What is Vector: Definition and 1000 Discussions

hi guys i saw this problem in my collage textbook on vector calculus , i don't know if the statement is wrong because it don't make sense to me so if anyone can help on getting a hint where to start i will appreciate it , basically it says : $$ \phi =\phi(\lambda x,\lambda y,\lambda...

[Ref. 'Core Concepts in Special and General Relativity' by Luscombe] Let ##M,M'## be manifolds and ##\psi:M\to M'## a diffeomorphism. Even if ##\psi## weren't a diffeomorphism, and instead just a smooth map, the coordinates of the pushback of ##\mathbf{t}\in T_p(M)##, would be related to the...

The components of a vector ##v## are related in two coordinate systems via ##v'^\mu = \frac{\partial x'^\mu}{\partial x^\sigma}v^\sigma##. When evaluating this at a specific ##x'(x_0) \equiv x'_0##, how should we proceed? ##v'^\mu(x'_0) = \frac{\partial x'^\mu}{\partial...

My solution is making an analogy of the ##\text{Relevant equations}## as shown above, starting from the equation ##\vec \omega = \frac{1}{2} \vec \nabla \times \vec v##. We have ##\vec B = \vec \nabla \times \vec A = \frac{1}{2} \vec \nabla \times 2\vec A \Rightarrow 2\vec A = \vec B \times...

Hey! 😊 Let $\mathbb{K}$ be a field, $1\leq n\in \mathbb{N}$ and let $V$ be a $\mathbb{K}$-vector space with $\dim_{\mathbb{R}}V=n$. Let $\phi :V\rightarrow V$ be a linear map. The following two statements are equivalent: - There is a basis $B$ of $V$ such that $M_B(\phi)$ is an upper...

Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...

Hi I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**. What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical...

That is, the triad forms a basis.

Given $\vec{r}=t^m* \vec{A} +t^n*\vec{B}$ where $\vec{A}$ and $\vec{B}$ are constant vectors, How to show that if $\vec{r}$ and $\frac{d^2\vec{r}}{dt^2}$ are parallel vectors , then m+n=1, unless m=n? I don't have any idea to answer this question. If any member knows the answer to this...

Say... A ball is moving to the right, and we want to say that it doesn't slip. My doubt is, in which case we put Vrot = - Vcm = - α*r or Vrot = Vcm = α * r

I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'. To give some background, I'm aware that basis vectors in tangent space are given by...

I am confused about the problem. I thought operators do not act on bra vectors, and the problem is equivalent to ##a^{\dagger} \left | \alpha \right > = \left ( \alpha ^{*} + \frac {\partial} {\partial \alpha} \right ) \left | \alpha \right > ##. Then, strangely, ##\left < \alpha \right |##...

Good Morning Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you. In it, someone wrote: " It is a theorem in mathematics that the set of all functions that are solutions of a...

A unit vector, ##\frac{\vec{v}}{|\vec{v}|}##, has dimensions of ##\frac{L}{L} = 1##, i.e. it is dimensionless. It has magnitude of 1, no units. For a physical coordinate system, the coordinate functions ##x^i## have some units of length, e.g. ##\vec{x} = (3\text{cm})\hat{x}_1 +...

Hi, Let f(t) be a differentiable curve such that $f(t)\not= 0$ for all t. How to show that $\frac{d}{dt}\left(\frac{f(t)}{||f(t)||}\right)=\frac{f(t)\times(f'(t)\times f(t))}{||f(t)||^3}\tag{1}$ My attempt...

Let r(t) be the position vector for a particle moving in $\mathbb{R^3}.$ How to show that $$\frac{d}{dt}(r \times (v\times r))=||r||^2 *a+ (r\cdot v)*v-(||v||^2+ r\cdot a)*r \tag{1}$$ Where r(t) is a position vector (x(t),y(t),z(t)), $v(t)=\frac{dr}{dt}=(x'(t),y'(t),z'(t))...

Hello, I have a question: Why is the y-component of the force at turning flight equal to the weight force? Here, Fs is equal to Fg. But why? I tried to explain it myself but I didn't get it

I'm having trouble finding the angle and displacement

Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector $$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t...

So I understand that time is now part of the four vector, and so dividing delta X by delta t (time according to me), would produce just c as the first dimension of the vector, which gives us no intuition as to how fast time is moving for the observer, so is not useful. I understand why we...

We have a basis {##\mathbf{e}_1##, ##\mathbf{e}_2##, ##\dots##} and the corresponding dual basis {##\mathbf{e}^1##, ##\mathbf{e}^2##, ##\dots##}. I learned that a vector ##\vec{V}## can be expressed in either basis, and the components in each basis are called the contravariant and covariant...

In the section 8-2 dealing with resolving the state vectors, we learn that |\phi \rangle =\sum_i C_i | i \rangle and the dual vector is defined as \langle \chi | =\sum_j D^*_j \langle j |Then, the an inner product is defined as \langle \chi | \phi \rangle =\sum_{ij} D^*_j C_i \langle j | i...

Hi, I would like to ask for a clarification about the difference between parallel transport vs Lie dragging in the following scenario. Take a vector field ##V## defined on spacetime manifold and a curve ##C## on it. The manifold is endowed with the metric connection (I'm aware of it does exist...

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I guess I will show my work for substantiating equation 1 and hopefully by doing so someone will be able to point out where I could generalize. ##\langle \vec{S}_{rad} \rangle = \frac{1}{2 \mu} \mathfrak{R} \left( \vec{E}_{rad} \times \vec{B}^*_{rad}\right) = \frac{1}{2 \mu} \mathfrak{R} \left(...

Why is electric current not a vector while electric current density is a vector? What's the intrinsic difference between the two through that surface integral?

Hello, A generic vector field ##\bf {F} (r)## is fully specified over a finite region of space once we know both its divergence and the curl: $$\nabla \times \bf{F}= A$$ $$\nabla \cdot \bf{F}= B$$ where ##B## is a scalar field and ##\bf{A}## is a divergence free vector field. The divergence...

Hello everyone, I'm stuck doing this problem, I've tackled the partial derivative but i can't figure out the derive for x component part, i solved the partial derivative part, i came to this result: What do can i do from here on, thank you!

In a spacetime diagram the spatialized time direction is the vertical y-axis and the pure space direction is the horizontal x-axis, ct and x, respectively. The faster you go and therefore the more kinetic energy you have, you'll have a greater component of your spacetime vector in the...

In this image of Introduction to Electrodynamics by Griffiths . we have calculated the vector potential as ##\mathbf A = \frac{\mu_0 ~n~I}{2}s \hat{\phi}##. I tried taking its curl but didn't get ##\mathbf B = \mu_0~n~I \hat{z}##. In this thread, I have calculated it like this ...

I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...

I am trying to solve the following problem from my textbook: Formulate the vector field $$ \mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}} $$ in spherical coordinates.My solution is the following: For the unit vectors I use the...

If this question is in the wrong forum please let me know where to go. For p, the vector space of polynomials to the form ax'2+bx+c. p(x), q(x)=p(-1) 1(-1)+p(0), q(0)+p(1) q(1), Assume that this is an inner product. Let W be the subspace spanned by . a) Describe the elements of b) Give a basis...

Let me define the letters before because they will be confusing: ##x##: 3-vector ##v##: 3-velocity ##a##: 3-acceleration ##X##: 4-vector ##U##: 4-velocity ##A##: 4-acceleration ##\alpha##: proper acceleration ##u##: proper velocity One can define the proper time as, $$d\tau = \sqrt{1 -...

Homework statement: Find the electric field a distance z from the center of a spherical shell of radius R that carries a uniform charge density σ. Relevant Equations: Gauss' Law $$\vec{E}=k\int\frac{\sigma}{r^2}\hat{r}da$$ My Attempt: By using the spherical symmetry, it is fairly obvious...

I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me. The vector field of A is written as follows, , and the divergence of a vector field A in spherical coordinates are written as...

Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...

Hi there, I have attached the problem I'm working with. I believe I must have the wrong idea of how to approach this question. My issue is with the stated width and calculating how long the boat will take to cross the river. It's using width; 110m and the boats velocity to determine how long...

Below is the attempted solution of a tutor. However, I do question his solution method. Therefore, I would sincerely appreciate it if anyone could tell me what is going on with the below solution. First off, the rotation of the matrix could be expressed as below: $$G = \begin{pmatrix} AB & -||A...

This exercise is located in the vector space chapter of my book that's why I am posting it here. Recently started with this kind of exercise, proof like exercises and I am a little bit lost Proof that given a, b, c real numbers, the set X = {(x, y) E R^2; ax + by <= c} ´is convex at R^2 the...

Do I have to write something like, $$\nabla' \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x'^m} + \frac{\partial J^m(t_r)}{\partial x'^m}$$ $$\nabla \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x^m} + \frac{\partial J^m(t_r)}{\partial x^m} = \frac{\partial J^m(t_r)}{\partial x^m}$$...

What I've done so far: From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1). We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z. We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...

[Moderator's Note: Spun off from previous thread due to increase in discussion level to "A" and going well beyond the original thread's topic.] A vector space has no origin to begin with ;-)). An affine space is a set of points and a vector space ##(M,V)##. Then you have a set of axioms which...

Hi, It is my first message :) I hope you are all fine and safe in these difficult days ! I cannot find the good orientation of the vector of friction. A circle moves in translation to the right and in the same time the wall rotates around A0. A0 is fixed to the ground. There is always the...

If propositions ##p,q\in{\mathscr L}_{\mathcal H}## (i.e., the lattice of subspaces of ##\mathcal H##) are incompatible, then ##\hat p\hat q\neq\hat q\hat p##. But since it's a lattice, there exists a unique glb ##p\wedge q=q\wedge p##. How are they mathematically related? In particular, I...

The proof that the set is a subspace is easy. What I don't get about this exercise is the dimension of the subspace. Why is the dimension of the subspace ##n-1##? I really don't have a clue on how to go through this.

Hi I found this paper on the measurement of unknown velocity vector of a closed space. Does it mean that it is possible to measure the unknown velocity vector of a closed space ? Can someone explain it to me

The first thing I did, was to find the equations for player A (p) and ball's (b) path (for each i and j component I used the equation I wrote in the relevant equations) and then I found the derivative of both equations so I could have the velocity: $$\vec{r}_p(t)=(6t^2+3t)\hat{i}+20\hat{j}...