All Questions
Tagged with differentiation differentiation or
259 questions
10
votes
2
answers
6k
views
Why do we write $(v\cdot \nabla) v$ instead of $v \cdot (\nabla v)$ for $v_j \frac{\partial}{\partial x_j} v_i$ in the material derivative?
Suppose I have a steady flow and I want to find the rate of change of pressure of a bit of fluid. This depends on the velocity of the fluid and the pressure gradient,
$$\frac{\mathrm{d}P}{\mathrm{d} ...
- 53.7k
8
votes
2
answers
3k
views
Intuitive analysis of gradient, divergence, curl
I have read the most basic and important parts of vector calculus are gradient, divergence and curl. These three things are too important to analyse a vector field and I have gone through the physical ...
- 741
8
votes
6
answers
1k
views
Mathematical Definition of Power [duplicate]
I am a high school student who was playing around with some equations, and I derived a formula for which cannot physically imagine.
\begin{align}
W & = \vec F \cdot \vec r
\\
\frac{dW}{dt} & = ...
- 199
8
votes
1
answer
2k
views
Why is that in the action principle, the Taylor's series is limited to the first order?
For the Hamilton's principle: $$\delta s =\int_{t_1}^{t_2}L(\mathbf {q+\delta q},\mathbf {\dot q+\delta \dot q},t) dt-\int_{t_1}^{t_2}L(\mathbf {q},\mathbf {\dot q},t) dt=0.\\$$
In the textbooks, ...
7
votes
3
answers
1k
views
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 ...
7
votes
1
answer
11k
views
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
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
4
votes
4
answers
747
views
Time derivative in Schrödinger equation
In quantum mechanics, a system is descibed by an element $|\psi\rangle\in\mathcal{H}$, where $\mathcal{H}$ is a Hilbert space.
Then on $\mathcal{H}$ (or on a dense subspace of $\mathcal{H}$), we can ...
- 149
4
votes
2
answers
993
views
Why do we consider potential energy function $U(x)$ differentiable?
Recently when skimming through my physics-text I encountered an interesting definition of Force
$$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$
We were taught that some functions are continuous but not ...
- 373
3
votes
1
answer
1k
views
What does $\overset\leftrightarrow{\partial_{\mu}}$ means?
I have a scalar complex field: $\phi(x) = \phi_{1} + i \phi_{2}\;$ so $\;\phi^{*}(x) = \phi_{1} - i \phi_{2}$ where $\phi_{1}, \; \phi_{2}$ are real scalar fields.
Then I have something like $\;\phi^{...
- 83
3
votes
2
answers
5k
views
Derivative of delta function
I am reading and following along the appendices of "The Physical Principles Of The Quantum Theory", and trying to learn how he derives Schrödinger's Equation from his Matrix Mechanics, but I have run ...
- 659
3
votes
1
answer
347
views
How come $\frac{\partial(\partial_{\beta}A_{\gamma})}{\partial(\partial_{\mu}A_{\nu})} = g_{\beta\mu}g_{\gamma\nu}$?
For context, this equation is used in the following (from Schwartz's QFT 3.44)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\...
- 2,022
2
votes
1
answer
420
views
Why does the integral symbol disappear when applying a functional derivative?
it is known that variation is defined by following:
but could anyone tell me why the integral symbol disappears after following functional derivative?
- 76
2
votes
1
answer
355
views
$\nabla$, $\cdot \nabla$, $\nabla \cdot$, $\nabla^2$ - What do they do? [closed]
I'm trying to teach myself Smoothed Particle Hydrodynamics. Unfortunately, my background is in electronics, so the Navier Stokes equations are somewhat alien to me, as is vector calculus. The video I'...
- 114
1
vote
7
answers
293
views
I'm having trouble understanding the intuition behind why $a(x) = v\frac{\mathrm{d}v}{\mathrm{d}x}$ [duplicate]
I was shown
\begin{align}
a(x) &= \frac{\mathrm{d}v}{\mathrm{d}t}\\
&= \frac{\mathrm{d}v}{\mathrm{d}x}\underbrace{\frac{\mathrm{d}x}{\mathrm{d}t}}_{v}\\
&= v\frac{\mathrm{d}v}{\mathrm{d}x}
...
- 339
1
vote
2
answers
3k
views
Leibniz Rule for Covariant derivatives
I recently came across a video by prof Fredrick Schuller on general relativity where he defines the leibniz rule to be,
$\nabla_X (T(\omega,Y))=\nabla_XT(\omega,Y)+T(\nabla_{X} \omega,Y)+T(\omega,\...
- 291
27
votes
3
answers
24k
views
Derivative with respect to a vector is a gradient?
I've encountered in some books (and even completed an exercise from the Goldstein by using it), a strange notation that seems to work exactly like a gradient, I have tried to look for an explanation ...
- 443
17
votes
7
answers
6k
views
What's the difference between average velocity and instantaneous velocity?
Suppose the distance $x$ varies with time as:
$$x = 490t^2.$$
We have to calculate the velocity at $t = 10\ \mathrm s$.
My question is that why can't we just put $t = 10$ in the equation $$x = 490t^2$...
14
votes
4
answers
22k
views
How do you do an integral involving the derivative of a delta function?
I got an integral in solving Schrodinger equation with delta function potential. It looks like
$$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$
I'm trying to solve this by ...
- 325
14
votes
3
answers
1k
views
What is meant by a partial derivative of a ket?
In my QM book I often see partial derivatives mixed with kets, like
$$
\frac{\partial}{\partial a} |\psi \rangle
$$
where $a \in \{x, y, z\}$. Here I'm assuming that $| \psi \rangle \in \mathbb{C}^n$ ...
- 337
12
votes
1
answer
1k
views
Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction
The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$
Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim \...
- 421
12
votes
6
answers
3k
views
Using differentials in physics [duplicate]
I was lately wondering about the use of differentials in physics. I mean, usually $dx$ is thought of as a small increment in $x$, but does this have any rigorous meaning mathematically.
Doubts started ...
- 139
11
votes
5
answers
15k
views
Gradient, divergence and curl with covariant derivatives
I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives.
...
- 312
8
votes
4
answers
1k
views
Struggling understanding definitions with infinitesimal quantities
Many quantities in physics are defined as ratio of infinitesimal quantities. For example: $$\rho(x)=\frac{dm}{dx}$$
or
$$P(t)=\frac{dW}{dt}$$
Are these quantities actually derivatives? I mean if we ...
- 1,688
7
votes
2
answers
5k
views
Meaning of time derivative of an operator
Today when my professor was deriving this equation:
$$\frac{\mathrm d\langle A\rangle}{\mathrm dt}=\frac{i}{\hbar}\langle\left[H,\,A\right]\rangle+\left\langle\frac{\partial A}{\partial t}\right\...
- 2,020
7
votes
2
answers
1k
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A confusion about notation in Goldstein
On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation
$$\dfrac{d}{dt}{\bf p}_i = {\bf F}_i^{(e)...
- 37.4k
7
votes
2
answers
3k
views
Covariant derivative of a covariant derivative
I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$.
This is something I've taken for granted a lot in calculations, namely I though that by the ...
- 623
6
votes
2
answers
5k
views
What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?
I am studying the Navier-Stokes equations and I have the equation in the form:
$$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
5
votes
2
answers
675
views
Is there a way to see that $ \nabla_\mu g_{\nu \rho} = 0 $ without explicit computation, where $\nabla_\mu$ refers to the covariant derivative?
In books, it is usually said that this is a consequence of the fact that parallel transport preserves dot product. How ?
5
votes
2
answers
1k
views
Partial derivatives vs total derivatives in thermodynamics
The specific heat of a system is defined as
$$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}.\tag{1}$$
Sometimes however, I find the same definition, but with total derivatives ...
- 428
4
votes
2
answers
1k
views
Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
4
votes
2
answers
2k
views
Difference tensor between two connections
I am using these supergravity lecture notes by Gary W. Gibbons. On page 18, the author claims that geodesics and autoparallels coincide for a theory with totally antisymmetric torsion, and proves it ...
- 330
4
votes
2
answers
758
views
Ordinary vs. partial derivatives of kets and observables in Dirac formalism
I'm a bit confused as to when ordinary and partial derivatives are used in the Dirac formalism.
In the Schrödinger equation, for instance, Griffiths [3.85] uses ordinary derivatives:
$$ i \hbar \...
- 380
3
votes
2
answers
2k
views
What does $\partial_{\mu}$ mean?
I've stumbled across the following notation a couple times reading physics articles on wikipedia:
$$\partial_{\mu}$$
But what does it mean? They don't clarify.
Source: https://en.wikipedia.org/wiki/...
3
votes
1
answer
504
views
$\delta$ differential notation
Various textbooks that I am currently consulting (including Spacecraft Dynamics and Control An Introduction - Anton H.J. De Ruiter | Christopher J. Damaren | James R. Forbes Section 1.4, page 32) use $...
- 258
3
votes
1
answer
593
views
Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective
I've been reading about connections on fibre bundles recently and it's made me think about the exact nature of the Christoffel symbols in GR.
If we have a vector bundle $E$ over $M$ and put a ...
3
votes
3
answers
2k
views
What is the physical meaning of the Levi-Civita connection?
I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry:
Let $M$ be a manifold with metric $g$. Then exists an unique torsion-free connection $\...
- 1,573
3
votes
1
answer
2k
views
How does one properly define the derivative of one operator-valued function?
In Quantum Mechanics we usually consider operator-valued functions: these are functions that take in real numbers and gives back operators on the Hilbert space of the quantum system.
There are ...
- 37.4k
3
votes
1
answer
767
views
A basic question about how to apply the gauge covariant derivative in Yang-Mills theory
I am sorry if this question is too stupid...
We know that Yang-Mills equation (without source) can be written as
$$D^\mu F_{\mu\nu}=0,\tag{1}$$
where $$D^{\mu}=\partial^\mu-ig A^{\mu}$$
and $$A^\mu=A^...
- 3,741
3
votes
0
answers
98
views
Why do we assume electromagnetic fields to be doubly differentiable? [duplicate]
It seems like the identities of curl of gradient, divergence of curl, and the simple derivations of electromagnetic waves from Maxwell equations all rely on the symmetry (interchangeability of their ...
- 1,169
3
votes
1
answer
1k
views
Erratum in Griffith's Introduction to Electrodynamics
Applying the divergence to Eq. $47$, we obtain
$$ \mathbf{\nabla} \cdot \mathbf{B} = \frac{\mu_{0}}{4\pi} \int \nabla \cdot \left( \mathbf{J} \times \ \frac{\hat{\mathbf{r}}}{r^2}\right) d\tau^{'}. \...
user240696
3
votes
2
answers
5k
views
Feynman's subscript notation
Consider this vector calculus identity:
$$
\mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) \...
- 163
3
votes
9
answers
4k
views
Can velocity be an undefined quantity?
We have the image below displaying the uniform velocity by time-distance graph. At every point velocity is constant but what if distance and time both become zero as at origin in the graph is? The ...
user66452
2
votes
3
answers
814
views
Notation in thermodynamics derivatives
In Yung Kuo Lim's book of exercises in thermodynamics and Stat. Physics I have found more than once the following notation for partial derivatives (ex. 1081 page 79):
$$ \left(\frac{\partial T}{\...
- 175
2
votes
1
answer
2k
views
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:
$$\frac{d}{dt}\langle Q\rangle=\frac{d}{dt}\langle \Psi|\hat Q\Psi\rangle=\langle \frac{\partial\Psi}{\partial t}|\hat Q\...
- 945
2
votes
1
answer
262
views
Exterior Derivative on Curved Manifold (SpaceTime)
Considering the 2-Form Gauge Potential $B_{\mu \nu}$, we can write $dB=H$.
In a flat manifold we have that $H_{\mu \nu \rho}=\partial_{\mu}B_{\nu \rho}+\partial_{\rho}B_{\mu \nu}+\partial_{\nu}B_{\rho ...
- 782
2
votes
1
answer
464
views
What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,703
2
votes
2
answers
320
views
Is the contracted Christoffel symbol a tensor?
The coordinate transformation law (from coordinates x to coordinates y) for the Christoffel symbol is:
$$\Gamma^i_{kl}(y)=\frac{\partial y^i}{\partial x^m} \frac{\partial x^n}{\partial y^k} \frac{\...
- 613
2
votes
2
answers
464
views
Does it make sense to speak in a total derivative of a functional? Part I
I would like to consider the problem of the total derivative of a given functional \begin{equation}
\mathcal{L}\bigg[\phi\big(x,y,z,t\big),\frac{\partial{\phi}}{\partial{x}}\big(x,y,z,t\big),\frac{\...
- 387
2
votes
2
answers
270
views
Does it make sense to speak in a total derivative of a functional? Part II
I am trying to derive the Noether theorem from the following integral action:
\begin{equation}
S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}%
\phi_{r},x\right) , \tag{II.1}\...
- 387