All Questions
Tagged with calculus differentiation
36 questions with no upvoted or accepted answers
3
votes
2
answers
160
views
Acceleration in terms of displacement
I am having problems understanding the derivation of acceleration in terms of displacement. The first step is fine:
$$a(x) = \frac{\mathrm dv(x)}{\mathrm dt}
= \frac{\mathrm dv(x)}{\mathrm dx} \frac{\...
- 131
2
votes
0
answers
91
views
Derivation of a partial derivative equation by Albert Einstein in Special Theory of Relativity
I was reading Albert Einstein's "On the Electrodynamics of Moving Bodies". In section "Kinematical Part", on $3 (Theory of the transformation of coordinates and times from a ...
- 21
1
vote
0
answers
85
views
Equations with fractional derivatives
Assume we have an equation which represents the flux of some quantity as $q = -D \dfrac{\partial T}{\partial x}$ (Eqn. 1), where the diffusion coefficient $D$, variable $T$, $x$ and $q$ have some ...
- 11
1
vote
2
answers
120
views
Differential form of Planck's Distribution Law interpretation
So I didn't encounter differentials that often until now, I was taught that the seperate parts of $dy/dx$ for example are not supposed to have any sort of independent existence - ok.
(Calculus, 4th ...
- 153
1
vote
1
answer
84
views
Finding back a simple SDE from its solution
I'm trying to self-learn Kurt Jacob's Stochastic Processes for Physicists: Understanding Noisy Systems. I've followed Chapter 3, where I saw how to derive that the solution to the SDE
$$
dx=\left(c+\...
- 11
1
vote
0
answers
46
views
Partial derivatives and the Joule Coefficient
The Joule Coefficient for a van der Waals gas can be shown to be
\begin{equation}
\left(\frac{\partial T}{\partial V}\right)_U=-\frac{a}{C_VV^2}
\end{equation}
where $U$ is the internal energy of the ...
- 2,456
1
vote
0
answers
64
views
How to transform a partial derivative to a directional derivative with respect to some affine parameter?
Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
- 105
1
vote
0
answers
173
views
Derivative of Function of Unitary matrices
I need some help in understanding derivative of function of matrices, Unitary matrices in my case.
I am studying lattice-qcd, there i need to take derivative of Wilson gauge action, $S[U]$ w.r.t link $...
1
vote
3
answers
307
views
What does the derivative of unit vector of velocity with respect to time represent?
let an object move with a constant accelration a. in my book,the following derivatve is said to be non-constant(variable).
$$\frac{d[\frac{v}{|v|}]}{dt}$$
what does this mean?
as far as i can think,it ...
- 41
1
vote
1
answer
141
views
What is the difference between zero and an infinitesimal number?
In a standard Atwood machine physics problem, the string going over the pulley is considered massless. So does that imply mass = 0 or mass = dm? General question: what is the difference between 0 and ...
- 87
1
vote
0
answers
51
views
Graph of $dQ/dt$ discontinuous
Charge is quantised then why how do we define $dQ/dt$ as current when graph of $Q$ v/s $x$ will be discontinuous and hence non-differentiable.
Is it an approximation we use?
- 21
1
vote
0
answers
55
views
Relation between computation of curl and divergence and their formal definitions
both divergence and curl of a vector field have a formal definition, however, we don't use these definitions when we compute the divergence or curl.
so can we just derive the computations from the ...
- 29
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 ...
- 1
0
votes
1
answer
136
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 ...
- 71
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 ...
- 21
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{...
- 41
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:
$$
...
- 405
0
votes
0
answers
50
views
Laplace transform: How to evaluate partial derivative in the denominator of a fraction?
I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
- 95
0
votes
1
answer
55
views
Cross factor for dependent terms in a differential?
How do you derive a cross factor to decouple differentials into independent differentials? For example:
$$ d(PV)= PdV+VdP $$
$$ PV=\int{PdV}+\int{VdP} $$
Obviously dP and dV are related. Do you simply ...
- 861
0
votes
0
answers
45
views
Physical and Diagrammatic representation of $a$=undefined when $v$=0 according to $a$=$vdv$/$dx$
$a$=acceleration
$v$=velocity
$x$=position along x axis
$t$=time instant
My teacher derived the $a$=$v$$dv$/$dx$ formula as follows
Assume a particle at time $t$ is at $x$ position having $v$ velocity
...
- 31
0
votes
1
answer
43
views
Are terms tangential acceleration and normal acceleration only used for instantaneous velocity?
Are terms tangential acceleration and normal acceleration only used
for instantaneous velocity?
- 23
0
votes
0
answers
70
views
What do equations involving infinitesimals say?
I am reading this note on the Bernoulli equations with the following derivations:
I am struggling to find a calculus based meaning for the above equations involving the infinitesimal $\delta V$: I ...
- 101
0
votes
1
answer
69
views
Use Index Notation properly when indices are already used in identifying which bases is the matrix metric calculated with respect to
I am wondering how to apply the usual linear algebra to the rather unfamiliar case of 'matrices' with indices in special relativity or even general relativity. In particular, consider$$f=\sqrt{-\det\...
- 862
0
votes
1
answer
742
views
What are the relationships between the motion-time graphs?
I was wondering if someone could explain the relationships between the three motion graphs (Position-Time, Velocity-Time, and Acceleration-Time). I believe that the slope of the P-T is Velocity and ...
0
votes
0
answers
36
views
How to evaluate a non-banal derivate?
I need to evaluate the following derivate:
$$\frac{dF}{d\Psi} = \frac{d}{d\Psi}\left[\beta\Delta\Psi+\alpha\left|\Psi\right|^2\Psi+\mu\Psi-i\vec{v}\cdot\bar{\nabla}\Psi\right]$$
where $\Psi$ is a ...
0
votes
0
answers
80
views
Why is cancellation of differnetial not allowed here?
This is about cancelation of differentials .I am learning basics of tesnor from "Mathematical Methods " by Boas. There I encountered this epression which author says are equal. $$ \frac{\...
- 128
0
votes
0
answers
145
views
Work-Energy Principle Derivation
I am currently in Mechanics I and both my professor and my book have derived the work principle in this way and I even asked about its derivation during class, but it has me puzzled.
I don't ...
0
votes
0
answers
156
views
Classical text of mathematics/infinitesimals for Landau-Lifshitz
I believe their is a pre- and post Weierstrass era of mathematics (loosely speaking). Afterwards there was epsilon-delta, before 'infinitesimals' (with certain rules, ideas and theorems, of course not ...
0
votes
0
answers
38
views
Why is linear approximation of contribution of electric field the same as if whole charge was concentrated at a single point?
I was reading about electric field of uniformly charged ring, of radius $R$, on the axis of the ring at the distance $d$ from the center of the ring and I am confused about usage of differentials. It ...
- 21
0
votes
1
answer
126
views
Multivariate analogue of triple product rule
I was doing some thermodynamic calculations and I found the need for a multivariable analogue of the triple product rule. Basically I had a set of $m$ functions $z_i(y, x_1, x_2, \ldots, x_m)$, and I ...
- 3,430
0
votes
0
answers
147
views
How to calculate the derivative of the angular momentum vector $ d\vec L = d(\hat I \vec \omega)?$
My last question, but also the most important one How to calculate the derivative of the angular momentum vector?
$$ d\vec L = d(\hat I \vec \omega)$$
I'm especially interested in derivative tensor to ...
0
votes
1
answer
158
views
Computation - can you compute the gradient, Laplacian, divergence and curl of any function?
In my physics class, we are currently studying gradient, Laplacian, divergence, and curl, and we have a problem that states to compute all four of these (I.e., (1) gradient, (2) Laplacian, (3) ...
- 151
0
votes
0
answers
94
views
Get rid of the derivatives and relativistic mass in Feynman lectures
i have a problem with get rid of the derivatives in Feynman lectures (chapter 15, Equivalence of mass and energy). The problem:
we have $\frac {d(mc^2)}{dt} = v\cdot \frac {d(mv)}{dt}$, then we ...
0
votes
0
answers
54
views
Mathematical Description of Time Speeding Up?
People are able to experience time speeding up or slowing down. This is confusing to me on a mathematical level because dT/dT is 1. Is there some way that makes sense for this not to be 1? The speed ...
Alephnull
-1
votes
2
answers
89
views
For how long is an objects velocity it's instantaneous velocity at time $t$?
Basically I'm asking if an object's instantaneous velocity at time $t$ is $8m/s$ and its instantaneous velocity at time $t^+$ (idk latex, but basically the t + an infinitely small number) is $10m/s$, ...
-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)\...
- 2,180