All Questions
Tagged with differentiation vectors
120 questions
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Mass Conservation in Kinetic Theory
In chapter 9 (The Boltzmann Equation) of Schwabl's 2006 text 'Statistical Mechanics', the author has the following statement of conservation of mass,
$$ \frac{\partial n}{\partial t} + \nabla \mathrm{...
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1
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59
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Need help in understanding Tangential Acceleration [closed]
I am studying Circular motion and I am confused about tangential acceleration and tangential velocity. I am studying uniform circular motion and it says the tangential acceleration is $0$ in uniform ...
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1
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69
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Is 4-velocity a Vector in the Sense of Covariant Derivative along Worldline
The definition of 4-velocity $U^{\mu} \equiv dx^{\mu}(\tau)/d\tau$, however, we've learnt that the covariant derivative for a vector along a curve parametrized by proper time is,
$$\frac{DA^{\mu}}{D\...
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Derivative of the product of a scalar function and a vector valued function
According to Berkeley Physics Course, Volume 1 Mechanics,
The time derivative of a vector valued function can be derived from the formula:
$$
\mathbf{r}(t) = r(t)\mathbf{\hat{r}}(t)
$$
where the ...
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2
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105
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Why must a constraint force be normal?
If we impose that a particle follows a holonomic constraint, so that it always remains on a surface defined by some function $f(x_1,x_2,x_3)=0$ with $f:\mathbb{R^3}\rightarrow\mathbb{R}$, we get a ...
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Problem with resources, Walter Lewin's third lecture
I've watched Walter's third lecture in 8.01 and I have a small problem with the last part, where he says that $$\vec r_t=x_t\cdot \hat x\ +\ y_t\cdot \hat y\ +\ z_t\cdot \hat z \\ \vec v_t=\frac{d\vec ...
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3
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176
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Where to apply $\nabla$ operator when taking curl of a cross product?
In my EM class we went over $$\nabla\times \frac{\vec{d}\times \vec{r}}{r^3}$$ which apparently can be breaken down to $$r(d\cdot \nabla)\frac{1}{r^3}-d(r\cdot\nabla)\frac{1}{r^3}+\frac{\nabla\times(d\...
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343
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Isomorphism of the tangent space and the space of directional derivatives [closed]
I have already constructed the tangent space to a manifold, denoted $T_pM$, and I have a good basis for it $\{\hat e_{(\mu)}\}$. (I followed the method of equivalence classes of curves tangent at $p$....
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1
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103
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Time derivative of a "general" vector $\vec A$ in an accelerating frame: what about e.g. velocity $\vec v$?
According to Morin "Classical Mechanics" (Section 10.1, page 459), the derivative of a general vector $\vec A$ in an accelerating frame may be given as
$$\frac{d\vec A}{dt}=\frac{\delta \vec ...
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Applications of time derivative of unit vector
A math methods textbook I'm currently reading went into great detail deriving the following expression for the time derivative of a generic unit vector $\hat{r}$.
$$
\frac{d\hat{r}}{dt} = \frac{1}{r^2}...
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Why is it wrong to find centripetal acceleration using change of velocity over change of time?
This question asks to find the centripetal acceleration by giving the initial and final velocity over the change of time.
As shown, my book combined two rules to find the acceleration. I utterly ...
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91
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From where does the expression of the tangential accerelation come from?
I've seen so many times that the expression of the tangential acceleration is known to be: $$a_t=\ddot{s}$$ but from the expression of the acceleration in spherical coordinates, in the tangential ...
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Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?
Hello fellow physicists,
I was trying to understand some behavior on rotating objects, specifically about the formula $\vec{v} = \vec{\omega} \times \vec{r}$.
The Book (Marion, J. B. (1965). Classical ...
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5
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348
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
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Schwartz "QFT and the Standard Model", eq. 15.59, derivative trick, deriving with a dot product
$$\frac{\partial }{\partial s}M(s)= \frac{p^{\mu}}{2s}\frac{\partial }{\partial p^{\mu}}M(s)\tag{15.59}$$
$$\ s=p^{2}$$
How does the derivative with respect to $s$ turn into the expression on the ...
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What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
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Problem with the constant magnitude of vectors if the change in the same vector is perpendicular to it [duplicate]
Note: I am merely a highschool student attempting to self-study Classical Mechanics, some of the assumptions I make are perhaps wrong, so please bear with me. Thank you.
This while can be condensed ...
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103
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Conceptual confusion about the formula for parallel transport
I am examining the covariant derivative of a vector according to the formula $$\nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$ and also operating under the ...
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407
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Vector addition for differentials in the context of electric potential
Recently, my professor drew the following diagrams to explain $\vec{ds}$ (in the context of electric potential, where $$V=-\int\vec{E}\cdot\vec{ds}$$
He showed us the following diagrams and ...
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414
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Why does tangential acceleration become 0 when the velocity is max? [closed]
I know that tangential acceleration equal to zero when the circular motion is uniform, but why is it equal to zero, when the velocity is max or min? Because there is no relation between the value of ...
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Are rates a scalar, a vector or both?
Are all rates in physics a scalar, a vector or both?
It seem to me like all rates in science are vectors.
Examples of rate that are vectors are rate of charge flow, rate of heat transfer, rate of mass ...
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Having trouble deriving the exact form of the Kinematic Transport Theorem
The Kinematic transport theorem is a very basic theorem relating time derivatives of vectors between a non rotating frame and another one that's rotating with respect to it with a uniform angular ...
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Are terms tangential acceleration and normal acceleration only used for instantaneous velocity?
Are terms tangential acceleration and normal acceleration only used
for instantaneous velocity?
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113
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If a body moves along a path (any path, not just circular) with constant speed, is it's tangential acceleration necessarily zero?
If a body moves along a path (any path, not just circular) with constant speed, is it's tangential acceleration necessarily zero?
I could only find general proofs for the case of circular motion and ...
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Proving the relation $\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}$ (quantum mechanics exercise) [closed]
I'm trying to prove this relation in my quantum mechanics exercise book
$$\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}.$$
Here's my attempt:
Expand the Laplacian ...
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113
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How to define differentiation of a time-dependent vectors with respect to a specific reference frame in a coordinate-free manner?
It is usual in classical mechanics to introduce the derivative of a time-dependent vector with respect to a reference frame. This is accomplished through the use of a basis that is fixed with respect ...
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Gradient of scalar field
On page183 of Rayd'inverno "An introduction to relativity" he says that the right term in parenthesis is a gradient of some scalar field i.e.
When $$\partial_a (\frac{\ X_b}{\ X^2})=\...
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Why is the magnitude of velocity negative in this example?
Magnitudes are positive values, but when I take, for example: the magnitude of a position vector: $r = 3 - 0.04t^2$ and try to take the derivate of it, the result will be $v = -2 * 0.04t$ which is a ...
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What is the General formula of gradient of $r^n$? [closed]
so, the question is that r is the separation vector from a fixed point $(x',y',z')$ to the point $(x,y,z)$ and let $r$ be its length.
the answer to the question of what is the general formula of $$\...
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Spherical and Cartesian forms of divergence [closed]
Suppose the electric field found in some region is $$\overrightarrow{E} = ar^3\vec{e}_r$$ in coordinates
spherical (a is a constant). What is the charge density?
So, using the spherical form of ...
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1
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94
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Simple difference between module of velocity and time derivative of module of position [duplicate]
What is the conceptually difference between the two:
$$\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|$$ ...
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What do you call $ \frac{d^2 r}{dt^2}$ in polar coordinates? [duplicate]
In polar coordinates, one finds centripetal acceleration as:
$$ a_c = \frac{d^2 r}{dt^2}- \frac{v^2}{r}$$
Where $|r|$ is distance from center to particle, $v$ is tangential velocity.
My question is ...
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Meaning of the transpose of a gradient
Sometimes I encounter PDE's with a term like this
$\nabla \cdot c(\nabla \textbf{v} + (\nabla \textbf{v})^T)$
An example are the Navier-Stokes equations. Oftentimes this can be further simplified to $...
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353
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Why isn't tangential acceleration just always 0?
This is probably a very stupid question but I can't help me.
Tangential acceleration is $\vec{a_t}=\frac{dv}{dt}\frac{\vec{v}}{v}=\frac{\vec{v} \cdot \vec{a}}{v} \frac{\vec{v}}{v}$. Since $\vec{a}$ is ...
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Vector calculus of a potential energy formula under Galileo transformation
I'm currently studying MIT OCW 8.20 Introduction to Special Relativity. In pset 1, the following question is being asked: Suppose you have a potential of the form U($\vec{r_1}, \vec{r_2}$) = U(|$\vec{...
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46
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1/velocity for higher dimensions
I have a somewhat basic question. I am sorry if it trivial.
Denote the velocity by $v=\frac{dx}{dt}$ suppose that $x \in \mathbb{R}^n$ and I want to parametrize $t$ in $x$ and compute $\frac{dt}{dx}$. ...
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One object moves along the cycloid at a constant rate, how about its acceleration? [closed]
We know that the parametric equation:
$$x=R(\theta+\sin(\theta))$$
$$y=-R(1+\cos(\theta))$$
and the constant velocity $c$.
How do I prove that the acceleration of the object in the $y$ direction is ...
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Vector cross product formula without a second term (Spiegel, Theoretical Mechanics)
In Spiegel's Outline Of Theoretical Mechanics (more precisely in the Moving Coordinate Systems chapter, § "Derivative Operators") I find (both in the 1968 and the 1977 edition) the following ...
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Why intuitively is the tangent vector the derivative of velocity of position with respect to their modulus?
When trying to find the tangential velocity, many textbooks define the tangent direction as one of the following:
or
Intuitively, why is the tangent vector the derivative of the position with ...
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What does the derivative of tangent means? [closed]
While studying the circular motion I had to find the derivative of a tangent so I thought what the derivative of a tangent could probably mean since the derivative of position gives velocity.
Or think ...
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319
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What is the time derivative of the linear velocity vector $\vec{v}\,(t)$?
If $\vec{v}\,(t)$ denotes linear velocity, we can then write $\vec{v}\,(t)$ as $|v(t)|\hat{v}$. My question is what is $\displaystyle\frac{d\vec{v}\,(t)}{dt}?$
The answer I have seen to this question ...
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Why is the derivative of Vector equal to Derivative of its rectilinear components?
Take a vector $\mathbf A=t^4\mathbf i +t^2\mathbf j$, and call the unit vector along direction of $\mathbf A$ is $\mathbf k$, so the magnitude of this vector $\mathbf A$ along $\mathbf k$ will be $\...
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Find the distance travelled between $t=0$ and $t=5$ [closed]
The position vector of a particle is given as $\vec r = \frac43 t^{3/2}\hat i - \frac{1}{2} t^2\hat j + 2 \hat k$, $t$ is in seconds. Find the distance travelled between $t = 0$ and $t = 5$ seconds.
...
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What is the meaning of the del operator in this equation?
$$\frac{\partial \left(\rho_m \vec{v}_m \right)}{\partial t} + \nabla \cdot \left(\rho_m \vec{v}_m\vec{v}_m \right) \\ = - \nabla P_m + \nabla \left(\mu_m \nabla \vec{v}_m \right) + \nabla \left(\...
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Time derivatives of the unit vectors in cylindrical and spherical
In cylindrical and spherical coordinates, the position vectors are given by $\mathbf{r}=\rho \widehat{\boldsymbol{\rho}}+z \hat{\mathbf{k}}$ and $\mathbf{r}=r \hat{\mathbf{r}}$, next to next, and ...
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180
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Avoiding a confusion with dot product
Some days ago I have asked a question about a formula for power, many generous people have answered my question and clarify for me that the correct formula of work is
$$\mathrm{d}W= \mathbf{F}\cdot \...
1
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3
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324
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Derivative with respect to vector of a function depending on vectors
I've been trying to understand this concept for hours without any success. I found similar questions on this forum (Derivative with respect to a vector is a gradient?) but I still don't understand.
...
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Given that $m \dot v \cdot v = 0$ , how is it equal to $m \frac{d}{dt} (v \cdot v)/2$? [closed]
While studying about scalar triple product in vector algebra, I stumbled upon the following question with the solution.
I want know how is $m \dot v \cdot v $ = $m \frac{d}{dt} (v \cdot v)/2$?
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2
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What is meant when we say that a differential takes on a certain value?
As far as i understand it, total differentials are linear maps that map vectors to numbers. In thermodynamics we encounter statements that a we have reached equilibrium when a total differential of a ...
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Is the relation "slope=velocity" mathematically valid?
$\text{Slope= tan(angle with respect to positive X-axis)= scalar output}$
$\text{velocity= a vector }$
Source: Hugh D Young_ Roger A Freedman - University Physics with Modern Physics In SI Units (...
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