All Questions
Tagged with differentiation mathematical-physics
15 questions
16
votes
5
answers
9k
views
Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)
We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is
$$\nabla^2\frac{q}{r}~...
- 3,802
6
votes
5
answers
4k
views
What is the meaning of following expression $C=\frac{\delta Q}{dT}$ mathematically?
Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics):
Many text books (even Wikipedia) writes wrong expressions (from ...
- 2,101
8
votes
1
answer
712
views
When motion begins, do objects go through an infinite number of position derivatives?
This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
- 83
11
votes
2
answers
1k
views
The derivation of fractional equations
Recently I saw some physical problems that can be modeled by equations with fractional derivatives, and I had some doubts: is it possible to write an action that results in an equation with fractional ...
- 1,247
12
votes
1
answer
900
views
Can You Obtain New Physics from the use of Fractional Derivatives?
I was curious if anyone could give me an example of the use of fractional derivatives in physics and explain what they offer that "conventional" mathematics does not (in terms of new physics and not ...
- 808
3
votes
2
answers
814
views
D'Alembertian of a Dirac delta function of a spacetime interval (i.e. with support on the 3+1D light-cone)
How one differentiates a delta-function of a spacetime interval? Namely,
$$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$
Somewhere I saw that the result ...
4
votes
2
answers
993
views
Why do we consider potential energy function $U(x)$ differentiable?
Recently when skimming through my physics-text I encountered an interesting definition of Force
$$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$
We were taught that some functions are continuous but not ...
- 373
12
votes
1
answer
1k
views
Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction
The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$
Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim \...
- 421
3
votes
1
answer
2k
views
How does one properly define the derivative of one operator-valued function?
In Quantum Mechanics we usually consider operator-valued functions: these are functions that take in real numbers and gives back operators on the Hilbert space of the quantum system.
There are ...
- 37.4k
2
votes
2
answers
270
views
Does it make sense to speak in a total derivative of a functional? Part II
I am trying to derive the Noether theorem from the following integral action:
\begin{equation}
S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}%
\phi_{r},x\right) , \tag{II.1}\...
- 387
18
votes
1
answer
3k
views
Is there a "covariant derivative" for conformal transformation?
A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$:
$$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$
It's fairly ...
15
votes
4
answers
3k
views
What is the relation between (physicists) functional derivatives and Fréchet derivatives
I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\...
- 2,526
10
votes
1
answer
806
views
Physical intuition/interpretation of fractional derivatives/integrals?
Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them:
Velocity is the derivative of position
...
- 421
6
votes
1
answer
2k
views
Functional derivative commutes with total derivative
I have a question about a rule from the calculus of variations.
Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
- 1,555
6
votes
2
answers
2k
views
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