesons more massive than pseudoscalar mesons? Not any sort of set problem, just reading but I can't find an answer or explanation Relevant Equations: * It's going to be something to do with the spin-spin interactions for J=0 and J=1. But then I don't see how...

I hope I'm asking this in the right place! I'm making my way through the tensors chapter of the Riley et al Math Methods book, and am being tripped up on their discussion of geodesics at the very end of the chapter. In deriving the equation for a geodesic, they basically look at the absolute...

1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector $$\nabla{f} = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} \frac{2\ln{(x)}}{x} \\ \frac{2\ln{(y)}}{y}...

So I heard a k-form is an object (function of k vectors) integrated over a k-dimensional region to yield a number. Well what about integrals like pressure (0-form?)over a surface to yield a vector? Or the integral of gradient (1-form) over a volume to yield a vector? In particular I’m...

The statement "at the initial moment of time v ⊥ u and the points are separated by a distance l " gives us a picture like the one which I have added in attachment. As the time passes velocity vector v would gradually change from fully vertical to fully horizontal in order to meet point B. Now...

Somewhat embarrassingly as a third year undergrad, this question has been completely stumping me for far too long now (2 hours). The solution is 1.42 Å and the working is given as |r2| = 2cos(30)*1/3(2.46) or alternatively |r2| = (1/2|a|)/cos(30) But I cannot grasp where this comes from...

I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms ... ... I need help Exercise 32 on page 46 ... ... Exercise 32 reads as follows: I have not been able to make much progress ... We have ...B_r(x) = \{ y \in M \ : \ d(x, y) \lt r \}...

i know its pretty basic but please give some insight for triangle law of vector addition and pythgoras theorem. becuase ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem

I want to render the Earth’s Magnetic field in a software and simulate solar wind electron interaction with it. How do I calculate the magnetic strength and vector orientation at each point around the Earth up to thousands of km? Is there a formula? Or do I need to download a vector field from...

Ax=6.3 cos 23; Ay=-6.3 sin 23; Bx= 5.7 cos 34; By=5.7 sin 34. Is this correct to calculate vector C magnitude which I got 7.7 units. Also is vector C in quadrant IV? I am not sure how to calculate the angle part of this question.

Hello, I am calculating the krauss operators to find the new density matrix after the interaction between environment and the qubit. My question is: Is there an operational order between matrix multiplication and tensor product? Because apparently author is first applying I on |0> and X on |0>...

referring to the image in fig 1 there is a rail carriage subject to an unknown velocity vector Vu (velocity unknown). Vu has a constant velocity Vu in the direction as shown. In the ceiling of the carriage is a light shown in blue and a columnator on the floor. The rail carriage is sitting on...