All Questions
Tagged with differentiation differentiation or
1,900 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
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
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
65
votes
4
answers
14k
views
Lie derivative vs. covariant derivative in the context of Killing vectors
Let me start by saying that I understand the definitions of the Lie and covariant derivatives, and their fundamental differences (at least I think I do). However, when learning about Killing vectors I ...
- 28.6k
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 ...
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
51
votes
3
answers
38k
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?
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
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
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
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
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
34
votes
7
answers
5k
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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
32
votes
6
answers
8k
views
Why are Killing fields relevant in physics?
I'm taking a course on General Relativity and the notes that I'm following define a Killing vector field $X$ as those verifying:
$$\mathcal{L}_Xg~=~ 0.$$
They seem to be very important in physics ...
- 1,573
27
votes
3
answers
24k
views
Derivative with respect to a vector is a gradient?
I've encountered in some books (and even completed an exercise from the Goldstein by using it), a strange notation that seems to work exactly like a gradient, I have tried to look for an explanation ...
- 443
26
votes
21
answers
5k
views
What happens when a car starts moving? The last moment the car is at rest versus the first moment the car moves
Imagine a car that's at rest and then it starts moving. Consider these two moments:
The last moment the car is at rest.
The first moment the car moves.
The question is: what happens between these 2 ...
- 371
26
votes
4
answers
6k
views
With what velocity are we moving along the time dimension?
Does the question make sense? Velocity along time axis means $v_t=\mathrm dt/\mathrm dt$? If it doesn't, please explain where the flaw is. Taking time as measure like length? Or do we need to ...
- 659
25
votes
3
answers
3k
views
Why don't we see the covariant derivative in classical mechanics?
I am wondering why I have seen the covariant derivative for the first time in general relativity.
Starting from the point that the covariant derivative generalise the concept of derivative in curved ...
- 873
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
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
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
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
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
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 ...
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
17
votes
7
answers
6k
views
What's the difference between average velocity and instantaneous velocity?
Suppose the distance $x$ varies with time as:
$$x = 490t^2.$$
We have to calculate the velocity at $t = 10\ \mathrm s$.
My question is that why can't we just put $t = 10$ in the equation $$x = 490t^2$...
16
votes
5
answers
9k
views
Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)
We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is
$$\nabla^2\frac{q}{r}~...
- 3,802
16
votes
4
answers
6k
views
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:
$$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \...
- 14.8k
15
votes
4
answers
3k
views
Why does the negative sign arise in this thermodynamic relation?
I can't understand why $\left(\frac{\partial P}{\partial V} \right)_T=-\left(\frac{\partial P}{\partial T} \right)_V\left(\frac{\partial T}{\partial V}\right)_P$. Why does the negative sign arise? I ...
- 303
15
votes
3
answers
44k
views
Derive vector gradient in spherical coordinates from first principles
Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient.
I've derived the spherical unit vectors but now I don't understand how to transform ...
- 498
15
votes
4
answers
5k
views
Hamilton equations from Poisson bracket's formulation
Referring to Wikipedia we have that the equation of motion for a $f(q, p, t)$ comes from the formula
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} f(p, q, t) = \frac{\partial f}{\partial q} \frac{\...
- 315
15
votes
4
answers
7k
views
Conserved quantities and total derivatives?
I am having a bit of a crisis in understanding of the physical meanings of total derivatives.
When a quantity $\rho$ (be it a vector or a scalar) is said to be conserved, then (mathematically) $$\...
- 25.3k
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:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\...
- 2,526
15
votes
2
answers
4k
views
Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?
Why is the following equation true?
$$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$$
where $\mathbf{v}_i$ is velocity, $\mathbf{r}_i$ is the ...
- 1,483
15
votes
5
answers
2k
views
What does it mean for a physical quantity if its mixed second partial derivatives are not equal?
This goes for every problem (either in electromagnetism or fluid dynamics) that has to do with vector fields. Say we have a fluid flowing in a closed circular pipe (or an electromagnetic field, the ...
- 7,860
14
votes
4
answers
22k
views
How do you do an integral involving the derivative of a delta function?
I got an integral in solving Schrodinger equation with delta function potential. It looks like
$$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$
I'm trying to solve this by ...
- 325
14
votes
3
answers
1k
views
What is meant by a partial derivative of a ket?
In my QM book I often see partial derivatives mixed with kets, like
$$
\frac{\partial}{\partial a} |\psi \rangle
$$
where $a \in \{x, y, z\}$. Here I'm assuming that $| \psi \rangle \in \mathbb{C}^n$ ...
- 337
14
votes
3
answers
13k
views
How does uncertainty/error propagate with differentiation?
I have a noisy temperature (T) vs. time (t) measurement and I want to calculate dT/dt. If I approximate $dT/dt = \Delta T/\Delta t$ then the noise in the derivative gets too high and the derivative ...
- 566
13
votes
7
answers
3k
views
Can we divide a vector by another vector? How about this: $a = vdv/dx?$
My physics teacher told us that we can’t divide vectors, that vector division has no physical meaning or significance. How about this: $$a = vdv/dx.$$
It says acceleration vector equals velocity (as ...
- 876
13
votes
3
answers
2k
views
Why, when going from special to general relativity, do we just replace partial derivatives with covariant derivatives?
I've come across several references to the idea that to upgrade a law of physics to general relativity all you have to do is replace any partial derivatives with covariant derivatives.
I understand ...
- 1,953
13
votes
1
answer
2k
views
Geometric meaning of spin connection
A very short question: Does the spin connection that we encounter in General Relativity $$\omega_{\mu,ab}$$ have a geometric meaning? I am supposing it does because it comes from mathematical terms ...
- 1,539
13
votes
3
answers
5k
views
Bianchi identity of a non-Abelian gauge theory?
How can one prove the Bianchi identity of a non-Abelian gauge theory? i.e.
$$
\epsilon^{\mu \nu \lambda \sigma}(D_{\nu}F_{\lambda \sigma})^a=0
$$
- 139
13
votes
1
answer
836
views
A formal procedure for thermodynamic relations
This is my third time taking a thermodynamics course (two in undergrad, one in grad), and I've finally become frustrated enough about something to post on here.
A lot of thermodynamic questions want ...
- 3,009
12
votes
6
answers
3k
views
Using differentials in physics [duplicate]
I was lately wondering about the use of differentials in physics. I mean, usually $dx$ is thought of as a small increment in $x$, but does this have any rigorous meaning mathematically.
Doubts started ...
- 139
12
votes
1
answer
2k
views
How can I compute the derivative of delta function using its Fourier definition?
I am wondering if it's possible to compute the derivative of the Dirac Delta function using the definition obtained from Fourier transformation: $$\delta(x-x')=\frac{1}{2\pi}\int e^{-ik(x-x')}dk.$$
...
- 185
12
votes
2
answers
3k
views
Do contravariant and covariant partial derivatives commute in GR?
I'm considering something like this: $\partial_{\mu}\partial^{\nu}A$ . I feel like we should be able to commute the derivatives so: $\partial_{\mu}\partial^{\nu}A = \partial^{\nu}\partial_{\mu}A$.
...
user163020
12
votes
1
answer
1k
views
Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction
The Schrodinger equation in Hilbert space is expressed as : $$\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
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
11
votes
4
answers
3k
views
When the direction of a movement changes, is the object at rest at some time?
The question I asked was disputed amongst XVIIe century physicists (at least before the invention of calculus).
Reference: Spinoza, Principles of Descartes' philosophy ( Part II: Descartes' Physics, ...
user229097