All Questions
Tagged with differentiation mathematics
95 questions
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Is this mathematically correct that gradient of deformation gradient is equal to deformation gradient?
The deformation matrix is defined as follows, where $x$ is the current location and $X$ is the reference location. It shows the relationship between current $x$s with regard to original $X$s,
$$F = \...
- 21
5
votes
4
answers
386
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Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
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0
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33
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Smoothness (differentiability class) of physical quantities
The concept of differentiability is fundamental to Physics. For instance, already second Newton's law
$$\mathbf{F} = m \frac{\mathrm{d}^2 s}{\mathrm{d}t^2}$$
involves the second derivative of space ...
- 1
1
vote
6
answers
539
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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
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2
answers
94
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Why do I get two different expression for $dV$ by different methods?
So, I was taught that if we have to find the component for a very small change in volume say $dV$ then it is equal to the product of total surface of the object say $s$ and the small thickness say $dr$...
- 45
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0
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92
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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
-2
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1
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213
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Exact definition of divergence. Is it really the dot product of nabla operator with a vector? [closed]
I was trying to understand the derivation for divergence in cylindrical and spherical coordinate system, and I am a bit confused here.
https://www.gradplus.pro/deriving-divergence-in-cylindrical-and-...
- 33
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73
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When can I commute the 4-gradient and the "space-time" integral?
Let's say I have the following situation
$$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$
Would I be able to commute the integral and the partial derivative? If so, why is ...
- 21
1
vote
1
answer
69
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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 ...
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1
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32
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Differentiation of a product of functions
If I have three (vector)functions, all dependent on different (complex)variables:
\begin{equation}
a = X^{\mu_1}(z_1, \bar{z}_1),
b = X^{\mu_2}(z_2, \bar{z}_2),
c= X^{\mu_3}(z_3, \bar{z}_3)
\end{...
- 41
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2
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84
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Solving a PDE using $x-vt$ as a variable
So I was reading this Landau and Lifshitz paper:
https://doi.org/10.1016/B978-0-08-036364-6.50008-9
The article can also be found without a paywall by just searching its title, "On the Theory of ...
1
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3
answers
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 $$r(d\cdot \nabla)\frac{1}{r^3}-d(r\cdot\nabla)\frac{1}{r^3}+\frac{\nabla\times(d\...
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2
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62
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Can the different differentiation notations be equated and do they have an integral definition? [closed]
Are these all equivalent and is there an extension of this to other notation?
Does anyone have a clear and concise chart equating the different notation dialects?
I am also curious if there are more ...
2
votes
1
answer
156
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Dirac Delta applied to the gradient of a function
The supplementary section of a paper I am reading uses the "substitution" property of the Dirac delta function :
$$\mathbf{v}\left(\mathbf{x}_j\right)\delta\left(\mathbf{x}-\mathbf{x}_j\...
- 49
2
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1
answer
74
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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
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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
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3
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410
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Is the derivative of the adjoint the adjoint of the derivative? [closed]
Does $\frac{d}{dx}\langle u|= (\frac{d}{dx}|u\rangle)^\dagger$? Here $|u\rangle$ is any vector in a Hilbert space and $\dagger$ is the adjoint/conjugate-transpose.
It seems to me that this should not ...
1
vote
0
answers
118
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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
$$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
- 343
1
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2
answers
133
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Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
- 535
0
votes
1
answer
47
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Is the ${u_{vs}}=0$ when $P$ is a constant?
In Callen's book Thermodynamics and I (second edition) p.124
To corroborate equation 4.22 show that $${\left( {\frac{{\partial P}}{{\partial s}}} \right)_T} = - {\left( {\frac{{\partial T}}{{\...
- 413
1
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1
answer
170
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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 ...
- 1,254
2
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1
answer
64
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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:
$$
\delta Q=dU+pdV=\bigg(\frac{\...
- 131
2
votes
1
answer
194
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Derivation of Leibniz Rule for Exterior Derivative
I was reading Sean Carrol's GR book, when on page 85 he introduces the Leibniz rule analogue for exterior derivatives:
$$\text d(\omega\wedge\eta) = (\text d\omega)\wedge\eta + (-1)^p\omega\wedge(\...
- 129
1
vote
2
answers
70
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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
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1
answer
65
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"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
4
votes
2
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196
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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
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1
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601
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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
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0
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52
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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
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0
answers
51
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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
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0
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50
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Why area under a curve equals the sum of values of function/quantities taken elementally? [duplicate]
Background:
I was taught basic formulas of differentiation and integration when I started learning physics. However, it wasn't taught through intuitions and concepts. It was like : " Hey these ...
- 325
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0
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44
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Does this particular notation for derivatives imply anything in particular? [duplicate]
In some physics textbooks (and in those of other sciences that use physics, like soil science), I've seen some derivatives written as:
$$\frac{\delta f}{\delta t} $$
Which is a bit strange. Does this ...
- 3,623
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1
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57
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What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
- 85
1
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1
answer
115
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Divergence not defined
I’m currently working on the practice problems in Introduction to Electrodynamics by Griffiths. I got confused by the solution to this problem.
What does “ill-defined divergence” even mean? I ...
- 353
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2
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68
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Why isn't $C_v=\left( \frac{\partial U}{\partial T}\right) _v$ equivalent to $C_v=\left( \frac{\Delta U}{\Delta T}\right) $?
This might be a naive question but I just saw
$$c_v=\frac{1}{n}\left( \frac{\partial U}{\partial T}\right)_V \approx \frac{1}{n}\left( \frac{\Delta U}{\Delta T}\right)$$
Refearing to the LHS as the ...
- 237
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4
answers
1k
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Is it possible to simplify this operator? Special case of Hadamard's formula
Let $D=\frac{d}{dx}$ be the derivative operator and $f(x)$ be a cubic polynomial. Is it possible to simplify the following differential operator? $T=e^{D^2}f(x)e^{-D^2}$.
I tried to use Hadamard's ...
- 213
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1
answer
60
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Discussion about Taylor expansion
I have a function like
$$\mathcal{F}(\phi_{e}(r),\phi_{c}(r),\phi_{a}(r)) = \exp\left({-\beta\left(\phi_{e}(r) + \phi_{c}(r)\cos{\theta}+\phi_{a}(r)\sin{\theta}\right)}\right).$$
If we assume $\beta\...
user316761
1
vote
1
answer
87
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Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform
Initially I asked this question on mathoverflow. I however thought physicists may face this sort of problem more than mathematicians (I am an engineer). Due to that, I decided to ask here as well.
...
- 213
3
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1
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307
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Functional derivatives: Fréchet, Gateaux derivatives and Euler-Lagrange operators
What is the relationship between Fréchet derivatives, Gateaux derivatives and the usual Euler-Lagrange operator (ELO)? Is the ELO the Fréchet derivative, the Gateaux derivative or does it depends on ...
- 6,727
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7
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255
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Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
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1
answer
110
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Taking the second time derivative of a scalar field
Given some scalar field $\phi(x,y,x,t)$, taking its first total derivative we get:
$$\frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\...
- 613
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4
answers
127
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Standing Wave Equation: Why does assuming a small slope $du/dx$ not make $d^2u/dx^2$ negligible as well?
Referencing the above image, just change the label for $y$-axis to $u$-axis.^
Following the derivation of the standing wave equation: https://www.youtube.com/watch?v=IAut5Y-Ns7g&t=1324s
So if ...
- 719
1
vote
2
answers
202
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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 ...
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2
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Explain this equation mathematically
$$\Bigl( \frac{\partial S}{\partial T} \Bigr)_H = \Bigl( \frac{\partial S}{\partial T} \Bigr)_M + \Bigl( \frac{\partial S}{\partial M} \Bigr)_T \Bigl( \frac{\partial M}{\partial T} \Bigr)_H$$
How can ...
- 446
0
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3
answers
79
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Function Values Surrounding Stationary Points
Taylor, in his widely read book "Classical Mechanics," writes on page 218 that
When $df/dx = 0$ at a point $x_0$, but we don't know which of the 3 possibilities obtains, we say that $x_0$ ...
- 463
0
votes
1
answer
162
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Representing $\frac{\mathbf{r-r'}}{|\mathbf{r-r'}|^3}$ in polar coordinates
In his book introduction to electrodynamics, Griffiths uses derives the identity
$$\nabla \cdot \frac{\mathbf{\hat{r}}}{r^2} = 4\pi\delta^3(\mathbf{r})$$
Using the formula for divergence in polar ...
- 323
0
votes
0
answers
80
views
Why is cancellation of differnetial not allowed here?
This is about cancelation of differentials .I am learning basics of tesnor from "Mathematical Methods " by Boas. There I encountered this epression which author says are equal. $$ \frac{\...
- 128
2
votes
0
answers
395
views
Dirac delta function representations in physics
The most common representation of the Dirac delta function in physics is
$$\delta(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \,e^{ikx}.$$
My question is in which sense is it a faithful representation ...
- 578
2
votes
1
answer
58
views
Complex Tensors and Metric [closed]
It has been given that in 2-dimensions, consider the metric: $$\ ds^2 = e^{\phi(z,\overline z)}dzd\overline z .$$
Show that $$\ t_{z...z;z} = (\partial_{z} - n\partial_{z} \phi)t_{z....} .$$
I can't ...
- 23
1
vote
1
answer
41
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Exponentiation of linear combination of commuting Vector fields proof other direction
I have already asked this question, but I forgot to include the if and only if. So the question has already been answered in one direction, but I have to prove the converse as well.
So If
$$e^{a\...
- 1,518
0
votes
1
answer
55
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Exponentiation of linear combination of commuting Vector fields
I have to prove the formula:
$$e^{a\partial/ \partial\lambda +b \partial / \partial\mu}=e^{a\partial/ \partial\lambda}e^{b\partial/ \partial\mu}$$
if $\partial/ \partial\lambda$ and $\partial/ \...
- 1,518