All Questions
Tagged with differentiation differentiation or
1,900 questions
-1
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1
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96
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Finding solution to this differential equation
In this paper http://arxiv.org/abs/hep-th/9506035 equation (3.11) was written as: $$\frac{\partial L}{\partial u}\frac{\partial L}{\partial v} = -1$$
The author then said p.9 that "approximate ...
- 215
0
votes
1
answer
268
views
Gradient of two-particle system
I'm working on problem 5.1a from Griffiths Intro to QM and given that:
$$\mathbf R \equiv \frac{m_1\mathbf{r_1} + m_2 \bf r_2}{m_1+m_2}$$
and $\bf r \equiv \bf r_1 - \bf r_2$ I need to show that,
$$\...
- 207
2
votes
2
answers
318
views
Derivation of velocities in the Coriolis force
In Fitzpatrick's Newtonian Dynamics book on the Coriolis force, he states
\begin{align}
v_{x'}&\simeq V_0\cos\theta-2\Omega t V_0\sin\lambda~\sin\theta \tag{433}\\
v_{y'}&\simeq-V_0\sin\...
- 1,629
4
votes
2
answers
476
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Why doesn't this multiplication of Grassmann variables give the expected result?
Would anyone be able to tell me how srednicki goes from step $(44.29)$ to $(44.30)$?
Here is the paragraph:
Now let us introduce the notion of complex Grassmann variables via
$$\begin{align}
\...
- 41
2
votes
1
answer
180
views
Why do derivatives act on vector fields on a worldsheet?
The covariant derivative of a vector $A^{\mu}$ at a point $x$ is defined as
$$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$
where Greek symbols are ...
- 142
1
vote
4
answers
407
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Rotation systems. Problem interpreting an equation
In this equation:
$$
\mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf v}{dt}\right)_i=\left[\left(\frac{d}{dt}\right)_r+\boldsymbol\Omega\times\right]\left[...
- 1,629
7
votes
1
answer
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Total and partial derivatives in thermodynamics and Maxwell relations
Consider the expression
$$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$
I'm trying to understand how to derive an expression for $\left( \frac{\...
- 935
1
vote
2
answers
371
views
Why is $\nabla\cdot(\hat{\bf r}/r^2)$ giving 0 as answer? [closed]
While I was reading I encountered the statement $\nabla\cdot(\hat{\bf r}/r^2)$ (r cap divided by $r$ square) is 0. Can anyone explain proof of the statement why is it giving 0?
- 21
2
votes
3
answers
3k
views
Derivation of the Riemann tensor confusion
I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
- 3,089
1
vote
1
answer
3k
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Want to know about divergence [duplicate]
Can anyone please explain how to know whether a vector field has divergence or not by seeing its diagram?
I have read that a vector field must change for having divergence but why is divergence zero ...
- 21
3
votes
1
answer
138
views
Is this covariant derivative identity true?
Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$
If this is true, I'm ...
- 3,089
5
votes
1
answer
5k
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Second derivative of Dirac delta expression
I have come across the expression
$$ \int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$
where the prime represents the derivative.
Usually with derivatives of the Dirac delta distribution I'd partially ...
- 9,197
0
votes
1
answer
2k
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Covariant derivative of stress-energy tensor for a scalar field [closed]
In order to prove that $$\nabla ^\mu T_{\mu\nu} =0$$ I want to find the
covariant derivative of $$T_{\mu\nu} = \partial_\mu\phi \partial_\nu \phi -\frac{1}{2}g_{\mu\nu}(g ^{\lambda\sigma}\partial_\...
- 1,539
2
votes
0
answers
184
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Problems while doing $\dfrac{\partial}{\partial(\partial_\mu \phi)}$ and $\dfrac{\partial}{\partial(\partial_\mu A_\mu)}$
In David Tong's lectures, he gives two Lagrangians as examples to derive the equations of motion:
$$\mathcal{L} = \dfrac{1}{2}\eta^{\mu\nu}\,\partial_\mu\phi\,\partial_\nu \phi-\dfrac{1}{2}m^2\phi^2, ...
- 178
4
votes
4
answers
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Why can't impulse be instantaneous?
We know from 2nd law of motion that $$\vec{F} = \frac{d\vec{p}}{dt}.$$
Now, a rate of change can be instantaneous. So, rate of change of momentum is instantaneous. But I doubt how can there be ...
user36790
3
votes
2
answers
2k
views
Physical meaning of harmonic function?
In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
- 1,558
4
votes
2
answers
18k
views
Why and when do we differentiate or integrate equations in physics? [closed]
I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like:
The object is moving in a positive ...
- 41
0
votes
1
answer
3k
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Total vs partial time derivative of action
I'm following Ref. 1 in my reasoning, struggling with action as a function of time.
Consider a Lagrangian
$$L=\dot x^2-x^2.\tag1$$
Solving the corresponding equations of motion with initial ...
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2
votes
2
answers
346
views
Determining Acceleration Based On Graph
I understand how to solve this problem, but I am unsure how to generate an equation for the graph (below). My current attempt involves using the mass provided along with the derivative of the line (...
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5
votes
5
answers
7k
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What is divergence?
What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
- 1,660
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votes
6
answers
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Why is curl of current density $\nabla \times \vec{J}$ equal zero?
I am revisiting the derivation for $\nabla \cdot \vec{B} = 0$ in magnetostatics for the field $\vec{B}(\vec{r})$ of a charge $q$ at position $\vec{0}$ with velocity $\vec{v}$. It proceeds like
\begin{...
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3
answers
2k
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Geometric meaning of parallel transport
The definition of parallel transport of a vector $v^b$ along a curve $C$ with tangent field $\it{t}^a$ is given by Wald's GR as $$t^a \nabla_a v^b = 0$$
Is it correct to think of $\nabla_a v^b$ as ...
- 763
-1
votes
1
answer
507
views
Finding the divergence of this force [closed]
I've got to find the divergence of this force,
$$
\mathbf F=\left(x^2+y^2+z^2\right)^n\left(x\hat e_x+y\hat e_y+z\hat e_z\right)
$$
I would know how to do it if the $n$ superscript wasn't there. Any ...
- 9
3
votes
3
answers
2k
views
Physical Meaning of Divergence of Convective Velocity Term
When taking the divergence of the convective velocity term, I get the following:
\begin{align}
\nabla\cdot\left[\mathbf u\cdot\nabla\mathbf u\right]&=\frac{\partial}{\partial x_i}\left[u_j\frac{\...
- 379
2
votes
1
answer
124
views
Can these two terms cancel out?
In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$
The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with $$J^b_{\mu\lambda}J^a_\...
- 1,539
1
vote
2
answers
589
views
Differentiation in general relativity
If we have:
$$ \frac{d\phi^a}{d\tau}= \frac{\partial \phi^a}{\partial x^\mu} \frac{dx^\mu}{d\tau} \tag{1}$$
Differentiating it, we get:
$$ \frac{\partial \phi^a}{\partial x^\mu}\frac{d^2x^\mu}{d\...
- 1,539
2
votes
3
answers
2k
views
Poincare invariant Lagrangians
The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean
$$
\partial_\mu \mathcal{L}=0~?
$$
If this is the case doesn't the ...
- 1,137
2
votes
2
answers
8k
views
Time derivative of angular velocity in rotating reference frame
I am going through a section in a textbook regarding the Newton Euler equations for a system of rigid bodies (robotics text). There is a particular line in the derivation I don't understand, I've ...
- 27
0
votes
1
answer
68
views
The vector r points from $P'(x',y',z')$ to $P(x,y,z)$ [closed]
For some reason this question is giving me a hard time :(
The vector $r$ points from $P'(x',y',z')$ to $P(x,y,z)$.
(a) Show that if $P$ is fixed and $P'$ is allowed to move, then $\nabla'(\frac{1}{r}...
- 817
6
votes
1
answer
2k
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Why do we need the material derivative?
I'm studying fluid mechanics, and I got the impression that the material derivative is nothing more than "differentiating along a path" and so I got confused on why do we need it. Basically, ...
- 37.4k
6
votes
2
answers
2k
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Relationship between Connection and Material Derivative
Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
- 37.4k
3
votes
2
answers
5k
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Prove that a derivative with respect to a covariant 4-vector is a contravariant vector operator
In special relativity, I know you can prove that the derivative with respect to a contravariant 4-vector component transforms like a covariant vector operator by using the chain rule, but I can't work ...
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1
vote
2
answers
4k
views
Is there a difference in handwritten nabla $\vec{\nabla}$ with an overset arrow and typeset nabla $\nabla$?
According to some physicist at KIT it is usual to write the following when using pen and paper:
whereas in typeset texts you write $\nabla$.
Is that true? Are there sources for this convention?
- 955
1
vote
4
answers
9k
views
Dot product of vector and its derivative with respect to time? How does $L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$? [closed]
How does:
$$L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$$
where L is a vector (I dunno how to make it bold in the equation).
How do they reach to this right hand side equation?
And what is ...
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0
votes
1
answer
34
views
Why does the power in an inductor equal what it does?
I understand that power is that rate at which work is done and that because of this the power in an inductor is equal to
$$P=\frac{d}{dt} \left(\frac12Li^2\right).$$
I also understand that the power ...
7
votes
2
answers
1k
views
A confusion about notation in Goldstein
On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation
$$\dfrac{d}{dt}{\bf p}_i = {\bf F}_i^{(e)...
- 37.4k
5
votes
1
answer
302
views
Why do we do partial and not covariant differentiation with $x^{\nu}$?
Why when taking the velocity vector we make
$$u^{\nu}=\frac{d}{d\tau}x^{\nu}$$
and not
$$u^{\nu}=\frac{\nabla}{d\tau}x^{\nu}$$
where in the last equation I meant the covariant derivative. Why?
- 6,137
2
votes
1
answer
2k
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How to get the time derivative of an expectation value in quantum mechanics?
The textbook computes the time derivative of an expectation value as follows:
$$\frac{d}{dt}\langle Q\rangle=\frac{d}{dt}\langle \Psi|\hat Q\Psi\rangle=\langle \frac{\partial\Psi}{\partial t}|\hat Q\...
- 945
1
vote
1
answer
794
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1D Smoluchowski diffusion equation in a linear potential
I am interested in solving a 1D Smoluchowski diffusion equation in a linear potential $U(x) = cx$ for a constant force $c$. This problem follows chapter 4 of the theoretical biophysics script by ...
- 43
2
votes
3
answers
126
views
Can we measure rates in real time?
I know what it means to say that my position is "X" at a particular moment in time. I can easily take a picture of my motion and observe my exact location at the instant the picture was taken. That is ...
- 3,966
3
votes
2
answers
1k
views
Any difference between thermodynamic double-derivative and derivative "at constant" value?
Reading about the Maxwell relations has left me confused, and I want a basic sanity check regarding the notation. The Wikipedia article breezes over the following switch of notation without really ...
- 21.3k
1
vote
3
answers
2k
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Why there is added a partial time derivative in formula for time derivative of potential energy? [duplicate]
In proving the total energy in conservative field is constant we have this equation(picture) why it added partial derivative? Why? I mean where it did come from?
- 21
1
vote
0
answers
95
views
Partial derivatives in Lagrangian formalism [duplicate]
Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant:
$$ \frac{\partial f}{\partial x} = y $$
Does this mean that in order to evaluate ...
- 959
7
votes
7
answers
5k
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Why do we need a metric to define gradient?
For me, the gradient of a scalar field (say, in three dimensions) is simply (formally)
$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} \...
- 4,152
3
votes
2
answers
1k
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Covariant derivative of a covariant tensor wrt superscript
Is it true that when you take the covariant derivative of a covariant tensor, do you always have to do with a subscript? What if you do it wrt a superscript?Does the first term (with the partial ...
- 2,778
5
votes
4
answers
5k
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How can there be really any instantaneous velocity?
I have read about Zeno's arrow paradox that tells us there is no motion of the arrow at a particular instant of its flight. It can be inferred that there can be no velocity at any instant. Moreover we ...
user36790
4
votes
2
answers
271
views
Conventions regarding partial derivatives
Look at this expression:
$$\frac{\partial}{\partial t} (V-\mathbf{v}\cdot\mathbf{A}).$$
This expression occurs in Griffiths EM book (4th ed, p.444). $V=V(\mathbf{r},t)$is the scalar potential, $\...
- 1,417
1
vote
2
answers
547
views
Taylor series: Epsilon not differentiated? [closed]
Why isn't epsilon differentiated with respect to time? (see my question on the right)
- 43
1
vote
3
answers
245
views
Apparent dimensional mismatch after taking derivative
Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them.
...
- 415
1
vote
1
answer
253
views
Curl of a vector field with two different systems of coordinates
Let
$$\mathbf{H} = H_x \mathbf{u}_x + H_y \mathbf{u}_y + H_z \mathbf{u}_z$$
be a vector field whose components are defined with respect to the unit vectors $\mathbf{u}_x$, $\mathbf{u}_y$ and $\...
- 777