Vector Definition and 1000 Threads

I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation...

Well, I understand that according to the conservation of momentum the total momentum of a system is conserved for objects in an isolated system, that is the sum of total momenta before the collison is equal to the sum of momenta after the collision. In this case, the momentum of the object...

hi i was recently introduced to the Dirac notation and i guess i am following it really well , but can't get my head around the idea that the bra vector said to live in the dual space of the ket vectors , i know about linear transformation and the structure of the vector spaces , and i realize...

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...

Here is how my teacher solved this: I understand what the nabla operator does, ##∇\cdot\vec v## means that I am supposed to calculate ##\sum_{n=1}^3\frac {d\vec v} {dx_n}## where ##x_n## are cylindrical coordinates and ##\vec e_3 = \vec e_z##. I understand why ##∇\cdot\vec v = 0##, I would get...

##(\nabla\times\vec B) \times \vec B=\nabla \cdot (\vec B\vec B -\frac 1 2B^2\mathcal I)-(\nabla \cdot \vec B)\vec B## ##\mathcal I## is the unit tensor

I'm reading 'Core Principles of Special and General Relativity' by Luscombe - the part on parallel transport. I guess ##U^{\beta}## and ##v## are vector fields instead of vectors as claimed in the quote. Till here I can understand, but then it's written: I want to clarify my understanding of...

I've taken multivariable/vector calc and can do most of the basic operations and have an OK understanding of the fundamental concepts, but certainly can't "see it" like I can calc I and II. In those subjects, I often feel competent to take on any problem I come across because the concepts are...

For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...

I have not tried to make any calculation. It's nonsense, because I don't understand the statement. The first vector points to the west. Given a two dimensional coordinate system, the first vector is pointing to the left. I imagine geographical coordinates, north (+y), south (-y), west (-x), and...

I'm reading 'Core Principles of Special and General Relativity' by Luscombe, specifically the introductory section on problems with defining usual notion of differentiation for tensor fields. I'll quote the relevant part: Since the equation above is a notational mess, here's my attempt to...

Seems to me the answer is a specific vector: The second forms a plane, while the first X is just a vector. The intersection between the λX that generates the (properties of all vectors that lie in the...) plane (i am not saying X is the director vector!) How to write this in vector language?

I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about: For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?: d1 = Diagonal one = (a,b,c) d2 =...

I managed to expand a general expression from the alternatives that would leave me to the answer, that is: I will receive the alternatives like above, so i find the equation: C = -sina, P = cosa So reducing B: R: Reducing D: R: Is this right?

is it true that vectors are symbolised as an italic boldface 'a'

Hi PF! I am trying to multiply each component of B by the matrix A and then solve A\C. See the code below. A = rand(4); B = rand(5,1); C = rand(4,1); for i = 1:5 sol(:,i) = (B(i)*A)\C end But there has to be a way to do this without a for-loop, right? I'd really appreciate any help you have!

For my understanding, to move to the coolest place, it has to move in direction of -∇f(x,y) How can I find the value of 'k' to evaluate the directional derivative and what can I do with the vertices given.

I think that we can say that PPR = α*PRPS where PR and PS are the points where occurs the intersection on the line R and S. Obs: line r and s are found by knowing that the straight line intersection of two planes are n1 X n2 [cross product] Lr = (0,1,-2) + y(-1,1,1) Ls = (0,1,-1) + u(1,2,1)...

Answers are the following : (i) v=(2cost)i - (2sint)j -(1/2)k (ii)2.06m/s (iii)2m/s^2 horizontally towards the vertical axis, making an angle of pi/4 with both the I and j axes.

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...