The differentiation tag has no wiki summary.
32
votes
2answers
800 views
What is the meaning of the third derivative printed on this T-shirt?
Don't be a $\frac{d^3x}{dt^3}$
What does it all mean?
14
votes
3answers
1k views
What is the difference between implicit and explicit time dependence e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?
What is the difference between implicit and explicit time dependence e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?
I know one is a partial derivative and the other is a total ...
11
votes
5answers
2k views
Laplace operator's interpretation
What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
9
votes
2answers
247 views
Difference between $\Delta$, $d$ and $\delta$
I have read the thread regarding 'the difference between the operators between $\delta$ and $d$', but it does not answer my question.
I am confused about the notation for change in Physics. In ...
8
votes
1answer
199 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 ...
5
votes
6answers
2k 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 $f(T)$ ...
5
votes
3answers
414 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:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} ...
4
votes
7answers
401 views
Physical intuition for higher order derivatives
Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?
4
votes
1answer
129 views
Do partial derivatives commute on tensors?
For example; is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
4
votes
0answers
92 views
Is it correct to sum over either index of the metric the same way?
I don't know if the following is correct, i want to compute the following derivative
$$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} ...
3
votes
6answers
159 views
Is acceleration $a = s/t^2$, or $a = 2s/t^2$, or something third?
I'm having trouble understanding some of the stuff regarding movement in my introductory physics class (I never thought I'd say that...)
Acceleration is defined as $ a = \frac{s}{t^2}.$
Distance can ...
3
votes
2answers
192 views
What are $\partial_t$ and $\partial^\mu$?
I'm reading the Wikipedia page for the Dirac equation:
$\rho=\phi^*\phi\,$
......
$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$
with the conservation of probability ...
2
votes
1answer
63 views
Partial derivative potential energy of 'free' vibration
I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by:
$$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial ...
1
vote
2answers
611 views
Derivative of the product of operators
I'm asked to show that
$\frac{d(\hat{A}\hat{B})}{d\lambda} = \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$
With $\lambda$ a continuous parameter
Should I use the definition
...
1
vote
2answers
179 views
What does $\textbf{f} = -\boldsymbol{\nabla} u$ mean in practice and how is it computed?
In classical computer simulations such as molecular dynamics (MD) simulations, one integrates Newton's equations of motion to determine particle trajectories. If we think of Newton's Second Law as ...
1
vote
2answers
104 views
Are there general circuits that differentiate/integrate empirically?
Is it possible to construct simple circuits, that given a time-varying input, produce an output that represents the derivative or integral of the input with respect to time?
1
vote
1answer
122 views
$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}$
Please see the next link: http://www3.kis.uni-freiburg.de/~peter/teach/hydro/hydro02.pdf
In (2.13), he used:
$$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf ...
1
vote
1answer
60 views
Arbitrary tensor covariant derivative
what are the rules for performing covariant derivatives on tensors of arbitrary rank?
I found a few examples of Tensor derivatives:
$$\nabla_{c} T^a {}_{b} = \partial_{c}T^a {}_{b}+ \Gamma^a{}_{cd} ...
1
vote
2answers
65 views
What is path of light in the accelerating elevator?
Mathematically, (by mathematically I means by equations) what is path of light in the accelerating elevator?
What is the difference between an ordinary derivative and covariant derivative (which is ...
1
vote
2answers
127 views
Notation for differential operators and wave function math
I know that $[\frac {d^2}{dx^2}]\psi$ is $\frac {d^2\psi}{dx^2}$ but what about this one $[\frac {d^2\psi}{dx^2}]\psi^*$? Is it this like $\frac {d^2\psi\psi^*}{dx^2}$ or this like $\frac ...
1
vote
1answer
142 views
Clarification on a Goldstein formula steps (classical mechanics)
At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
1
vote
0answers
76 views
Implicit Differentiation, A doubt
$v=v_c(\tau, t)$ is a smooth function and suppose we have a relation $y_c(\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes ...
1
vote
2answers
243 views
Derivatives of operators
How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this?
For example: say you had the ...
0
votes
4answers
404 views
Which Schrodinger equation is correct?
In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrodinger equation is
$$H\Psi(x,t) = ...
0
votes
1answer
319 views
How to get the gradient potential in polar coordinate
In polar coordinate,
$$\nabla U = \frac{\partial U}{\partial r}\hat{\mathbf{r}} + \frac{1}{r}\frac{\partial U}{\partial \theta}\hat{\mathbf{\theta}} .$$
Can anyone show me how to get this result?
0
votes
2answers
181 views
Finding an equation for velocity and acceleration
I'm trying to derive an equation for the velocity and acceleration of an object undergoing simple harmonic motion.
I have the equation for displacement: $x = A\sin (2 \pi ft)$
If I differentiate the ...
0
votes
0answers
53 views
Why does the cross derivative of the partition function disappear here?
They state that the chemical potential in a canonical ensemble is given by:
$$\mu = -kT \frac{\partial{\ln Z(N,V,T)}}{\partial{N}} \tag{1}$$
But if I use the definition of chemical partial (which I ...
0
votes
1answer
45 views
Is there any case where one would use, snap, crackle or pop?
As we all know, if you differentiate distance with reference to time, you get speed, and likewise, differentiating speed you get acceleration. However, if you keep differentiating, to the rate of ...
0
votes
0answers
52 views
Definition of frequency domain coordinates
I am using the Fourier Transform in Optics to perform differentiation with a filter by making use of the relation
$\frac {\partial}{\partial x} f(x)=2\pi i \int^{\infty}_{-\infty} u F(u) \exp (2i\pi ...
0
votes
1answer
131 views
How to find the intrinsic covariant derivative component?
How to find the intrinsic covariant derivative component?
In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant ...
0
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0answers
87 views
Nicholas Kollerstrom article on the history of Calculus
Today, Newton´s birthday, I read an article posted in the arXiv by Nicholas Kollerstrom
http://www.arxiv.org/abs/1212.2666
That basically claims that Newton did not invent Calculus. The article does ...
0
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2answers
63 views
What is the common difference between partial time derivative and ordinary time derivative? [duplicate]
What is difference between partial and ordinary time derivative?
for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$?
where the $v$ is velocity.
-1
votes
0answers
55 views
Don't know what this means (Derivative) [migrated]
I was hoping to get a little help here :)
I have this equation:
${{\left. \frac{d}{d\varepsilon }f\left( \varepsilon \right) \right|}_{\varepsilon =\mu }}$
What I'm not sure about is what the ...
-1
votes
1answer
73 views
Elevator acceleration and velocity equation [closed]
I have a small question about the velocity and the acceleration of elevators.
I am looking for equations which is for a constant velocity and acceleration ( for exemple some elevators didn't have a ...
-4
votes
3answers
117 views
How to recognize broken candies from whole ones [closed]
Let's say I have a bag full of sugar candy. Some will be whole, some will be dent, some will be broken (in part, or half, etc).
Let's say I have a device with an input box where I empty the bag, and ...
