All Questions
Tagged with differentiation vectors
120 questions
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The time derivative of a vector not defined in terms of the time variable $t$
Recently I got a question where I needed to determine the time derivative of a position vector. However, the vector didn’t have the variable $t$ but instead had $x$, $y$, and $z$ as its terms, so I ...
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3
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785
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Derivative of cross product equation
Recently I got a problem that equated the time derivative of a cross product d/dt (P x Q) with a function of time (like t + t^2).
Ex. d/dt (P x Q) = 5t - 6t^2
My question is, how can you have an ...
- 193
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3
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193
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Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?
My teacher has proved the following:
$$ \mathit v^2 = \mathbf v·\mathbf v = \frac{d\mathbf r}{dt}·\frac{d\mathbf r}{dt} = \left(\frac {ds}{dt}\right)^2 \Rightarrow \mathit v = \frac{ds}{dt} $$
Because ...
- 23
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1
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What motivates defining vectors as first order differential operators?
I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
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1
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303
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How to ("geometrically") differentiate unit vectors of spherical coordinates?
I have been trying to derive the expressions of partial derivatives of unit vectors with respect to each other in the spherical coordinate system. I was able to get all of them except $\frac{\partial \...
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1
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6
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713
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Why does differentiating a scalar give a vector? [closed]
I was wondering why $F=-\frac{dU}{dr}$ would give me a vector quantity when a scalar quantity is differentiated. There are similar pre-existing queries but I think this issue has yet to be properly ...
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4
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413
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Suppose there is a vector $\vec v$ which is a function of time, then will $\dfrac{d}{dt}|\vec v|$ be a vector quantity or a scalar quantity?
Suppose there is a vector $\vec v$ which is a function of time, then will $\dfrac{d}{dt}|\vec v|$ be a vector quantity or a scalar quantity?
I think it should be scalar because, let's assume $\vec v=...
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152
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Product rule for 4-vectors and derivation of 4-force form
In deriving the form for the 4-force in special relativity, we begin with
$$\frac{d}{d\tau}p^{\alpha}=m\frac{d}{d\tau}u^{\alpha}=f^\alpha$$
where $\tau$ is the proper time, m is rest mass.
Since $...
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1
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What mean this momentum-derivative?
I'm working with quantum gravity. I have to make a Taylor-series. I got some help for this, but I have problem with understanding the formalism. So, I have the operator $A((P-p)^2)$, which needs to ...
- 3
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Derivative of a metric tensor
I would like to ask you a question - maybe simple - but bothering me.
We have two four-position vectors product in curvilinear coordinates given by
$(1) \quad X^{\alpha}g_{\alpha \beta}X^{\beta} = \...
- 9
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66
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Differentiating the four-velocity contracted with itself
Let us denote $u^\mu$ as the contravarient component of a four velocity at a point in some coordinate system for a pseudo-Riemannian manifold. I want to examine the following equation.
$$\partial_\nu(...
- 551
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2
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85
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How to Differentiate $\vec{r}$ in Polar Form?
Question
First off, I do realize that:
$$\vec{r} = r \hat{r}$$
$$\dot{\vec{r}} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$$
$$\ddot{\vec{r}} = \ddot{r} \hat{r} + \dot{r} \hat{r} + \dot{r} \dot{\...
- 59
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1
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197
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4-velocity lowering index question
The 4-velocity in contravariant form is given by
$$V^\mu=\frac{dx^\mu}{d\tau}$$
for some general co-ordinates $x^\mu$ and proper time $\tau$.
Is the 4-velocity in covariant form given by
$$V_\nu=V^\...
- 5,991
1
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1
answer
50
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Path Coordinates: direction problem (doubt) in derivative of tangential vector
Why is the direction of derivative of tangential vector perpendicular to the direction of the tangential vector?
- 29
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1
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509
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Divergence of a vector multiplied by dot product [closed]
If I am correct, then
$\operatorname{div} [(\vec A\cdot \vec B)\vec C] = (\vec A \cdot \vec B) \operatorname{div} \vec C + \vec C \cdot \nabla (\vec A\cdot\vec B)= (\vec A \cdot \vec B) \operatorname{...
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1
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554
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Meaning of normal acceleration?
acceleration means the rate of change in velocity (vector quantity) and the differentiation means to divide a certain quantity into small elements (i.e $dx$) as we do to find the acceleration at any ...
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1
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78
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Why do we neglect $\Delta t^2(\frac{d\vec{r}}{dt}\frac{d\vec{\hat{r}}}{dt})$ at Taylor Expansion?
I'm just started to Ankara University Physics Department two weeks ago. I have missed my 2 hours of PHY105 course that is the last week Wednesdey. The subject that i missed was Derivatives of Vectors. ...
- 709
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Non-uniform circular motion with constant radius of curvature [closed]
$\let\oldhat\hat
\renewcommand{\vec}[1]{\mathbf{#1}}
\renewcommand{\hat}[1]{\oldhat{\mathbf{#1}}}$
Suppose we have a car moving on a circular track of radius $b$ and speed $v=ct$, where $t$ is time ...
-1
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2
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158
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What does it mean for velocity to be a derivative of position, if position a point, not a function? [closed]
So in mass-spring simulation I encountered that one simulates particles by using positions and velocities of particles etc.
People may say that velocity is the derivative of position.
But isn't "...
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6
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2
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1k
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Terminology for time derivative of speed (not velocity)
Is there any standard terminology for the derivative of the magnitude of velocity with respect to time (suitable for use in first-year Calculus)? The word ‘acceleration’, in its technical sense, is ...
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225
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Derivative with respect to vector
How in Lagrangian and Hamiltonian mechanics we take derivatives with respect to velocity and momentum respectively if they are vectors? Can we take derivative with respect to a vector?
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2
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1
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Work-Kinetic energy theorem derivation
So I came across this derivation in the book Classical Mechanics by Herbert Goldstein. I don't follow from the second step onwards. I understand that there's a dot product, but how do you compute it? ...
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1
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1
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415
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Vector calculus notation, maybe?
I just got a new book on turbomachinery that uses some notation I'm not familiar with.
$$ \nabla \lor \vec{W} = -2\vec{\Omega} $$
The del-(something)-vector format makes me think its vector calculus....
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1
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204
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Can you apply product rule to arg of a bra-ket?
I found the following expression in a paper:
$$
\frac{d}{dt}\arg\langle\phi_+|\dot{\phi_-}\rangle
$$
where the $\arg$ term is the argument of the complex number given by inner product between two ...
- 591
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74
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Does the proper four-acceleration $A^{\mu} = (0,0)?$
Let the proper four-position vector $x^{\mu}(\tau) = (0, \tau)$. Differentiating this successively wrt $\tau$ I get the four-velocity $u^{\mu}(\tau) = (0, 1)$ and then the four-acceleration $A^{\mu}(\...
- 3,148
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7
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Can we divide a vector by another vector? How about this: $a = vdv/dx?$
My physics teacher told us that we can’t divide vectors, that vector division has no physical meaning or significance. How about this: $$a = vdv/dx.$$
It says acceleration vector equals velocity (as ...
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1
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Is $\nabla=\nabla'$? Nabla operator acting on $r^n$
I have been taught that
$$\nabla r^n =\text{gradient of }r^n =n r^{n-1}\ \hat{\boldsymbol r}$$
but in introduction to electrodynamics by Griffith (4th edition) on page 173, $\nabla' r^n$ is given by $-...
2
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1
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278
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Dot product in cylidrical coordinates
I'm given the vector:
$$\vec{V}{(r,θ,z)}=\frac{1}{r}\hat{e_r} + (r\cosθ)\hat{e_θ}+\frac{z^2}{r^2}\hat{e_z}$$
I want the scalar product ${\vec{\nabla}}\cdot{\vec{V}}$
We know that in cylindrical ...
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3
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1k
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How to determine the direction of instantaneous acceleration in a 2D motion? [duplicate]
How do we determine the direction of instantaneous acceleration when the body is moving in a plane (or a 3D space)? This question has been truly bothering me for nearly two weeks. I looked it up, ...
- 876
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2
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84
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Acceleration in a non-inertial reference frome - derevation
The general velocity equation for a point B in on body rotating and translating about point A with respect to the inertial reference frame say 'xyzo' can be expressed as,
$\vec{r_{B/o}} = \vec{r_{A/o}...
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3
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84
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How is velocity defined in circular motion in central force field?
In my view the velocity is change of displacement in the increasing direction of displacement. Now in circular motion in central force field the particle is changing its direction then how is the ...
- 1,159
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1
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149
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Basic vector calculus: Show that $\nabla \vec{r} = \vec{1}$ [closed]
Show that $\nabla \vec{r} = \vec{1}$
My instructor in my E & M class put the $r$ and $1$ in bold. I am not sure what a bold one means. From my work I get $1ii + 1jj + 1zz$.
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2
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273
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How can I show that the acceleration vector for uniform circular motion undergoes uniform rotation?
Does it suffice to show that the dot product between the acceleration vector and the derivative of the acceleration vector = 0?
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4
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2
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261
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The acceleration of circulation motion
We know that in circular motion the position vector is $r\hat{r}$. Then the velocity is the time derivative of it. So it gives $$dv/dt = r d\hat{r}/dt + \frac{dr}{dt} .\hat{r}.$$ now I can't ...
- 1,159
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Which relation is correct for resultant instantaneous velocity in 2d?
Please forgive me if the following question sounds silly and I can't exactly pin point where exactly the problem is but there is some problem with my understanding of vectors.
In Cartesian ...
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Time derivative of vector in rotating frame with angular velocity about a rotating axis
In general, I know that if you have a vector $\vec{F}$ in a rotating frame, and the frame has an angular velocity $\vec{\Omega}$ that the time derivative of $\vec{F}$ in a fixed frame would be $$\frac{...
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3
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1k
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Feynman's Four Gradient in Special Relativity
I am having trouble understanding Feynman's explanation of four gradient.
In section 25-3 of Vol. 2 of the Feynman lectures, he explains why the four gradient is not $(\frac{\partial}{\partial t},\...
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1
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130
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On the derivative of a vector function
In "An Introduction to Mechanics" by Kleppner and Kolenkow, in the section on the time derivative of a vector:
Given $A(t)$ is a vector valued function, then,
$$\Delta A = A(t + \Delta t) - A(t)$$
...
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2
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5k
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How is dot or cross product possible using the del operator?
Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
- 1,432
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1
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651
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What is difference between $d\vec{l}$ and $\vec{dl}$? [closed]
What is difference between $d\vec{l}$ and $\vec{dl}$? $d$ means differential as usual.
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482
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Partial Derivative of a scalar (absolute distance) with respect to its position vector
Imagine we want to take the partial derivative of a quantity, we will call it $\rho_i = f(F(r_{ij}))$ with respect to a particle's position vector, $\vec{r}_k$.
In mathematical terms, this would be ...
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2
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2k
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Why the unit vector is represented as a partial derivative in GR?
Can someone give a good intuitive explanation why we represent the unit vector as a partial derivative in GR and what does it mean?
- 595
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5
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7k
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Direction of velocity vector in 3D space
According to a well-known textbook (Halliday & Resnick), the direction of a velocity vector, $\vec v$, at any instant is the direction of the tangent to a particle's path at that instant, as is ...
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4
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Does $\nabla \cdot \vec{E}$ have the same meaning of dot product $ \vec{a}\cdot \vec{b}=|a| |b| \cos(θ)$?
When the del operator is involved, does the meaning of dot and cross product of vectors change? That is, $\nabla \cdot \vec h$ is defined as below:
$$\nabla \cdot \vec h = \frac{\partial h_x}{\...
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1
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259
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Confusion in adding constant to magnetic vector potential
The $x$-component of $B$ is:
$B_x=\dfrac{\partial {A_z}}{\partial y}-\dfrac{\partial {A_y}}{\partial z}
=\dfrac{\partial {(A_z+C_1)}}{\partial y}-\dfrac{\partial {(A_y+C_2)}}{\partial z}$
where $...
- 137
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1
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Kinetic energy derivation: Why is $\frac{d \mathbf v}{dt} \cdot \mathbf v= \frac 12 \frac{d}{dt}(v^2)~?$
In Goldstein's Classical Mechanics 3rd edition, page 3, the Kinetic energy is derived by considering the work done on a particle by an external force $\mathbf F$ from point $1$ to point $2$ $$W_{12}=\...
user138458
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4
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318
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Why is $\vec{v}\cdot d \vec{v} = v dv$? [closed]
Can someone help me understand why is this true:
$$\vec{v} \cdot d \vec{v} = v \cdot dv$$
where $v$ is speed? I found somewhere that $\vec{v} \cdot d \vec{v}=|\vec{v}||d \vec{v}| \cos \phi$. And I ...
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2
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134
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Generalization of $F=mv\frac{dv}{dx}=\frac{m}{2}\frac{d}{dx}(v^2)$ to 3-dimensions in a compact notation
Starting from $F=ma=m\frac{dv}{dt}$, in 1-dimension, it is easy to show that $$F=mv\frac{dv}{dx}=\frac{m}{2}\frac{d}{dx}(v^2).\tag{1}$$ Can we generalize this formula in 3-dimensions? In 3D, $$\textbf{...
- 12.3k
4
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2
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861
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Integration of tangential acceleration with respect to time
Here, by tangential acceleration, I mean the component of acceleration along the velocity vector.
What do you get when you integrate tangential acceleration with respect to time? What does the '$v$' ...
- 1,106
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2
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471
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Why exactly in general relativity are tangent vectors defined as maps from functions to $\mathbb{R}$?
I am basing this on the lectures from the hereaus international winter school on gravity and light.
If $M$ is the manifold of physical spacetime, then at any point $p \in M$, we have a tangent space ...
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