All Questions
Tagged with differentiation differentiation or
1,900 questions
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Which quantity gives the resistance of a component?
In a current vs potential difference graph, we can obtain the value of the resistance of the component. There are books that say gradient-inverse is the resistance and also books that say the value of ...
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Covariant derivative-Differential
I was trying to prove that the derivative-four vector are covariant. This can be proved only if you consider the time and space derivatives to be
$\dfrac{\partial}{\partial t^\prime}=\dfrac{\partial}{...
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Difference between $\Delta$, $d$ and $\delta$
I have read the thread regarding 'the difference between the operators $\delta$ and $d$', but it does not answer my question.
I am confused about the notation for change in Physics. In Mathematics, $\...
- 899
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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:
$$\nabla_{c} T^a {}_{b} = \partial_{c}T^a {}_{b}+ \Gamma^a{}_{cd} T^...
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Why does the cross derivative of the partition function disappear here?
They state that the chemical potential in a canonical ensemble is given by:
$$\mu = -kT \frac{\partial{\ln Z(N,V,T)}}{\partial{N}} \tag{1}$$
But if I use the definition of chemical partial (which I ...
- 1,271
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Is there any case where one would use, snap, crackle or pop? [duplicate]
As we all know, if you differentiate distance with reference to time, you get speed, and likewise, differentiating speed you get acceleration. However, if you keep differentiating, to the rate of ...
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Is there a "covariant derivative" for conformal transformation?
A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$:
$$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$
It's fairly ...
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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.
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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 ...
- 11
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The role of the affine connection the geodesic equation
I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is ...
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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?
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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 ...
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A formal procedure for thermodynamic relations
This is my third time taking a thermodynamics course (two in undergrad, one in grad), and I've finally become frustrated enough about something to post on here.
A lot of thermodynamic questions want ...
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$\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 ...
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3
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Bianchi identity of a non-Abelian gauge theory?
How can one prove the Bianchi identity of a non-Abelian gauge theory? i.e.
$$
\epsilon^{\mu \nu \lambda \sigma}(D_{\nu}F_{\lambda \sigma})^a=0
$$
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Do derivatives of operators act on the operator itself or are they "added to the tail" 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 ...
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How is the second-order covariant derivative of a scalar computed?
What is second-order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric takes the form $$ds^2=dr^2+g(r)d\theta^2$$ and $f$ is a ...
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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:
$$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial x_i\...
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Do partial derivatives commute on tensors?
Do partial derivatives commute on tensors? For example, is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
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Differentiation and delta function
Need help doing this simple differentiation.
Consider 4 d Euclidean(or Minkowskian) spacetime.
\begin{equation}
\partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ?
\end{equation}
where $a_\mu$ is a constant ...
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Are there general circuits that differentiate/integrate empirically?
Is it possible to construct simple circuits, that given a time-varying input, produce an output that represents the derivative or integral of the input with respect to time?
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Can one raise indices on covariant derivative and products thereof?
Can the following be true?
$g^{\sigma\rho}\nabla_{\rho}\nabla_{\mu} = \nabla^{\sigma}\nabla_{\mu}$
$g^{\sigma\rho}\nabla_{\nu}\nabla_{\sigma} = \nabla_{\nu}\nabla^{\rho}$
$g^{\sigma\rho}\nabla_{\nu}\...
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How to find the intrinsic covariant derivative component?
How to find the intrinsic covariant derivative component?
In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant ...
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Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta}...
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Finding an equation for velocity and acceleration [closed]
I'm trying to derive an equation for the velocity and acceleration of an object undergoing simple harmonic motion.
I have the equation for displacement: $x = A\sin (2 \pi ft)$
If I differentiate the ...
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2
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Notation for differential operators and wave function math
I know that $[\frac {d^2}{dx^2}]\psi$ is $\frac {d^2\psi}{dx^2}$ but what about this one $[\frac {d^2\psi}{dx^2}]\psi^*$? Is it this like $\frac {d^2\psi\psi^*}{dx^2}$ or this like $\frac {\psi^*d^2\...
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Physical intuition for higher order derivatives
Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?
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Clarification on a Goldstein formula steps (classical mechanics)
At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
- 1,143
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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Which Schrödinger equation is correct? [duplicate]
In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is
$$H\Psi(x,t) = i\hbar\frac{\partial}{\partial ...
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What does $\textbf{f} = -\boldsymbol{\nabla} u$ mean in practice and how is it computed?
In classical computer simulations such as molecular dynamics (MD) simulations, one integrates Newton's equations of motion to determine particle trajectories. If we think of Newton's Second Law as ...
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What is the relation between (physicists) functional derivatives and Fréchet derivatives
I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\...
- 2,526
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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 ...
- 3,217
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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 $f(T)$ ...
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Derivatives of Dirac delta function and equation of continuity for a single charge
For a single charge $e$ with position vector $\textbf R$, the charge density $\rho$ and and current density $\textbf{j}$ are given by:
\begin{equation} \rho(\textbf{r},t)= e\,\delta^3(r-\textbf{R}(t))...
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What's the physical meaning of vector Laplacian of Electric field intensity?
Could someone explain to me the physical meaning of vector Laplacian of Electric field intensity?
Where vector Laplacian means: $$\nabla^2 \mathbf{E} = \nabla(\nabla \cdot \mathbf{E}) - \nabla \times ...
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Derivative of the product of operators and Derivative of exponential
I'm asked to show that
$$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$$
With $\lambda$ a continuous parameter. Should I use the ...
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Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?
Why is the following equation true?
$$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$$
where $\mathbf{v}_i$ is velocity, $\mathbf{r}_i$ is the ...
- 1,483
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Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)
We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is
$$\nabla^2\frac{q}{r}~...
- 3,802
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What is the difference between implicit, explicit, and total time dependence, e.g. $\frac{\partial \rho}{\partial t}$ and $\frac{d \rho} {dt}$?
What is the difference between implicit, explicit, and total 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 ...
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4
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Covariant derivative for spinor fields
scalars (spin-0) derivatives is expressed as:
$$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}.$$
vector (spin-1) derivatives are expressed as:
$$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \...
- 14.8k
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Time evolution in quantum mechanics
We know that an operator A in quantum mechanics has time evolution given by Heisenberg equation:
$$
\frac{i}{\hbar}[H,A]+\frac{\partial A}{\partial t}=\frac{d A}{d t}
$$
Can we derive from this ...
- 1,434
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2
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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
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5
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How do I calculate the perturbations to the metric determinant?
I am trying to calculate $\sqrt{-g}$ in terms of a background metric and metric perturbations, to second order in the perturbations. I know how to expand tensors that depend on the metric, but I don't ...
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4
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What is the physical meaning of the connection and the curvature tensor?
Regarding general relativity:
What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)?
What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
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9
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Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
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