Vector Definition and 1000 Threads

Homework Statement My problem is: For the logarithmic spiral R(t) = (e^t cost, e^t sint), show that the angle between R(t) and the tangent vector at R(t) is independent of t. Homework Equations N/A The Attempt at a Solution The tangent vector is just the vector that you get when you take the...

Homework Statement Suppose that ##T_i## is the contravariant component of a vector field ##\mathbf{T}## that is constant along the trajectory ##\gamma.## Show that intrinsic derivative is ##0.## Homework Equations $$\frac{\delta T_i}{\delta t} = \frac{dT^i}{dt}+V^j\Gamma^i_{jk}T^k$$ The...

Dear forum, I am trying to understand what a separable vector space is. I know we can perform the tensor product of two or more vector space and obtain a new vector space. Is that vector space separable because it is the product of other vector spaces? thanks

Hello, I try to understand the following demonstration of an author (to proove that dot product is conserved with parallel transport) : ------------------------------------------------------------------------------------------------------------------------ Demonstration : By definition, the...

Homework Statement Darryl drives his load of tomatoes 14.0 km [E], 6.0 km [N], 12.0 km [N 15° E], and then 2.0 km [N 65° E]. This takes him 42 minutes. Calculate Darryl’s distance and displacement. Then draw a diagram to show your work. [/B]Homework Equations Not really sure.[/B]The Attempt...

Homework Statement A golf ball is struck at ground level. The speed of the golf ball as a function of the time is shown in the figure, where t = 0 at the instant the ball is struck. The scaling on the vertical axis is set by va= 16 m/s and vb= 29.3 m/s. (a) How far does the golf ball travel...

I hear the phrase vector mass being used a lot in discussions of various models in particle physics. But I am not exactly sure what it means. There is no Wikipedia article on this term. Is vector mass somehow different from scalar mass? The only possible mass terms in the Lagrangian arise due...

Homework Statement [/B] A plane leaves the airport in Galisteo and flies 180 km at 67.0 degrees east of north; then it changes direction to fly 255 km at 49.0 degrees south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, how...

Hey! :o Let $V$ be the real vector space $\mathbb{R}[X]$ and $M \subset \mathbb{R}$ a set with $d$ elements. Let $$U_1 := \{ f \in \mathbb{R}[X] | \forall m \in M : f(m) = 0\}, \ \ U_2 := \{ f \in \mathbb{R}[X] \mid \deg(f) \leq d − 1\}$$ be two vector spaces of $V$. Let $\Phi: V\rightarrow...

Im trying to solve the following problem from the book 'Learning with kernels', and would really appreciate a little help. Background information - Let $\{(x_{1},y_{1}),...,(x_{N},y_{N})\}$ be a dataset, L a Loss function and $H(k)$ a reproducing kernel Hilbert space with kernel $k$. The...

Homework Statement Suppose that a particle follows the path r(t) = 2cos(t)i + 2sin(t)j Give an equation (in the form of a formula involving x and y set equal to 0 ) whose whose solutions consist of the path of the particle. Homework Equations None that come to mind The Attempt at a Solution I...

I'm going through a basic introduction to tensors, specifically https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf and I'm confused by the author when he defines vectors as directional derivatives at the bottom of page 3. He defines a simple example in which ƒ(x^j) = x^1 and then...

I have some problems in calculating the flux of Poynting's vector through a surface. I have that: S=c/4π⋅H^2⋅n (S: Poynting's vector, n: versor of the direction of the propagation of the em wave) H=1/cR A' but: A=(1/cR) d'; A'=(1/cR) d'' ==> H= (1/cR)^2 d''...

So, the Wiki page on the Poynting vector has this image: I remember hearing/reading somewhere that the energy transmission in a circuit like this is actually not traveling through the wire, but that it actually happens through the electromagnetic field, I.e. essentially the Poynting vectors...

Homework Statement As part of a longer problem: "Find necessary and sufficient conditions for the point with positionvector r to lie inside, or on, the tetrahedron formed by the vertices 0, a, b and c." Homework Equations I am not sure... vector addtion? The Attempt at a Solution I don't...

Homework Statement Homework Equations The Attempt at a Solution So I began by subtracting. (205-160)=55 i (495+128)=623 j Both of these vectors are in the positive direction. So if I divide the vector by its magnitude I should get an answer of 1 in the positive direction for both i and...

Why is ##\theta## a scalar whereas ##d\theta## a vector ?

Surface S and 3D space E both satisfy divergence theorem conditions. Function f is scalar with continuous partials. I must prove Double integral of f DS in normal direction = triple integral gradient f times dV Surface S is not defined by a picture nor with an equation. Help me. I don't...

Homework Statement Recall that F is the vector space of functions from ℝ to ℝ, with the usual operations of addition and scalar multiplication of functions. For each of the following subsets of F, write down two functions that belong to the subset, and determine whether or not the subset is a...

Homework Statement [/B] 1. Suppose {v1, . . . , vk} is a linearly independent set of vectors in Rn and suppose A is an m × n matrix such that Nul A = {0}. (a) Prove that {Av1, . . . , Avk} is linearly independent. (b) Suppose that {v1, . . . , vk} is actually a basis for Rn. Under what...

As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative. The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...

Homework Statement Show that (ap1,bp2) is not a vector unless a = b.(The 1 and 2 are superscripts) In Einstien's Gravity in a Nutshell, p 43, Zee states the above is not a vector because it doesn't transform like a vector under rotation. When I use the usual rotation matrix for rotation about...

In string theory, if we have NN BCs along ##X^i, i = 1, \ldots, n-1## and DD BCs along ##X^a, a = n, \ldots, 25## then you get, from ##\alpha^{i,a}_{-1}|0,p\rangle ##, ##n## massless vectors and ##24-n## massless scalars. I understand that for the first excited level, ##M^2=0## and so we have...

Hello Forum and happy new year, Aside from a rigorous definitions, a linear vector space contains an infinity of elements called vectors that must obey certain rules. Based on the dimension ##N## (finite or infinite) of the vector space, we can always find a set of ##n=N## linearly independent...

see post 3

$\tiny{s6.194.4.12.4.29}$ $\textsf{a. Find a nonzero vector orthogonal to plane through the points: }$ $\textsf{b. Find the area of the triangle PQR}$ \begin{align} \displaystyle &P(1,0,0)& &Q(0,2,0)& &R(0,0,3)\\ %&=\color{red}{\frac{1209}{28} } \end{align} $\textit{do what first?}$

Hello Forum, The state of a quantum system is indicated by##\Psi## in Dirac notation. Every observable (position, momentum, energy, angular momentum, spin, etc.) corresponds to a linear operator that acts on ##\Psi##.Every operator has its own set of eigenstates which form an orthonormal basis...

Finding length of vector with unknown variable. Purely for study purposes. Find the smallest possible length of the vector →v . Let vector V = (-2/3,b,16/7) Equation for finding length of vector : Sqrt(a^2+b^2+c^2)Question would be quite straight forward had there been no unknown variable but...

I am a little bit confused on Center of Mass and Center of Gravity about what are they vectors or scalars . As I may think they are neither because they are simply two points . Am I saying right or wrong .

I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j. What would be the effect on a vector, like 3i + 4j...

Given the Fourier conjugates ##\vec{r}## and ##\vec{k}## where ##\vec{r} = [r_1,r_2,r_3]## and ##\vec{k} = [k_1,k_2,k_3]## , are ##r_1## /##k_1##, ##r_2##/##k_2##, ##r_3##/##k_3## also Fourier conjugates, such that: ##\begin{equation} \begin{split} f(\vec{r})&=[f_1(r_1),f_2(r_2),f_3(r_3)] \\...

A car is driven eastward for a distance of 50 miles, then northward for 30 miles, and then in a direction 30 degree east of north for 25 miles. Draw the vector diagram and determine the total displacement of the car from its starting point. The relevant equations I used here are: i) rx= ax +...

Homework Statement Find the value of ez if e = 1/3i + 2/3j + ezk Homework Equations Any help? The Attempt at a Solution Have no idea where to start

Hi everyone please help how can i find the force in Q(a)(ii)?

How do you solve this question I just need a solution

Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...

Homework Statement Given the velocity vector in the polar coordinates, ##\vec v=-awsin{wt}\vec e_r + awcos{wt}\vec e_\theta## determine the average velocity vector and velocity intensity over a time period ##[0, \pi/2w]## Homework Equations 3. The Attempt at a Solution [/B] For the first part...

1. Homework Statement I am looking at problem 2.2 pictured above. I have solved all portions of the question except the last part, which asks for the angle between the normal vector to the surface and the z-axis. I am aware that the normal vector is simply equal to the gradient of the surface...

Homework Statement The direction of vectors A and B are given below for several cases. For each case, state the direction of A X B. a) A points east, B points south. b) A points east, B points straight down. c) A points straight up, B points north. d) A points straight up, B points straight...

I recognize the rate of change of a vector in an inertial frame S can be related to the rate of change of the vector in a rotating frame S0 by the equation below taken from my textbook, where Ω is the angular velocity vector. $$\Big(\frac{dQ}{dt}\Big)_{S_{0}}= \Big(\frac{dQ}{dt}\Big)_{S} +...

I assume this is a simple summation of the normal components of the vector fields at the given points multiplied by dA which in this case would be 1/4. This is not being accepted as the correct answer. Not sure where I am going wrong. My textbook doesn't discuss estimating surface integrals...

Hey! :o I want to check if the following are true. $V_1=\{a\in \mathbb{R}\mid a>0\}$ with the common multiplication as the vector addition and the scalar multiplication $\lambda \odot v=v^{\lambda}$ is a $\mathbb{R}$-vector space. $V_2=\{(x,y)\in \mathbb{Q}^2 \mid x^2=-y^2\}$ with the...

In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...

Homework Statement Bead on spoke: constant speed ##u## along spoke it starts at center at ##t=0## angular position is given by ##\theta=\omega t##, where ##\omega## is a constant Homework Equations ## \frac{d\hat r}{dt} = \dot \theta \hat \theta ## (1) ## \frac{d\hat \theta}{dt} = -\dot...

Sorry may i ask a question here~ i don't understand how did GRIFFITHS prove the statement from(C) to (B) What is his logic in this case? and What is Ai Aii Bi Bii in the integral? thank you

Homework Statement If an object had been projected horizontally with the same magnitude as in the depicted situation, how would the motion compare with that of the object in the diagram? (I have drawn the diagram in my attachment and have done questions c and d but I don't understand question...

When I induce magnetic flux through a closed loop, I should expect the lines of flux produced by current in that loop to oppose the change of flux through that loop. But what happens when that loop, say a rectangular loop, is curved into the shape of the letter J (like a candy cane) and my flux...

http://www.physicspages.com/2014/08/22/electromagnetic-waves-in-vacuum/ Sorry i have no idea how to get the last step in the vector form how to convert and mix it?thank you

Homework Statement Prove that the relationship between the dimensionless amplitude a of a wave and the intensity I of the wave is given by: Homework Equations B = curl (A) B: Magnetic Field A = [mass of electron * c / e ]*a*e^[i*(kx-wt)] where k is the wavenumber and is given by: 2pi /...

Homework Statement Loop of current ##I## sitting in the xy plane. Current goes in counter clockwise direction as seen from positive z axis. Find: a) the magnetic dipole moment b) the approximate magnetic field at points far from the origin c) show that, for points on the z axis, your answer is...