All Questions
Tagged with calculus differentiation
317 questions
-2
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0
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Use of $dv/ds$ in defining acceleration [duplicate]
We can write acceleration as either
$dv/dt$ or $v dv/ds$.
And surprisingly the work-energy theorem arrives from the second definition.
I feel it would be fundamentally understanding towards work ...
3
votes
1
answer
479
views
Second derivative of unit vector
We know that the second derivative of unit vector (the vector from a point toward the source) is proportional to the Electric field caused by the source in a particular point.
If we imagine that our ...
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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
1
vote
1
answer
481
views
Doubt in Verlet's Algorithm
In studying the temporal evolution of a system according to the deterministic model, we begin by considering a Taylor series expansion for the displacement $r$. First, we consider a positive variation ...
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2
votes
1
answer
96
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Why take the derivative of variables such as area, mass, and radius?
I'm taking a module on stars and the solar system; I've attached notes from our first lecture- hydrostatic equilibrium. I'm confused about the notation $\mathrm{d}$ for $\mathrm{d}A, \, \mathrm{d}r$, ...
1
vote
2
answers
44
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Perfect gas relation in differential form [closed]
I have a problem to understand the transformation of the perfect gas relation:
$$ \rho\cdot R\cdot T = P $$
into its differential form:
$$\frac {dp}{p} = \frac {d{\rho}}{\rho} + \frac {d{T}}{T}$$
How ...
- 21
0
votes
0
answers
33
views
Smoothness (differentiability class) of physical quantities
The concept of differentiability is fundamental to Physics. For instance, already second Newton's law
$$\mathbf{F} = m \frac{\mathrm{d}^2 s}{\mathrm{d}t^2}$$
involves the second derivative of space ...
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-2
votes
1
answer
84
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Where did $1/2$ of this come from? [duplicate]
Work done by an external force $F$ upon a particle displacing from point 1 to point 2 is defined as
$$
W_{12} = \int_1^2 F \cdot dr
\, .$$
Kinetic energy and work-energy theorem: According to Newton's ...
1
vote
0
answers
85
views
Equations with fractional derivatives
Assume we have an equation which represents the flux of some quantity as $q = -D \dfrac{\partial T}{\partial x}$ (Eqn. 1), where the diffusion coefficient $D$, variable $T$, $x$ and $q$ have some ...
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0
votes
1
answer
90
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Derivative of the product of a scalar function and a vector valued function
According to Berkeley Physics Course, Volume 1 Mechanics,
The time derivative of a vector valued function can be derived from the formula:
$$
\mathbf{r}(t) = r(t)\mathbf{\hat{r}}(t)
$$
where the ...
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1
vote
2
answers
120
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Differential form of Planck's Distribution Law interpretation
So I didn't encounter differentials that often until now, I was taught that the seperate parts of $dy/dx$ for example are not supposed to have any sort of independent existence - ok.
(Calculus, 4th ...
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0
votes
1
answer
53
views
Why is linear charge density $dq/dl$ and not $q/l$?
If linear charge density is charge per unit length then shouldn't it be $q/l$. Why is it $dq/dl$ instead? Wouldn't that mean it is only being calculated for a small element and not the whole length?
1
vote
1
answer
117
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How to "rectify" this wave-equation derivation for Longitudinal waves?
To derive the differential equation for longitudinal waves, my professor proceeded like this:
We are using the concept of $N$-coupled oscillators. Consider a slab of length $l$ and cross sectional ...
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1
vote
6
answers
539
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Is integration physical, but differentiation is not? [closed]
There are electrical (e.g. analogue computers), and even mechanical (ball-pen) methods to generate the integral of a given function.
On the other hand, naively differentiating a physically given ...
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2
votes
1
answer
247
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How to calculate jerk in uniform circular motion?
We can calculate the centripetal acceleration in circular motion by the equation v^2/r. But how do we calculate the jerk (which is acceleration over time)?
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0
votes
2
answers
81
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What is $F$ and what is $P$ in the sentence "$F × dP$ is a total differential"?
The physicist Emilio Segrè, as a student, attended lessons of Calculus given by Francesco Severi and of Analytical Mechanics given by Tullio Levi-Civita. Segrè wrote in his autobiography1
For many ...
1
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1
answer
76
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How do force and mass work with all derivatives of position?
I think if $F(t) = kt^0$ then $$x(t) = x_0 + v_0t + \frac{k}{m}\frac{t^2}{2!},$$ and if $F(t) = kt^1$ then $$x(t) = x_0 + v_0t + \frac{k}{m} \frac{t^2}{2!} + \frac{k}{m} \frac{t^3}{3!},$$ and so on, ...
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7
votes
3
answers
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
0
votes
1
answer
134
views
Differential form of Lorentz equations
A Lorentz transformation for a boost in the $x$ direction ($S'$ moves in $+x$, $v>0$) is given by:
$$ t'=\gamma\left(t-v\frac{x}{c^2}\right),~x'=\gamma(x-vt)$$
In the derivation of the addition of ...
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0
votes
2
answers
94
views
Why do I get two different expression for $dV$ by different methods?
So, I was taught that if we have to find the component for a very small change in volume say $dV$ then it is equal to the product of total surface of the object say $s$ and the small thickness say $dr$...
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0
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7
answers
104
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How does the result of derivative become different from average ratio calculation?
Lets give an example. Velocity, $v=ds/dt$. If we know the value of $s$ (displacement) and $t$ (time), we can instantly find the value of $v$. But then this $v$ will be the average velocity.
Now ...
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-1
votes
2
answers
80
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Problem with resources, Walter Lewin's third lecture
I've watched Walter's third lecture in 8.01 and I have a small problem with the last part, where he says that $$\vec r_t=x_t\cdot \hat x\ +\ y_t\cdot \hat y\ +\ z_t\cdot \hat z \\ \vec v_t=\frac{d\vec ...
0
votes
0
answers
73
views
When can I commute the 4-gradient and the "space-time" integral?
Let's say I have the following situation
$$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$
Would I be able to commute the integral and the partial derivative? If so, why is ...
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1
vote
1
answer
69
views
Material to Study the Definition, Algebra, and Use of Infinitesimals in Physics [closed]
This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or ...
0
votes
1
answer
32
views
Differentiation of a product of functions
If I have three (vector)functions, all dependent on different (complex)variables:
\begin{equation}
a = X^{\mu_1}(z_1, \bar{z}_1),
b = X^{\mu_2}(z_2, \bar{z}_2),
c= X^{\mu_3}(z_3, \bar{z}_3)
\end{...
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0
votes
1
answer
78
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What does this equation for density mean?
What does this equation for density mean?
$$\rho = \lim_{\Delta V\to\varepsilon^3} \ \frac{\Delta m}{\Delta V}$$
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0
votes
1
answer
36
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Space-for-time Derivative Substitution in Solving for Elliptical Orbit
I am currently working on a simulation of the Newton's Cannonball thought experiment, in which a stone is launched horizontally from atop a tall mountain at a high speed (in the absence of air) and ...
0
votes
0
answers
56
views
Partial derivative operator
It's mentioned in this paper that if $\partial^i \partial^j$ applied on an equation like:
$$
x \delta_{ij} + (\nabla^2 \delta_{ij}- \partial_i \partial_j) y =0
$$
It yields a couple of equations:
$$
...
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0
votes
2
answers
234
views
Why is the differential form of Gauss's Law equivalent to the integral form?
I can understand the Differential form of Gauss's Law ∇⋅𝐄= $\frac{ρ}{ɛ_0}$
as saying that the source of electric field vectors or flow disperse(The divergence of the electric field) is equal to the ...
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0
votes
1
answer
89
views
In $a = dv/dt$, is $a$ the net acceleration? [closed]
While going through the calculus approach to accelerate, we have,
$$a = dv/dt, $$
I think, here, v and a should be in the same axis,
is my process correct?
in a planar motion in two dimensions, it ...
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-1
votes
3
answers
96
views
Proof that small change in temperature leads to small change in entropy
I have been trying to find a mathematical proof (or even from a reliable source) which verifies that/proves that:
A small change in temperature leads to a small change in entropy.
However, I was ...
1
vote
1
answer
142
views
First law of thermodynamics: Can we always speak in terms of infinitesimal changes?
While reading lecture notes for the course on thermodynamics I have encountered some tiny details that seem extremely important for the understanding of the topic. However, something seems amiss so, I ...
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0
votes
1
answer
105
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Derivation of the state equation of a van der Waals gas. Can I invert the derivative to help me?
The state equation of a van der Waals gas is
$$\left(P+\frac{a}{v^2}\right)(v-b)=RT$$
with $a,b$ and $R$ constant. Find $$\frac{\partial v}{\partial T}\bigg\rvert_P.$$
Finding $\frac{\partial v}{\...
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1
vote
2
answers
142
views
Average velocity showing different results
I was solving a question, in which, a particle has travelled a distance $s$, with initial velocity $0$ and constant acceleration.
So the equation of motion becomes,
$$ v = a t \tag{1} $$
and
$$ v = \...
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0
votes
1
answer
98
views
Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
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1
vote
3
answers
293
views
Are we allowed to cancel the units of a derivative?
Since the volume of a sphere $v(r)=\frac{4}{3} \pi r^{3} \left[m^{3}\right]$, its derivative relative to the radius is:
$$
\frac{dv}{dr} =4\pi r^{2} \left[\frac{m^{3}}{m}\right]
$$
Which is also a ...
-2
votes
2
answers
62
views
Can the different differentiation notations be equated and do they have an integral definition? [closed]
Are these all equivalent and is there an extension of this to other notation?
Does anyone have a clear and concise chart equating the different notation dialects?
I am also curious if there are more ...
1
vote
2
answers
71
views
How to calculate wave equation from a stretched string?
I am reading "Introduction to Electrodynamics" [Griffiths] and in section 9.1.1, there is an explanation for why a stretched string supports wave motion. It begins as follows:
It identifies ...
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votes
3
answers
96
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Why is it wrong to find centripetal acceleration using change of velocity over change of time?
This question asks to find the centripetal acceleration by giving the initial and final velocity over the change of time.
As shown, my book combined two rules to find the acceleration. I utterly ...
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0
votes
0
answers
50
views
Laplace transform: How to evaluate partial derivative in the denominator of a fraction?
I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
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-5
votes
1
answer
141
views
Why is the regular Taylor expansion so similar to the construction of the Hamiltonian from a Lagrangian/Legendre transform?
An example of a first order Taylor expansion of a function with two variables is given by:
$$f\left(x,y\right)=f\left(x_0,y_0\right)+\left(x-x_0\right)\frac{\partial f}{\partial x}+\left(y-y_0\right)\...
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5
votes
5
answers
442
views
Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?
Hello fellow physicists,
I was trying to understand some behavior on rotating objects, specifically about the formula $\vec{v} = \vec{\omega} \times \vec{r}$.
The Book (Marion, J. B. (1965). Classical ...
0
votes
1
answer
94
views
What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?
For each of the two reference books the constant equations are as follows:
$$
\boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
- 105
0
votes
1
answer
55
views
Cross factor for dependent terms in a differential?
How do you derive a cross factor to decouple differentials into independent differentials? For example:
$$ d(PV)= PdV+VdP $$
$$ PV=\int{PdV}+\int{VdP} $$
Obviously dP and dV are related. Do you simply ...
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2
votes
5
answers
345
views
Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
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0
votes
2
answers
42
views
Velocity to Acceleration negative line [closed]
Is the velocity line in below 0 is a different acceleration line?
For example from 0 - 6s and from 10 - 17s.
It has the same slope.
1
vote
1
answer
79
views
Weird derivative with respect to inverse temperature identity in Tong's statistical physics lecture notes
While reading David Tong's Statistical Physics lecture notes (https://www.damtp.cam.ac.uk/user/tong/statphys.html) I came across this weird identity in page 26 (at the end of the 1.3.4 free energy ...
- 61
1
vote
1
answer
48
views
Equality of variables for small values of time, when the time derivative of the variables are equal to one another
It is given that $\frac{ds}{dt} = \frac{d\theta}{dt}$, i.e. the time derivative of s and $\theta$ are equal to each other.
Does it follow that for small values of $t$, $\Delta s ≈ \Delta \theta$? My ...
-2
votes
2
answers
122
views
Why does $\vec{a}=\vec{\omega}\times \vec{r}$ as well as the velocity does?
Today I came in class and in one of the problems the teacher used $\vec{a}=\vec{\omega}\times \vec{r}$ which made me very confused because I don't know where it comes from, it seems pulled out of thin ...
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1
vote
1
answer
100
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Variation of Torsion-Free Spin Connection
In the book 'Supergravity' by Freedman and van Proeyen, in exercise (7.27) it is written
To calculate [the variation $\delta\omega_{\mu ab}$ of the torsion-free spin connection], consider the ...
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