Vector Definition and 1000 Threads

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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...

The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in...

For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds: $$ df = \sum^n \frac{\partial f}{\partial x_i} dx_i $$ If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at...

What is the notation to show element-wise square root of a vector or matrix?

I calculated force vector by differentiating momentum vector.Since acceleration and velocity vectors are at45°,therefore force and momentum vector are at 45°.But i am not able to find the time at which it will take place.I tried F vector.P vector=FPcos45° but i am not getting from it.I also used...

If within a volume v ,there exists 10 velocity fields at different points then can anyone please suggest how to compute ##\int_v(\nabla•v)## within the volume?? using matlab For exm if the velocity vector field be ##v=x\hat x+y\hat y+z\hat z## and for x=1 to 10,y=1 to 10 and z= 1 to 10 the 10...

To be honest i don't know from where to start. I know how i can test the stokes theorem if i have a cylindrical shape and a cylindrical vector or spherical vector and a spherical shape but here I am out of ideals. The first thing i tried was to compute the left part of the stokes theorem but i...

Hi PF! Given a list of numbers, how do I select the element that has the smallest real part? I don't just want the real part though, I want the entire component. I googled this and tried a few things but nothing works.

https://www.physicsforums.com/attachments/9050 ok I think I got (a) and (b) on just observation but (c) doesn't look like x,y,z will be intergers so ?

Determine if the set of vectors $\begin{bmatrix} x\\y\\5 \end{bmatrix}\in \Bbb{R}^3$ form a vector space ok if I follow the book example I think this is what is done $\begin{bmatrix} x_1\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5...

I am trying to update a vector. I have tried this but it's not working. Any ideas? \[Lambda] = {1, 2} For[\[CapitalLambda] = 0.1, \[CapitalLambda] <= 0.2, \[CapitalLambda] += 0.1, \[Lambda] = Catenate[{\[Lambda], {\[CapitalLambda]}}] // Print]

Hi i am e-pie's brother and he let me use his account.Since $$\vec{\omega}$$ is always perpendicular to the plane.Shouldn't this be $$0^o$$?

The magnitude of each force is shown below: F1 = 10 N F2 = 20 N F3 = 40 N R = \sqrt {Rx^2 + Ry^2} R = \sqrt {-10^2 -15^2} = 18N θ = tan^{-1} \frac{Ry}{Rx} θ = tan^{-1} \frac{Ry}{Rx} = 56 To express the direction of R, we need to calculate the direction angle (i.e. the counterclockwise angle...

Summary: I've posted this in a few forums but still confused on this problem. If the plane is moving at 200km/h and the wind pushes the plane with a velocity of 85km/h, then the resultant velocity would be 217km/h, and using sine inverse, 217.sin(theta=85, I got 23°, which is B, but the answer...

Dear everyone. I'm doing an assignment on vectorfields and for most of the assignment I have to deal with tensors and tensornotation. The first assignment asks me to express the following vector and matrixproducts in tensornotation. $$\overline c = \overline a + \overline b \\ d=(\overline a +...

We have the retarded magnetic vector potential ##\mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'## And its curl ##\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}'...