What is Vector: Definition and 1000 Discussions

Given the equation ##\frac{xy} 3##. It is a fact that the gradient vector function is always perpendicular to the contour graph of the origional function. However it is not so evident in the plot above. Any thought will be appreciated.

We have a scalar potential $$\Phi(\vec{r})=\frac{q}{4\pi\epsilon_0} \left( \frac{1}{r} - \frac{a^2\gamma e^{-\gamma t}\cos\theta}{r^3}\right)$$ and a vector potential $$\vec{A}(\vec{r})=\frac{a^2qe^{-\gamma t}}{4\pi\epsilon_0r^4}\left(3\cos\theta\hat{r} + \sin\theta\hat{\theta} \right) .$$ how...

Solution: u = [-2,3,1] Po = (6,0,0) & P = (4,2,3) PoP = v = [-2,2,3] Therefore, the answer is [6,0,0] + r[-2,3,1] + q[-2,2,3]; r, q are real numbers I don't understand why (6,0,0) is used as the point in the vector equation, since it only lies on the [-2,2,3] vector, not the u = [-2,3,1]...

Hi PF, I've one question about vector spaces. There is only one way to define the operations of a vector space? For example if V is a vector space there is other way to define their operations like scalar multiplication or the sums of their elements and that the result is also a vector space?

Hello! So I need to find the potential function of this Vector field $$ \begin{matrix} 2xy -yz\\ x^2-xz\\ 2z-xy \end{matrix} $$ Now first I tried to check if rotation is not ,since that is mandatory for the potentialfunction to exist.For that I used the jacobi matrix,and it was not...

Hello! I am suspossed to write (sketch) this particular vector field. $$V2(r) = \frac{C}{\sqrt{x^2+y^2+z^2})^3} * (x,y,z) $$ Note that the x y z is suspossed to be a vector so they would be written vertically (one over the other) but I don't know how to write vectors and matrices in LaTeX,so...

Hi If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b. Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...

Here is my attempt at the vector diagram: Could anyone give me any clues as to where to go from here? Is this diagram correct? I tried finding θ using inverse tan 50/15 but I don't think I can do that because that's mixing up velocity and displacement. EDIT: I copied and pasted the incorrect...

I read this in the wiki article about Wick rotation: Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no...

The term for the electromagnetic interaction of a Fermion is ##g \bar{\Psi} \gamma_\mu \Psi A^\mu##, where ##g## is a dimensionless coupling constant, ##\Psi## is the wave function of the Fermion, ##\gamma## are the gamma matrices and ##A## is the electromagnetic field. One can quite simply see...

I have to perform a calculation on my data. Here is an example of data from just one time step (data from other time steps would appear as additional rows). X Y Z Total 2 2 1 3 Total = SQRT(X2 + Y2 + Z2). The calculation I have to do is: (N • N), where "N" is an average. I tried...

Hi guys, I am losing my mind over this passage... I cannot understand how to get from the first expression with the cross products to the second ##\dot{\textbf{r}}(\textbf{r}\cdot \textbf{r})-\textbf{r}(\textbf{r}\cdot\dot{\textbf{r}})##

Hi, I am looking for a short document discussing the usage of the wave vector. Any recommendations? Thank you!

Hi I was always under the impression that i could write a2 = a.a = a2 Equation 1 where a⋅ is a vector and a is its modulus but when it comes to the kinetic energy term for a particle in plane polar coordinates I'm confused ( i apologise here as i don't know how to write time derivative with...

i need clarity on part (f) only...we have two values for ##t## i.e ## t=2.79## and ##t=2.15##, ...the mark scheme says solution is: why ##2.15## only, i have tried substituting the two values back into the problem and they both satisfy part ##e##

Hi guys, I'm having trouble computing a pass 1 to 106.15. It's in the pictures. So, what a have to do is the derivative of ##f## with respect to time and coordinates. Then I need to rearrange the terms to find the equation 106.15. I am using the following conditions. ##r## vector varies in...

Let ##P## be an uncountable locally finite poset, let ##F## be a field, and let ##Int(P)=\{[a,b]:a,b\in P, a\leq b\}##. Then the incidence algebra $I(P)$ is the set of all functions ##f:P\rightarrow F##, and it's a topological vector space over ##F## (a topological algebra in fact) with an...

I identified $$(\Phi_{SN})_{*})$$ as $$J_{(\Phi_{SN})}$$ where J is the Jacobian matrix in order to $$(\Phi_{SN})$$, also noticing that $$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$, I wrote...

Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right. I also tried to compute the right-hand side of...

I am trying to draw the Poynting vector field for a single electron in free space between two capacitor plates. The electron is moving (and accelerating) to the positive plate at the right. I expected the Poynting vector field lines to converge to the electron, because that is where the work...

Hello, This question is with regards to the discussion around page 56 (1971 Edition) in Anthony French's Newtonian Mechanics. He is discussing the choice of a coordinate system where the axes are not necessarily perpendicular to each other. Here is the summary of what I read (as applied to...

By considering a vector triangle at any point on its circular path, at angle theta from the x -axis, We can obtain that: (rw)^2 + (kV)^2 - 2(rw)(kV)cos(90 + theta) = V^2 This can be rearranged to get: (r thetadot)^2 + (kV)^2 + 2 (r* thetadot)(kV)sin theta = V^2. I know that I must somehow...

Majoring in electrical engineering imply studying Griffiths book on electrodynamics, so I have begun reading its first chapter, which is a review of vector calculus. A list of vector calculus identities is given, and I would like to derive each one, with one of them being ##\nabla \cdot (A...

Not HW, but seems to fit here. I compute $$n.S = \frac{(-1+\cos(c s))}{c^2} \sin(c s) \neq 0$$ I use the following in Mathematica: r[s_, \[Alpha]_] := Sin[Cos[\[Alpha]] s]/Cos[\[Alpha]] z[s_, \[Alpha]_] := (1 - Cos[Cos[\[Alpha]] s])/Cos[\[Alpha]] x[s_, \[CurlyPhi]_, \[Alpha]_] := r[s...

Hi, In Problem 9.12 of Griffiths Introduction to Electrodynamics, 4th edition (Problem 9.11 3rd edition), in the problem, he says that one can calculate the average energy density and Poynting vector as using the formula I don't really understand how to do...

Given ##f(\vec{x})##, where the Fourier transform ##\mathcal{F}(f(\vec{x}))= \hat{f}(\vec{k})##. Given ##\vec{x}=[x_1,x_2,x_3]## and ##\vec{k}=[k_1,k_2,k_3]##, is the following true? \begin{equation} \begin{split} \mathcal{F}(f(x_1))&= \hat{f}(k_1) \\ \mathcal{F}(f(x_2))&= \hat{f}(k_2) \\...

I am trying to derive the tangential acceleration of a particle. We have tangential velocity, radius and angular velocity. $$v_{tangential}= \omega r$$ then by multiplication rule, $$\dot v_{tangential} = a_{tangential} = \dot \omega r + \omega \dot r$$ and $$a_{tangential} = \ddot \theta r +...

Considering an stopped object in a horizontal plane, the frictional force between the object and the plane would be the product of the friction coefficient (static or kinetic if there was movement between the surfaces) by normal. Since the normal in this case would be given by N (vector) = - mg...

So say I have a bubble embedded in a spacetime with metric: $$ds^2 = -dt^2 + a(t) ( dr^2 + r^2 d\Omega^2_2) $$ how do I compute the normal vector if I assume the wall of the bubble the metric represents follows a time-like trajectory, for any ##a(t)##? Since we are interested in dynamical...

1- Write down the complete MAXWELL equations in differential form and the material equations. 2- An infinitely extensive area is homogeneously filled with a material with a location-dependent permittivity. There are charges in the area. Give the Maxwell equations and material equations of...

I didnt understand the question. The magnitude is 2(pie)/wavelength. I get 78500 rad/cm which is pretty wrong as guess. Where would angle come into picture? Ref: https://www.millersville.edu/physics/experiments/062/index.php Shouldn't direction be like i + j + k ? So will it be like: |k|cos30...

I am having trouble with finding the x and y components of V3 . According to various different websites the correct way to find the components of V3 is Vx=10*cos(100) and Vy=10*sin(100). I can see where the 100 comes from, the previous vector was already traveling 30 degrees and now V3 swung...

Prove, by giving counterexamples, that vector subtraction is not commutative and not associative. ok I read all I could on trying to understand this but seem to not see something simple if we have the example of $u=\begin{bmatrix}2\\-3\\4\\2\end{bmatrix}...

I got the attached photo from someone who solves physics problems on youtube. As you can see their final answer is 6.7i+16j. I understand how she got these values but I came out with something slightly different. I solved for the x and y components on the opposite side of each vector. So...

A = |A|cos33.7 A = |A|0.83195

1.)##\dot{\vec{r}}=\dot{x}\hat{i}+\dot{y}\hat{j}+\dot{z}\hat{k}=\dot{r}\hat{r}## since the unit vector is constant 2.) ##\dot{r}\hat{r}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{\sqrt{x^2+y^2+z^2}}\frac{\dot{x}x+\dot{y}y+\dot{z}z}{\sqrt{x^2+y^2+z^2}}##...

How do I write the following expression $$\epsilon_{mnk} J_{1n} \partial_i\left[\frac{x_m J_{2i}}{|\vec{x}-\vec{x}'|}\right]$$ back into vectorial form? Einstein summation convention was used here. Context: The above expression was derived from the derivation of torque on a general current...

I am looking at the following document. In section 2.3 they have the formula for the pushforward: f*(X) := Tf o X o f-1 I am having trouble trying to reconcile this with the more familiar equation: f*(X)(g ) = X(g o f) Any help would be appreciated.

Hello (Everyone here has been so helpful -- thank you. Things I thought I knew, I now doubt; and this is so helpful to have this group.) There is an current discussion on Yaw. I am enjoying that. And that raised an issue for me. However, I do NOT want to hijack that thread, so I am posting...

I am considering using a pair of point charges: positive and negative electric charge to model a magnetic dipole's magnetic field by just average the electric field vectors between the two charged particles where they overlap. Will that work? In this case the + field will be vectors pointing...

why the general wave vector q (in the proof of Bloch theorem in Ashcroft Mermin) is represented by k-K, where k is in the 1st BZ ? why not q=k+K ( usual vector form) what is special about k-K?

I have been studying overloading [ ]. I understand how it works from the book. I experiment using vector of structure and obviously I can't make it work. I wonder if it is possible to overload [] on vector of structure. Here is what I have and it's not even close to working, what can I do to...

I am looking at antenna theory and just came upon scalar fields. I found an site giving an example of a scalar field as measuring the temperature in a pan on a stove with a small layer of water. The temperature away from the heat source will be cooler than near it but it doesn't have a...

Summary:: I am suppose to show that this columns matrix does not transform as a vector. In another words, it is not in fact a vector. I think this become trivial if we get the rotation matrix composed of Euler angles. But, i think that it is not the best way to solve this problem, and i...

Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...

I don't know what is the answer, so i am not sure when to stop the computation or not. The far i reached was ## <k|j> \sum_{j} c_{j}|k>##. That is, the action of the projector operator is, obviously, project the state in |k>. Now, the coefficients was changed. So now what i have to do?

I'm having difficulties solving this. For finding a unit vector that is orthogonal to two unit vectors I understand we use the cross product and such. However, I am confused about how to approach this problem as it has a third vector. We can let x = (x1, x2, x3, x4) be a vector orthogonal to u...

$$\nabla \times B(r)=\frac{\mu _0}{4\pi} \int \nabla \times J(r') \times \frac{ (r-r')}{|r-r|^3}dV'$$ using the vector identity: $$\nabla \times (A \times B) = (B \cdot \nabla)A - B(\nabla \cdot A) - (A \cdot \nabla )B + A(\nabla \cdot B)$$ ##A=J## and ##B=\frac{r-r'}{|r-r'|^3}## since...

Hi, In $\mathbb{R^3} || v-w ||^2=||v||^2 + ||w||^2 - 2||v||\cdot ||w||\cos{\theta}$ But can we say $||v+w||^2=||v||^2 +||w||^2 + 2||v|| \cdot||w|| \cos{\theta}$ where v and w are any two vectors in $\mathbb{R}^3$