Differentiation is the set of techniques and results from Differential Calculus, concerning the calculation of derivatives of functions or distributions.
Questions tagged [differentiation]
1953 questions
159
votes
9 answers
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 them as independent variables?
grizzly adam
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109
votes
4 answers
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 does not have such a nice form. But I could define…
Sam Jaques
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69
votes
6 answers
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 the given spot?
I can grasp the meaning of…
Džuris
- 3,227
67
votes
4 answers
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 discovered I don't really have an intuitive…
Javier
- 28,811
62
votes
2 answers
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, $\delta$ and $\Delta$ essentially refer to the same…
Yuruk
- 909
60
votes
3 answers
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 total derivative. But physically I cannot…
58
votes
7 answers
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 Lagrangian is a function of the state of a particle,…
Trevor Kafka
- 1,903
51
votes
3 answers
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?
VodkaTampons
49
votes
4 answers
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 ($R^i_{\ jkl}$), Ricci tensor ($R_{ij}$) and Ricci scalar…
Sklivvz
- 13,829
42
votes
3 answers
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 explanation of the notation, which leads to a poor…
1__
- 1,644
41
votes
6 answers
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} $
$$\Gamma ^{\gamma} _{\beta \mu} = \frac{1}{2}…
Aftnix
- 949
38
votes
5 answers
Equivalence between Hamiltonian and Lagrangian Mechanics
I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me.
The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian is defined via the Legendre transform
$$H(p,q,t) =…
Mark
- 559
- 5
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35
votes
2 answers
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 all.
$$\frac{\partial y}{\partial x}, \frac{\delta…
Steeven
- 53,191
34
votes
7 answers
The usage of chain rule in physics
I often see in physics that, we say that we can multiply infinitesimals to use chain rule. For example,
$$ \frac{dv}{dt} = \frac{dv}{dx} \cdot v(t)$$
But, what bothers me about this is that it raises some serious existence questions for me; when we…
Clemens Bartholdy
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32
votes
6 answers
Why are Killing fields relevant in physics?
I'm taking a course on General Relativity and the notes that I'm following define a Killing vector field $X$ as those verifying:
$$\mathcal{L}_Xg~=~ 0.$$
They seem to be very important in physics but I don't understand why yet because that…
S -
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