Questions tagged [vectors]
Geometric object with magnitude (length) and direction.
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What is a Tensor, intuitively?
Before reporting this a duplicate, this post explains the computations of tensors nicely, but I still have questions regarding why they are needed. Also, what makes a tensor actually tensor? The way ...
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Necessary and sufficient conditions for periodic motion
Let us fix a reference frame $S$ with origin in $O$ in the euclidean space $\Bbb R^3$, then let us also define a periodic motion in the following manner:
A motion is periodic if and only if the time-...
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Three perpendicular vectors, one has unknown directional cosines [closed]
I need help figuring out this textbook excersise. I know the relations:
$$\hat{a} \cdot \hat{c} = 0$$
$$\hat{b} \cdot \hat{c} = 0$$
and
$$\alpha{}^2 + \beta{}^2 + \gamma{}^2= 1$$
but am still a bit ...
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What direction should i exactly put for negative displacements?
If I have A....p....B....d....C points
If I am initially on B and walk towards c, it's a positive displacement. Example: BC=10m east and then all of sudden I change my vector and walk to d. Is ...
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Change in kinetic energy when force is perpendicular to velocity
I am a high school student and a have a fundamental doubt.
It is said that when force is applied perpendicular to velocity, there is no change in kinetic energy since there is no change in speed. For ...
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How to remember which way the magnetic field goes?
For the case of a current in a wire, we learn the direction of the associated magnetic field by learning the right hand rule. But what about a bar magnet and the names 'north' and 'south' for the ends ...
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Doubt regarding simplification of Lense-Thirring precession frequency formula
The standard result for Lense-Thirring precession frequency in the weak-field approximation is given by
$$\vec{\Omega}_{LT}=\frac{1}{r^3}[3(\vec{J}\cdot\hat{r})\hat{r}-\vec{J}].$$
I am trying to ...
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In an $n$ particle system, why is the Hamiltonian summed over $n$?
Suppose I am working in a system consisting of $n$ particles. Thus the phase space will be $\mathbb{R}^{6n}$, and both the momentum and position space will be $\mathbb{R}^{3n}$ each.
Then, for some ...
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Velocity of points in a rigid body
I'm trying to derive the following statement:
Let $\mathcal{B}$ be a rigid body. Then there is an unique vector $\vec{\omega}$ such that for every pair of points $P,Q\in \mathcal{B}$ the following ...
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The meaning of negative sign in vectors?
if the velocity vector $\overrightarrow{v}=v\widehat{i}$ ,, we define the accelration vector as it's the time derivative of the velocity vector $\overrightarrow{a}=(\frac{dv }{dt})\...
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Is the space-part of a four-vector temporally connected to the time-part and vice-versa?
This question made me think about four-vectors. All four-vectors, be it the archetypical time/position vector or the charge-density/current one, the energy/3D-momentum or electric/magnetic fields or ...
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What specifically about the torque vector is perpendicular? Is the torque vector like this only so that it works smoothly with linear algebra?
What specifically about the torque vector is perpendicular?
Is the torque vector like this only so that it works smoothly with linear algebra?
The only explanation I get usually is "because it's ...
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Notation for contracting vectors using metric tensors
If we take a vector $A$, which has three components, my understanding is that we can write this using Einstein notation as $A_{u}$ where this is actually $A_1+A_2+A_3$. We can also write $g^{uv}A_v = ...
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Clarifications on proving lightlike vectors must be orthogonal with themselves
I'm trying to prove that lightlike vectors in Minkowski space must be orthogonal to themselves, and I have two questions about this.
I tried two different approaches:
If lightlike, $ds^2=0$
By ...
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Abstract definition of four-vector
It is a long time that I am looking for an abstract definition of four-vectors. This is the definition that I have reach to so far:
A four-vector is an element of the representation space of the ...
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How can I see the Calculation process of the skyrmion number?
I'm having a problem calculating a skyrmion number.
below equations are from Wikipedia - Magnetic skyrmion
I know that I should use the chain rule with the polar, spherical coordinate system.
But I ...
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Why do coordinates have to be inverted to form a dual basis?
I am reading Guidry's Modern General Relativity, and there is a definition for dual basis vectors that are as follows:
If we have three coordinates $x(u,v,w)$, $y(u,v,w)$, and $z(u,v,w)$ where we ...
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How time-like unit four-vector is tangent to the observer's world-line?
I just have started studying special relativity and moving to general relativity
Special relativity only deals with inertial frame (non accelerated frame) but there are no inertial frame in the curved ...
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Do we have to use the metric constructed from the tangent basis to form a line element?
I'm working through Guidry's Modern General Relativity, and there is a problem where it asks to construct the metric using the tangent and dual basis vectors provided. I have done this, and my metrics ...
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Simple difference between module of velocity and time derivative of module of position [duplicate]
What is the conceptually difference between the two:
$$\frac{d|\vec{r}|}{dt}=\frac{\vec{r}\cdot\frac{d\vec{r}}{dt}}{|\vec{r}|}\neq|\dot{\vec{r}}|\equiv \bigg|\frac{d\vec{r}}{dt}\bigg|$$ ...
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Which force should I break into components?
One really common problem in almost all physics textbooks is that of the wedge. So let us say that a block is resting on a wedge that is inclined at angle x from the horizontal. The block has mass mg. ...
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Which of these is the beta factor in special relativity? [closed]
Obviously, $\beta=v/c$. But in this case, I'm not too sure what $v$ represents. I've mostly done 1-D special relativity and therefore it is pretty clear in those cases. However say there are two ...
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Vector form of centrepetal force
We know the centripetal force $F_c$ had magnitude $m\omega^2r$. But let's try to write it in vector form.
First of all,since it is directed along the radius,the unit vector in radial direction in this ...
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Peskin & Schroeder Chapter 16.1 derivation
I am trying to derive the term shown in chapter 16.1 in Peskin & Schroeder. We have defined our polarisation vectors as
$$
\epsilon^+_\mu(k)=\left(\frac{k^0}{\sqrt{2}|\vec{k}|},\frac{\vec{k}}{\...
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Polarization vector basis in Peskin & Schroeder
I am studying chapter 16.1 of Peskin & Schroeder and I am trying to understand how the chosen polarization vector basis works. It is given by the following:
$$
\epsilon_i^T\cdot\epsilon_j^{*T}=-\...
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Why does four-momentum have the same transformation matrix as spacetime coordinates?
I will outline my question in 1+1D for brevity. We can passively transform our coordinate system using a Lorentz boost; $\Lambda^{\bar{\nu}}_{\mu}x^{\mu}=x^{\bar{\nu}}$. I've seen that, by stipulating ...
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Normal to the hypersurface
I have a given metric and I want to choose a spacelike hypersurface and find the normal to that hypersurface. I know that if the hypersurface is spacelike, then the normal is timelike.
The given ...
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Minkowski's equation of motion
I'm trying to prove $f^{\mu}U_{\mu}=0$ for four-force $f^{\mu}=c\frac{dP^{\mu}}{ds}$ and four-velocity $U_{\mu}$. I start by using the chain rule, $f^{\mu}=c\frac{dP^{\mu}}{dt}\frac{dt}{ds}=\gamma\...
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Contravariant Vector Component Transformation from Polar to Cartesian
I am new to tensors and I have just learned that the contravarient components of a vector transforms in the following way (using Einstein summation convention)
$$A^{'i}=\frac {\partial x^{'i}}{\...
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Test for Lorentz invariance of scalar function
This is a simplification of a previous question. I have a function of two 4-vectors and two factors. One factor is on-shell, that is $q_0=\sqrt{M^2+q^2}$. The other factor is off-shell, that is the ...
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Prove dot product between two timelike vectors in Minkowski spacetimw [closed]
In the Minkowski 4-dimensional space-time $(\mathbb{M}^4,\eta)$ the dot product is:
$a\cdot b = -a^0b^0 + a^1b^1 + a^2b^2 + a^3b^3 ~\qquad~ a = (a^0,a^1,a^2,a^3) ~~,~~ b = (b^0,b^1,b^2,b^3)$
Now ...
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How do we assume the direction of $u_{\theta}$ and $u_{r}$ in polar coordinate systems? [closed]
Is there a way to correctly predict the direction of the unit radial vector and the unit transverse vector in problems like the one below or is it just better to take a guess and solve the problem ...
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2D rotation dynamics/control systems as a complex number
I have a dynamic system (it's a rocket in a 2D plane), that I'd like to model the orientation of using complex numbers to remove the need for trig functions in my ode.
I'm having trouble defining the ...
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About Contact, Normal and Friction force
I'm currently studying Friction $f_a$ and Normal force $N$, and I read that those two forces are nothing but the parallel and perpendicular components of a "general" force called Contact ...
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Meaning of "$=$" in $\vec{F}=m\vec{a}$ (for example)
I don't understand how the two could really be one and the same. E.g. we can exert forces $\vec{F}$ and $-\vec{F}$ on a body and it's acceleration will not change. I don't think it makes sense to say ...
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Why does Griffiths Example 5.5, assume the distance from wire to be $1$?
I was under the impression the magnitude of the cross product of cursive $r$ and $dl^{'}$ would be $|\overrightarrow{r}||\overrightarrow{dl^{'}}|sin\theta$. But he simply writes it as $|\...
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Difference between point force and force
In high school, we used to draw a free diagram, and we are asked what are the forces acting on the object. When we represent these forces, we represent them using vectors going out from a point called ...
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Please explain statement in a book on Loop Quantum Gravity
In a book by Carlo Rovelli on Covariant Loop Quantum Gravity, I struggle to understand a statement on Tetrahedron as follows:
What is the dimension of the matrix?
How to derive the given matrix ...
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On the superposition of forces [closed]
Isn't force just a vector quantity? Don't vectors of the same kind add according to the superposition principle? So why don't all forces obey the superposition principle?
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Show that the contraction of a covector and a vector is Lorentz invariant
I just got Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity a couple of weeks ago, and I have resolved to go through the entire book. In the first chapter, he prompts the ...
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About Volterra's displacement equation on dislocation: cancellation of a surface integral on stress
In the theory of dislocations, the displacement induced by a dislocation in an anisotropic solid media can be expressed by Volterra's displacement equation as follows:
$$
u_j(\mathbf{x}) = \int\int\...
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Angular velocity and axis of rotation
If the angular velocity is along the axis of rotation, then why angular velocity has different components in space and body axis. Let's say $\boldsymbol{\omega}$ is the angular velocity, if it is in ...
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Formula for the n-th iteration of rotational operator
Hello i was wondering since we have a formula for $$\operatorname{rot}(\operatorname{rot}(A))=\operatorname{grad}(\operatorname{div}(A))-\Delta(A)$$
Is there any nice generalisation of it ?
Thanks
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When defining a coordinate system, does it matter if it is right- or left-handed?
When you are defining a coordinate system when solving a problem, do the coordinates need to be right-handed to obtain a correct solution? I feel like the answer is no because the directions of ...
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How would I normalize this ket vector? [closed]
So I am given the vector:
$$|Ψa⟩ = |x⟩ + |y⟩ − |z⟩$$
And I need to normalize it. I know that I have to take the dot product of the vector with itself (and it needs to equal 1) but how would I do this ...
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Showing that 4-electric field tensor is spacelike [duplicate]
I have this question in my school assignment.
Given 4-electric field as the following: $$E^\alpha = F_{\alpha\beta}U^\beta$$
where $U^\beta$ is the 4-velocity of the observer and $F_{\alpha\beta}$ is ...
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What do you call $ \frac{d^2 r}{dt^2}$ in polar coordinates? [duplicate]
In polar coordinates, one finds centripetal acceleration as:
$$ a_c = \frac{d^2 r}{dt^2}- \frac{v^2}{r}$$
Where $|r|$ is distance from center to particle, $v$ is tangential velocity.
My question is ...
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How to calculate velocity vector from scalar angular velocity and position vector in 2D?
I would like to know, if I have an angular velocity as scalar, how can I calculate the velocity vector.
I know that the product of angular velocity and the length of the distance gives the speed, but ...
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Does it actually make sense to talk about velocity of the point of contact of a wheel rolling without slipping?
I was reading this answer where I saw the following gif:
We can see that when a point becomes point of contact (i.e: touching the ground), the curve of it's motion has a cusp. To my knowledge, a cusp ...
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Why can't I use vector addition in this way here? [duplicate]
I know that this exact question has been asked here a number of times, but none of the answers sit right with me. The question says that the ends of the strings are pulled with a velocity of "u&...
user331392