All Questions
Tagged with differentiation differentiation or
259 questions
154
votes
9
answers
19k
views
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 ...
- 2,235
38
votes
5
answers
9k
views
Equivalence between Hamiltonian and Lagrangian Mechanics
I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me.
The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the ...
- 559
61
votes
2
answers
98k
views
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
60
votes
3
answers
28k
views
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 ...
35
votes
2
answers
5k
views
Symbols of derivatives
What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for ...
- 52.4k
18
votes
2
answers
16k
views
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 ...
- 1,156
25
votes
2
answers
2k
views
Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are ...
- 281
10
votes
3
answers
14k
views
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 ...
- 2,937
47
votes
4
answers
16k
views
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_{\ ...
- 13.7k
15
votes
2
answers
4k
views
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
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
$$\nabla^2\frac{q}{r}~...
- 3,802
39
votes
5
answers
47k
views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta}...
- 929
20
votes
9
answers
5k
views
Why is small work done always taken as $dW=F \cdot dx$ and not $dW=x \cdot dF$?
I was reading the first law of thermodynamics when it struck me. We haven't been taught differentiation but still, we find it in our chemistry books. Why is small work done always taken as $dW=F \cdot ...
- 339
8
votes
3
answers
3k
views
Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?
The Euler-Lagrange equation for particles is given by
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$
and for fields it is
$$ \partial_{\mu} \frac{\...
- 4,064
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
5
votes
1
answer
13k
views
Divergence of Electric Field Due to a Point Charge [duplicate]
I am trying to formally learn electrodynamics on my own (I only took an introductory course). I have come across the differential form of Gauss's Law.
$$ \nabla \cdot \mathbf E = \frac {\rho}{\...
- 504
68
votes
6
answers
48k
views
Laplace operator's interpretation
What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
- 3,217
13
votes
1
answer
2k
views
Geometric meaning of spin connection
A very short question: Does the spin connection that we encounter in General Relativity $$\omega_{\mu,ab}$$ have a geometric meaning? I am supposing it does because it comes from mathematical terms ...
- 1,539
10
votes
6
answers
3k
views
Physical intuition for higher order derivatives
Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?
- 202
6
votes
2
answers
4k
views
Advection operator
How are exactly $u_j\partial_ju_i$ and $u_i\partial_j u_i$ related?
And what is their relation to ($\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\boldsymbol{u}\cdot(\nabla\boldsymbol{u})$ ?
I ask ...
- 1,813
34
votes
7
answers
5k
views
The usage of chain rule in physics
I often see in physics that, we say that we can multiply infinitesimals to use chain rule. For example,
$$ \frac{dv}{dt} = \frac{dv}{dx} \cdot v(t)$$
But, what bothers me about this is that it raises ...
- 8,040
24
votes
4
answers
14k
views
Why is the covariant derivative of the determinant of the metric zero?
This question, metric determinant and its partial and covariant derivative,
seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
- 749
8
votes
1
answer
712
views
When motion begins, do objects go through an infinite number of position derivatives?
This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
- 83
6
votes
2
answers
2k
views
Derivative with respect to a spinor of the free Dirac lagrangian
When we derived Dirac Equation starting form the lagrangian, our QFT professor said:
"let's take the free lagrangian $$\mathscr L = i\bar\Psi\gamma^\mu\partial_\mu\Psi - m\bar\Psi\Psi$$ and perform
$...
- 523
3
votes
1
answer
670
views
Physical interpretation of total derivative
Can I get some help interpreting the following?
"Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call ...
- 1,081
2
votes
4
answers
793
views
Why do we use different differential notation for heat and work?
Just recently started studying Thermodynamics, and I am confused by something we were told, I understand we use the inexact differential notation because work and heat are not state functions, but we ...
- 283
20
votes
3
answers
22k
views
D'Alembertian for a scalar field
I have read that the D'Alembertian for a scalar field is
$$
\Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu).
$$
Exactly when is this correct? Only for $...
- 15.3k
11
votes
2
answers
1k
views
The derivation of fractional equations
Recently I saw some physical problems that can be modeled by equations with fractional derivatives, and I had some doubts: is it possible to write an action that results in an equation with fractional ...
- 1,247
5
votes
5
answers
7k
views
What is divergence?
What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
- 1,660
5
votes
4
answers
5k
views
How can there be really any instantaneous velocity?
I have read about Zeno's arrow paradox that tells us there is no motion of the arrow at a particular instant of its flight. It can be inferred that there can be no velocity at any instant. Moreover we ...
user36790
1
vote
4
answers
407
views
Rotation systems. Problem interpreting an equation
In this equation:
$$
\mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf v}{dt}\right)_i=\left[\left(\frac{d}{dt}\right)_r+\boldsymbol\Omega\times\right]\left[...
- 1,629
57
votes
7
answers
10k
views
Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,883
42
votes
3
answers
4k
views
Partial derivative notation in thermodynamics
Most thermodynamics textbooks introduce a notation for partial derivatives that seems redundant to students who have already studied multivariable calculus. Moreover, the authors do not dwell on the ...
- 1,634
15
votes
4
answers
3k
views
Why does the negative sign arise in this thermodynamic relation?
I can't understand why $\left(\frac{\partial P}{\partial V} \right)_T=-\left(\frac{\partial P}{\partial T} \right)_V\left(\frac{\partial T}{\partial V}\right)_P$. Why does the negative sign arise? I ...
- 303
15
votes
4
answers
7k
views
Conserved quantities and total derivatives?
I am having a bit of a crisis in understanding of the physical meanings of total derivatives.
When a quantity $\rho$ (be it a vector or a scalar) is said to be conserved, then (mathematically) $$\...
- 25.3k
12
votes
1
answer
900
views
Can You Obtain New Physics from the use of Fractional Derivatives?
I was curious if anyone could give me an example of the use of fractional derivatives in physics and explain what they offer that "conventional" mathematics does not (in terms of new physics and not ...
- 808
9
votes
4
answers
5k
views
Hermitian adjoint of 4-gradient in Dirac equation
I'm having issues deriving the Dirac adjoint equation, $$\overline{\psi}(i\gamma^{\mu}\partial_{\mu}+m)=0.\tag{1}$$
I started by taking the Hermitian adjoint of all components of the original Dirac ...
- 401
9
votes
5
answers
7k
views
How to interpret the derivative of the Dirac delta potential?
I met a Hamiltonian containing the derivative of the Dirac delta potential:
In order to do it we use a method described in [9]. We define a formal Hamiltonian
$$
\tag{2}\tilde{H}_{abcd}=-\frac{{\...
- 4,152
6
votes
2
answers
2k
views
Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$
If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$.
In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
- 27.2k
5
votes
1
answer
1k
views
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
4
votes
2
answers
293
views
Do integrals of position make any sense? Do they have an application? [closed]
I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
- 41
4
votes
2
answers
1k
views
Error analysis via two different methods
We have a quantity $a$ expressed in terms of two quantities $b $ and $c$ as $a = b/c$.
It seems to me that there are two ways of estimating the error on $a$, the "physics" ...
- 109
4
votes
3
answers
568
views
Notation confusion about time derivative of a vector in a rotating frame
As far as I can tell, this question, or similar ones, have been asked a number of times:
Derivation of the time-derivative in a rotating frame of refrence
Time derivatives in a rotating frame of ...
- 860
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 ...
2
votes
1
answer
172
views
How does instantaneous velocity cause displacement in just one point? [closed]
I have a question.
Falling object graph is curve shape right?
And instantaneous velocity is tangent line but how does this velocity make displacement in distance? Because suppose instantaneous ...
- 311
2
votes
4
answers
5k
views
What is the physical significance of curl $\nabla\times\boldsymbol{V}$?
What is the physical significance of curl $$\nabla\times\boldsymbol{V}~?$$ I mean I read 'curl V represents the rotation of the vector $V$. My question what is it about the term $\nabla\times\...
- 300
0
votes
1
answer
483
views
Least action principle : is $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $ always true?
(Just some recalls)
We have an action on which we want to apply Least action principle.
$$ S=\int_{t_i}^{t_f} L(q,\dot{q},t)dt$$
We assume that $t \mapsto q(t)$ is the function that will extremise ...
- 1,560
107
votes
4
answers
11k
views
Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
- 1,357
24
votes
5
answers
16k
views
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 ...
- 541
11
votes
1
answer
1k
views
Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?
If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$
with $g\in G$.
Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = (\partial_{\mu}+A_{\mu})\...
- 25.3k