All Questions
Tagged with calculus differentiation
318 questions
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?
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
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
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
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 $$x = 490t^2$...
15
votes
3
answers
44k
views
Derive vector gradient in spherical coordinates from first principles
Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient.
I've derived the spherical unit vectors but now I don't understand how to transform ...
- 498
14
votes
4
answers
22k
views
How do you do an integral involving the derivative of a delta function?
I got an integral in solving Schrodinger equation with delta function potential. It looks like
$$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$
I'm trying to solve this by ...
- 325
11
votes
4
answers
3k
views
When the direction of a movement changes, is the object at rest at some time?
The question I asked was disputed amongst XVIIe century physicists (at least before the invention of calculus).
Reference: Spinoza, Principles of Descartes' philosophy ( Part II: Descartes' Physics, ...
user229097
11
votes
2
answers
3k
views
Kinematic equation as infinite sum
I'm not sure exactly how to phrase this question, but here it goes:
$v=\dfrac{dx}{dt}$ therefore $x=x_0+vt$
UNLESS there's an acceleration, in which case
$a=\dfrac{dv}{dt}$ therefore $x=x_0+v_0t+\...
10
votes
7
answers
1k
views
What is the instant velocity? [duplicate]
The velocity is the variation rate of the position correct? So does it make sense to talk about velocity without time?
- 117
9
votes
4
answers
2k
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Can I find the acceleration or velocity when my displacement-time graph is discontinuous?
Today, I encountered the problem where I was asked to find the velocity and acceleration from displacement-time graph but the displacement-time graph was discontinuous. So I am unable to find the ...
- 123
9
votes
2
answers
958
views
Do partial derivatives of different coordinate systems commute?
Consider an arbitrary set of coordinates $x^\mu$ and another set of coordinates $y^{\mu}$, which is a (lorentzian) transformation from $x^\mu$ given by $y^\mu = f(x^\mu)$.
So I want to know whether $\...
9
votes
1
answer
600
views
Inverse of the covariant derivative
Given the covariant derivative of some tensor, for the sake of this example a covariant vector:
$$\nabla_\mu A_\nu$$
Is there a well-defined inverse operation on the covariant derivative such that it ...
- 613
8
votes
2
answers
1k
views
What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?
I'm trying to find out
$$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$
I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
- 12.1k
8
votes
4
answers
1k
views
Struggling understanding definitions with infinitesimal quantities
Many quantities in physics are defined as ratio of infinitesimal quantities. For example: $$\rho(x)=\frac{dm}{dx}$$
or
$$P(t)=\frac{dW}{dt}$$
Are these quantities actually derivatives? I mean if we ...
- 1,688
8
votes
3
answers
478
views
Name this Mulltivariable Calculus Theorem
In Robert Wald's book General Relativity a multivariable calculus theorem is cited on page 16, which states:
If $F:\mathbb{R}^n\mapsto \mathbb{R}$ is $C^{\infty}$ then for each $a=(a^1,...,a^n) \in ...
- 636
7
votes
3
answers
1k
views
In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$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)$$
Should ...
6
votes
6
answers
1k
views
Question about derivation of kinematics equations
Apologies if this has been asked before, but I browsed the sub and couldn't find something specific.
I understand the derivation for one of the equations as follows:
\begin{gather}
\frac{dv}{dt} = a ...
- 69
6
votes
2
answers
2k
views
How to deal with differentials? [duplicate]
I am currently working on this. More specifically my question is about Problem 2.5 b). In the solution they get from
$$
Nd\mu=-SdT+VdP
$$
to
$$
N\Big(\frac{\partial\mu}{\partial N}\Big)_{T,V}=V\Big(\...
- 131
6
votes
2
answers
5k
views
What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?
I am studying the Navier-Stokes equations and I have the equation in the form:
$$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
6
votes
1
answer
640
views
Infinitesimal and approximations in physics
I'm a first year student studying physics. Solutions of many physics problems, which I've seen so far, are achived through solving this problem for infinitesimal part of problem's subject (some curve, ...
- 61
6
votes
3
answers
1k
views
Is $\dfrac{dx}{dt}$ a fraction or not?
I am new to calculus and during my mathematics class my sir defined $\dfrac{dx}{dt}$ as $$dx/dt=\lim_{t\to t_1}\dfrac{f(t)-f(t_1)}{t-t_1}$$ and my sir made a clear statement that
$\dfrac{dx}{dt}$ ...
- 480
6
votes
7
answers
255
views
Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
5
votes
2
answers
2k
views
How does instantaneous velocity or acceleration have any other numerical value than 0? [duplicate]
Instantaneous velocity is defined as the limit of average velocity as the time interval ∆t becomes infinitesimally small. Average velocity is defined as the change in position divided by the time ...
- 163
5
votes
5
answers
657
views
Is $dx$ always positive?
When we refer to change in a quantity, we define it to be (final-initial). If it is positive it indicates an increase from the initial value and negative indicates a decrease.
But when we take this to ...
- 71
5
votes
5
answers
7k
views
What is divergence?
What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
- 1,660
5
votes
5
answers
443
views
Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?
Hello fellow physicists,
I was trying to understand some behavior on rotating objects, specifically about the formula $\vec{v} = \vec{\omega} \times \vec{r}$.
The Book (Marion, J. B. (1965). Classical ...
5
votes
1
answer
1k
views
Is the inverse of the deformation gradient simply the deformation gradient of the inverse transformation?
If we have a continuum where the initial positions are denoted $X$ and the positions after some deformation are denoted $x$, the deformation gradient is defined:
$$ F = \frac{\partial x}{\partial X} $...
- 103
4
votes
6
answers
855
views
How to understand instantaneous velocity concept [duplicate]
When I started learning instantaneous velocity it didn't make sense to me. I don't understand in real life why we can't measure instantaneous velocity and therefore why we use this concept.
Or is this ...
- 311
4
votes
2
answers
5k
views
How is dot or cross product possible using the del operator?
Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
- 1,432
4
votes
2
answers
510
views
Converting differential to gradient
Landau & Lifschitz's fluid mechanics book proposes the following statement for an isentropic proccess:
$$dH=vdp \Rightarrow \nabla H=v\nabla p$$
What's the rigorous way to get this result (...
- 43
4
votes
2
answers
419
views
What does it mean when we say 'The difference between two quantities is of first order'?
This question is about the explanation below Eq.(6.19) of Modern Quantum Mechanics by Sakurai Nepolitano (2nd edition)
Let ${\bf j}(dx)$ be an operator that translates a point $x$ to $x+dx$.
jf(x) = ...
- 397
4
votes
1
answer
263
views
Taylor Series of a logarithmic function
I was reading Intro to Modern Statistical Mechanics by David Chandler, on page 63. He states the following:
we can expand $\ln\Omega(E-E_v)$ in the Taylor series $$\ln\Omega(E-E_v) = \ln\Omega(E) - ...
- 348
4
votes
2
answers
18k
views
Why and when do we differentiate or integrate equations in physics? [closed]
I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like:
The object is moving in a positive ...
- 41
4
votes
2
answers
196
views
In physical calculations, is the elimination of higher-order small quantities an approximation or a strict equality in mathematics?
Physics sometimes uses a technique called the method of differentials, which seems magical and not very systematic. This makes me unsure which variable I should take the differential of, and sometimes ...
- 119
4
votes
1
answer
1k
views
Change of variables in gradient
Take two coordinates with $\mathbf r$ and $\mathbf r'$ and take a function $f(|\mathbf r - \mathbf{r'}|)$. In many electromagnetism derivations I see a conversion like this
$$
\nabla_r f(|\mathbf r - \...
- 1,048
4
votes
2
answers
861
views
Integration of tangential acceleration with respect to time
Here, by tangential acceleration, I mean the component of acceleration along the velocity vector.
What do you get when you integrate tangential acceleration with respect to time? What does the '$v$' ...
- 1,106
3
votes
2
answers
739
views
Derivation of curl of magnetic field in Griffiths
Can someone please derive how $$\frac{d}{dx} f(x-x') = -\frac{d}{dx'} f(x-x')~?$$
In Griffiths electrodynamics, this is directly mentioned. I'm really confused, can someone elaborate!
- 453
3
votes
3
answers
296
views
If the displacement of an object is not differentiable at some point, say $x(t)=t\sin(1/t)$ at $t=0$, how is its instant $v$ defined? [closed]
If instant velocity at any given time $t_0$ is defined as the derivative of $x(t)$ at $t_0$, what if the derivative does not exist? How are we supposed to deal with $x(t)=|t|$ at $t=0$, or for more ...
- 39
3
votes
2
answers
5k
views
Feynman's subscript notation
Consider this vector calculus identity:
$$
\mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) \...
- 163
3
votes
2
answers
279
views
Why the inconsistency, chain rule in $\text dS$ and $\text dU$
So, we started the study of thermodynamics by introducing $\text dU$ in a logical way:
$$
\text dU = T \text dS - P\text dV + \mu \text dN . \tag1
$$
Later we started to see that all the properties ...
- 125
3
votes
1
answer
480
views
Second derivative of unit vector
We know that the second derivative of unit vector (the vector from a point toward the source) is proportional to the Electric field caused by the source in a particular point.
If we imagine that our ...
- 63
3
votes
2
answers
233
views
Generalization of straight line motion under constant acceleration
My question is that, we all know the three equations of straight line motion under constant acceleration,
\begin{align}
x & =x_{\rm o}+v_{\rm o}\,t+\tfrac12 \mathrm a\,t^2
\tag{1d-a}\label{1d-a}\\
...
3
votes
2
answers
133
views
Is this notation inconsistent? If not, can some explain why not?
Im working through a textbook section on particle kinematics. An example given is relating vertical velocity to horizontal velocity and states:
$y$ has a constant velocity of $10 \ \rm [m/s]$
$y=(0....
- 31
3
votes
1
answer
1k
views
Taylor expanding a function of an operator?
I am trying to understand the following description in my quantum mechanics textbook:
Let $F(\hat{A})$ be a function of an operator $\hat{A}$. If $\hat{A}$ is a linear operator, we can Taylor expand $...
- 273
3
votes
2
answers
201
views
What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
- 644
3
votes
2
answers
970
views
Differential operators in curvilinear coordinates
In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
3
votes
1
answer
92
views
Bianchi identity in EMT [closed]
$ ∇_a∇_b F_{ab} = 0 $ ($F_{ab}$ Faraday tensor in EMT.)
proof is given by
"To see this, assume a Minkowski spacetime for simplicity and adopt
Cartesian coordinates, so that the covariant ...
- 81
3
votes
1
answer
759
views
Neglecting second order differentials
I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials.
It's not the first time I have come ...
- 6,137
3
votes
2
answers
160
views
Acceleration in terms of displacement
I am having problems understanding the derivation of acceleration in terms of displacement. The first step is fine:
$$a(x) = \frac{\mathrm dv(x)}{\mathrm dt}
= \frac{\mathrm dv(x)}{\mathrm dx} \frac{\...
- 131