All Questions
Tagged with differentiation mathematics
95 questions
15
votes
5
answers
2k
views
What does it mean for a physical quantity if its mixed second partial derivatives are not equal?
This goes for every problem (either in electromagnetism or fluid dynamics) that has to do with vector fields. Say we have a fluid flowing in a closed circular pipe (or an electromagnetic field, the ...
- 7,860
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
13
votes
7
answers
3k
views
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 ...
- 876
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
12
votes
1
answer
2k
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How can I compute the derivative of delta function using its Fourier definition?
I am wondering if it's possible to compute the derivative of the Dirac Delta function using the definition obtained from Fourier transformation: $$\delta(x-x')=\frac{1}{2\pi}\int e^{-ik(x-x')}dk.$$
...
- 185
9
votes
1
answer
601
views
Inverse of the covariant derivative
Given the covariant derivative of some tensor, for the sake of this example a covariant vector:
$$\nabla_\mu A_\nu$$
Is there a well-defined inverse operation on the covariant derivative such that it ...
- 613
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
6
answers
8k
views
How is gradient the maximum rate of change of a function?
Recently I read a book which described about gradient. It says
$${\rm d}T~=~ \nabla T \cdot {\rm d}{\bf r},$$
and suddenly they concluded that $\nabla T$ is the maximum rate of change of $f(T)$ ...
- 872
6
votes
4
answers
1k
views
Is it possible to simplify this operator? Special case of Hadamard's formula
Let $D=\frac{d}{dx}$ be the derivative operator and $f(x)$ be a cubic polynomial. Is it possible to simplify the following differential operator? $T=e^{D^2}f(x)e^{-D^2}$.
I tried to use Hadamard's ...
- 213
6
votes
7
answers
255
views
Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
5
votes
4
answers
386
views
Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
5
votes
1
answer
5k
views
Second derivative of Dirac delta expression
I have come across the expression
$$ \int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$
where the prime represents the derivative.
Usually with derivatives of the Dirac delta distribution I'd partially ...
- 9,197
4
votes
1
answer
167
views
What motivates defining vectors as first order differential operators?
I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
- 73
4
votes
2
answers
196
views
In physical calculations, is the elimination of higher-order small quantities an approximation or a strict equality in mathematics?
Physics sometimes uses a technique called the method of differentials, which seems magical and not very systematic. This makes me unsure which variable I should take the differential of, and sometimes ...
- 119
3
votes
3
answers
278
views
Do there exist motions in which the second derivative of position is not well defined?
After some thought today, I realized that all the power of Newton's laws is fundamentally rooted in the fact that $\frac{d^2 x}{dt^2}$ is a sensible thing to write i.e: there exists a function $x$ ...
- 8,040
3
votes
1
answer
307
views
Functional derivatives: Fréchet, Gateaux derivatives and Euler-Lagrange operators
What is the relationship between Fréchet derivatives, Gateaux derivatives and the usual Euler-Lagrange operator (ELO)? Is the ELO the Fréchet derivative, the Gateaux derivative or does it depends on ...
- 6,727
3
votes
1
answer
74
views
Newton's axioms and collision
Newton's axioms for point particles states that the velocity of a point particle is differentiable. However when two object collide there is a jump in their respective velocities. So is "ideal" ...
3
votes
1
answer
76
views
Derivative with respect to a difference of independent variables
I am dealing with an equation from nonlinear acoustics (Khokhlova-Zabolotskaya-Kuznetsov equation) where a strange term (for me as a mathematician) is used.
The equation looks like this
$$ \frac{\...
- 241
2
votes
4
answers
733
views
Can a particle have no instantaneous velocity at all points of the path taken but a finite average velocity?
I have a question on kinematics.
Say the path traced by a particle is given by a Koch curve or Koch snowflake.
Now consider the particle starts from some arbitrary point $A$ on the curve and ...
- 4,984
2
votes
1
answer
194
views
Derivation of Leibniz Rule for Exterior Derivative
I was reading Sean Carrol's GR book, when on page 85 he introduces the Leibniz rule analogue for exterior derivatives:
$$\text d(\omega\wedge\eta) = (\text d\omega)\wedge\eta + (-1)^p\omega\wedge(\...
- 129
2
votes
1
answer
2k
views
Derivative of tensor product of quantum states
Recently I asked a question over at the math stack exchange:
https://math.stackexchange.com/q/3210375/.
However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
- 569
2
votes
2
answers
1k
views
Is curvature the exterior covariant derivative of the connection?
Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space.
We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
- 258
2
votes
2
answers
464
views
Does it make sense to speak in a total derivative of a functional? Part I
I would like to consider the problem of the total derivative of a given functional \begin{equation}
\mathcal{L}\bigg[\phi\big(x,y,z,t\big),\frac{\partial{\phi}}{\partial{x}}\big(x,y,z,t\big),\frac{\...
- 387
2
votes
2
answers
442
views
How do I find the function derivative $(\delta/\delta \phi) (\partial_\mu \phi)$?
The question is simple: How do I find the function derivative of $$(\delta/\delta \phi(x)) (\partial_\mu \phi(x))~?$$ As far as I can tell, I cannot use any of the standard computational rules for the ...
- 1,420
2
votes
1
answer
58
views
Complex Tensors and Metric [closed]
It has been given that in 2-dimensions, consider the metric: $$\ ds^2 = e^{\phi(z,\overline z)}dzd\overline z .$$
Show that $$\ t_{z...z;z} = (\partial_{z} - n\partial_{z} \phi)t_{z....} .$$
I can't ...
- 23
2
votes
1
answer
156
views
Dirac Delta applied to the gradient of a function
The supplementary section of a paper I am reading uses the "substitution" property of the Dirac delta function :
$$\mathbf{v}\left(\mathbf{x}_j\right)\delta\left(\mathbf{x}-\mathbf{x}_j\...
- 49
2
votes
1
answer
74
views
Closed interval in variation of a field
Let's suppose that $\psi$ is a field so that $$\psi\in\Gamma^\infty(\pi):=\{\psi\in C^\infty(M,E)\ |\ \pi\circ\phi=I_M\}$$ where $E$ is a spacetime bundle over spacetime $M$ and $\pi : E\rightarrow M$ ...
user361492
2
votes
1
answer
127
views
Sum of two state functions is not path independent
I am trying to explain the different physical meanings of the various thermodynamic potentials (before resorting to Legendre transforms, and without making appeals to statistical mechanics) and I ...
- 205
2
votes
1
answer
104
views
What is the meaning of the del operator in this equation?
$$\frac{\partial \left(\rho_m \vec{v}_m \right)}{\partial t} + \nabla \cdot \left(\rho_m \vec{v}_m\vec{v}_m \right) \\ = - \nabla P_m + \nabla \left(\mu_m \nabla \vec{v}_m \right) + \nabla \left(\...
- 55
2
votes
1
answer
535
views
Why is the first derivative of the time-dependent Schrödinger equation continuous? Where does it come from?
I was taught in first year physics that the first derivative of the time-dependent Schrödinger equation had to be continuous. However I was never taught (or at least I don't remember) the reason why.
\...
- 579
2
votes
1
answer
64
views
Expanding state variables and state functions of a thermodynamic system
In this Wikipedia article under the section "Heat capacities of a homogeneous system undergoing different thermodynamic processes" there is on line that says:
$$
\delta Q=dU+pdV=\bigg(\frac{\...
- 131
2
votes
0
answers
395
views
Dirac delta function representations in physics
The most common representation of the Dirac delta function in physics is
$$\delta(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \,e^{ikx}.$$
My question is in which sense is it a faithful representation ...
- 578
2
votes
2
answers
105
views
How to infer what integrals and derivatives signify and when to take them? [closed]
So I have very little background in physics since I'm a mathematical sciences major, but upon being exposed to some physics I've had some difficulties in understanding how to infer the derivatives and ...
- 459
1
vote
6
answers
539
views
Is integration physical, but differentiation is not? [closed]
There are electrical (e.g. analogue computers), and even mechanical (ball-pen) methods to generate the integral of a given function.
On the other hand, naively differentiating a physically given ...
- 177
1
vote
4
answers
420
views
What do $\nabla$ and $\frac{d }{d t}$ mean when they are by themselves?
In QM and QFT, I have seen some equations where they have just the derivative and/or the gradient without specifying what it is acting on.
Taken from wiki.
This does not make sense to me since I ...
- 2,042
1
vote
3
answers
176
views
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 $$r(d\cdot \nabla)\frac{1}{r^3}-d(r\cdot\nabla)\frac{1}{r^3}+\frac{\nabla\times(d\...
- 33
1
vote
2
answers
202
views
Why can I write $\frac{d}{dt}=\frac{d}{dt'}\frac{dt'}{dt}+\frac{d}{dx'}\frac{dx'}{dt}$?
I’m dealing with a Lorentz invariance problem, and in one of the solutions I’ve seen to prove the wave equation the term above was used. However I don’t really understand why it can be written that ...
- 11
1
vote
1
answer
170
views
What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
- 1,254
1
vote
1
answer
107
views
Show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$
Let us start from $\textbf{B}=\nabla \times \textbf{A}$ and write its components $B_k=\epsilon_{ijk}\partial_i A_j$.
I want to show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$. I can ...
- 551
1
vote
2
answers
133
views
Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
- 535
1
vote
1
answer
65
views
"To order $n$ of" arguments
Often one finds in physics textbooks that arguments will be made "to order $n$". I am not sure on what the procedure or argument ought to be when we have some denominator dependence though. ...
- 1,271
1
vote
1
answer
87
views
Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform
Initially I asked this question on mathoverflow. I however thought physicists may face this sort of problem more than mathematicians (I am an engineer). Due to that, I decided to ask here as well.
...
- 213
1
vote
1
answer
41
views
Exponentiation of linear combination of commuting Vector fields proof other direction
I have already asked this question, but I forgot to include the if and only if. So the question has already been answered in one direction, but I have to prove the converse as well.
So If
$$e^{a\...
- 1,518
1
vote
2
answers
153
views
Two total differentials with equal variable differentials. Why coefficients in front of differentials are equal?
Could you prove that inference like that is valid:
$$(1)
\left\{
\begin{array}{c}
dU=T dS-pdV \\
dU=\frac{\partial U}{\partial S}dS+ \frac{\partial U}{\partial V} dV
\end{array}
\right.
\implies
\...
- 321
1
vote
2
answers
70
views
Expressing infinitesimal physical quantities
In physics class, my teacher demonstrated that in polar coordinates, an infinitesimal area involving radial length dr and infinitesimal angle dθ is equal to rdr dθ, since the area is roughly a square ...
- 303
1
vote
0
answers
92
views
A preposterous abuse of notation involving Helmholtz decomposition theorem
Take what I am about to present with a light heart, since the mathmetically inclined may find it too out-of-the-world and devastating.
The above diagram (this is drawn by me, but the original is very ...
- 934
1
vote
1
answer
69
views
Material to Study the Definition, Algebra, and Use of Infinitesimals in Physics [closed]
This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or ...
1
vote
0
answers
118
views
Lie derivative of a one-form
I am going through Nakahara's textbook on geometry and topology in physics. Intuitively, I understand the definition of a lie derivative of a vector field
$$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
- 343
1
vote
0
answers
52
views
What is the relationship between $\partial_x^2\frac1r$ and $\delta^3(r)$? [closed]
We have the equation
\begin{equation}
\nabla^2\frac1r=-4\pi\delta^3(r).
\end{equation}
I first encountered this equation in electrodynamics. So what is $\partial_x^2\frac1r$ then? It looks like the ...
- 41
1
vote
0
answers
51
views
Why differentiation of Fourier operator is difficult? [closed]
I have a question when I read some papers about physics-informed neural networks.
In the paper of physics-informed neural operator, they said "it is non-trivial to compute the derivatives for ...
- 11