All Questions
Tagged with differentiation mathematics
95 questions
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
1
vote
4
answers
420
views
What do $\nabla$ and $\frac{d }{d t}$ mean when they are by themselves?
In QM and QFT, I have seen some equations where they have just the derivative and/or the gradient without specifying what it is acting on.
Taken from wiki.
This does not make sense to me since I ...
- 2,042
3
votes
3
answers
278
views
Do there exist motions in which the second derivative of position is not well defined?
After some thought today, I realized that all the power of Newton's laws is fundamentally rooted in the fact that $\frac{d^2 x}{dt^2}$ is a sensible thing to write i.e: there exists a function $x$ ...
- 8,040
0
votes
1
answer
101
views
When should we differentiate an equation? [closed]
If we think of a rectangle and differentiate the whole are that means I am just taking small piece of area of that rectangle. But, in physics we find a new equation by differentiating. Even, in ...
user286470
2
votes
1
answer
104
views
What is the meaning of the del operator in this equation?
$$\frac{\partial \left(\rho_m \vec{v}_m \right)}{\partial t} + \nabla \cdot \left(\rho_m \vec{v}_m\vec{v}_m \right) \\ = - \nabla P_m + \nabla \left(\mu_m \nabla \vec{v}_m \right) + \nabla \left(\...
- 55
1
vote
2
answers
153
views
Two total differentials with equal variable differentials. Why coefficients in front of differentials are equal?
Could you prove that inference like that is valid:
$$(1)
\left\{
\begin{array}{c}
dU=T dS-pdV \\
dU=\frac{\partial U}{\partial S}dS+ \frac{\partial U}{\partial V} dV
\end{array}
\right.
\implies
\...
- 321
-1
votes
3
answers
180
views
Avoiding a confusion with dot product
Some days ago I have asked a question about a formula for power, many generous people have answered my question and clarify for me that the correct formula of work is
$$\mathrm{d}W= \mathbf{F}\cdot \...
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
1
answer
74
views
Analogous notation to $\nabla$ but for gradient with respect to $\vec{k}$ not $\vec{x}$
$\nabla = \frac{\partial}{\partial x_i}$ so $\nabla F = (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z})$.
However, is there a similar equalivalent notion ...
- 2,654
2
votes
1
answer
535
views
Why is the first derivative of the time-dependent Schrödinger equation continuous? Where does it come from?
I was taught in first year physics that the first derivative of the time-dependent Schrödinger equation had to be continuous. However I was never taught (or at least I don't remember) the reason why.
\...
- 579
0
votes
0
answers
273
views
Best Calculus one book [duplicate]
I’m currently in my senior year of high-school. I’m planning to major in physics. I really enjoyed basic calculus but I really want to start studying it for real. I know university courses include ...
0
votes
2
answers
71
views
Why quantities in physics are always talking about rates? [closed]
I get the idea that physics wishes to study changes to discover new rules.
But why is everything related to rates? Acceleration,Velocity?
Could we use something else apart from these?
What can you ...
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
-1
votes
1
answer
45
views
Proof of the following derivate results
Anyone can help me proving the following
If x'=sx where s is a constant then
(d/dx') = (1/s)(d/dx)
- 155
0
votes
2
answers
55
views
Isn't the following addition wrong on manifold as done in Frankel book?
In ch-$4$ when calculating expression of Lie derivative using Hadamard's Lemma before $(4.4)$ Frankel's do following manipulation:
$$\lim_{t\rightarrow0}\frac{\textbf{Y}_{\phi_tx}(f)-\textbf{Y}_x(f)}{...
- 3,073
4
votes
1
answer
167
views
What motivates defining vectors as first order differential operators?
I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
- 73
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
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
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
0
votes
0
answers
38
views
Making the sloppy time-reversal transformation precise
I'm reading Harbovsky and Susskind's The Theoretical Minimum Vol I.
In Lecture 3: Dynamics, under Aristotle's Law of Motion is mentioned Aristotle's (fallacious) law $F(t) = m {dx\over dt}$. Later, ...
- 1,999
-1
votes
2
answers
110
views
Does incomplete differentials $\delta Q$ or $\delta W$ have potentials? [closed]
I am very confused because my text book have following formula.
$$dU = \delta Q \tag {1-1}$$
$$dU = \delta W \tag {1-1'}$$
Because these might mathematically mean "incomplete derivative = ...
- 290
0
votes
1
answer
2k
views
What is infinitesimal displacement? [duplicate]
This section is from the Openstax University Physics: Volume 1 online textbook.
In physics, work is done on an object when energy is transferred to
the object. In other words, work is done when a ...
- 1
1
vote
1
answer
107
views
Show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$
Let us start from $\textbf{B}=\nabla \times \textbf{A}$ and write its components $B_k=\epsilon_{ijk}\partial_i A_j$.
I want to show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$. I can ...
- 551
0
votes
1
answer
133
views
Why is force 0 either side of an inflexion point in neutral equilibrium?
In Tipler & Mosca 5th edition p173 it defines neutral equilibrium as a point in a U-x curve where $\frac{dU}{dx}=0$ and also $\frac{dU}{dx}=0$ for a small displacement either side of the point. ...
- 537
0
votes
3
answers
2k
views
How to derive kinematics equations using calculus? [closed]
I read derivation of kinematics equations using calculus:
$$a=\frac{\text dv}{\text dt}$$
$$\implies \text dv=a\text dt$$
$$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$
$$\implies v-v_0=at$$
$$\...
- 259
2
votes
1
answer
2k
views
Derivative of tensor product of quantum states
Recently I asked a question over at the math stack exchange:
https://math.stackexchange.com/q/3210375/.
However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
- 569
2
votes
2
answers
1k
views
Is curvature the exterior covariant derivative of the connection?
Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space.
We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
- 258
2
votes
2
answers
464
views
Does it make sense to speak in a total derivative of a functional? Part I
I would like to consider the problem of the total derivative of a given functional \begin{equation}
\mathcal{L}\bigg[\phi\big(x,y,z,t\big),\frac{\partial{\phi}}{\partial{x}}\big(x,y,z,t\big),\frac{\...
- 387
13
votes
7
answers
3k
views
Can we divide a vector by another vector? How about this: $a = vdv/dx?$
My physics teacher told us that we can’t divide vectors, that vector division has no physical meaning or significance. How about this: $$a = vdv/dx.$$
It says acceleration vector equals velocity (as ...
- 876
1
vote
0
answers
55
views
Relation between computation of curl and divergence and their formal definitions
both divergence and curl of a vector field have a formal definition, however, we don't use these definitions when we compute the divergence or curl.
so can we just derive the computations from the ...
- 29
-1
votes
3
answers
2k
views
What is the significance of the second derivative of a function? [duplicate]
Basically, I just want to know the significance of the 2nd derivative of a function, or what does it tell us.
- 411
0
votes
0
answers
81
views
Why does the commutativity of partial differentiation hold for Hamiltonian?
When I was going through canonical transformation I came across some equation like
$$\dfrac{\partial^2 H}{\partial q \partial p} = \dfrac{\partial^2 H}{\partial p \partial q}$$ where $H$ is the ...
- 147
-1
votes
1
answer
111
views
What is $\delta t$? [duplicate]
I'm confused whether it's difference between two times (i.e final and initial) or it represents very small time.
- 17
-5
votes
2
answers
184
views
$\sin x/x$ uncertainty at $x=0$, and some confusions [closed]
Think about the function; $$\sin(x)\over x$$ We say frequently at $x=0$ this function is in a uncertain form $0\over 0$. But we conclude $$\lim_{x\to 0}\frac{sin(x)}{x}=1$$Then we say the function ...
0
votes
2
answers
320
views
Laplacian in polar and spherical cordinates
why does the radial dependence of Laplacian in spherical and polar coordinate vary? ie, in polar coordinates if there is no $\theta$ dependence the laplacian goes as $\frac{1}{r} \frac{\partial}{\...
- 117
2
votes
2
answers
442
views
How do I find the function derivative $(\delta/\delta \phi) (\partial_\mu \phi)$?
The question is simple: How do I find the function derivative of $$(\delta/\delta \phi(x)) (\partial_\mu \phi(x))~?$$ As far as I can tell, I cannot use any of the standard computational rules for the ...
- 1,420
3
votes
1
answer
74
views
Newton's axioms and collision
Newton's axioms for point particles states that the velocity of a point particle is differentiable. However when two object collide there is a jump in their respective velocities. So is "ideal" ...
2
votes
2
answers
105
views
How to infer what integrals and derivatives signify and when to take them? [closed]
So I have very little background in physics since I'm a mathematical sciences major, but upon being exposed to some physics I've had some difficulties in understanding how to infer the derivatives and ...
- 459
0
votes
1
answer
2k
views
Use of infinitesimals in physics [duplicate]
I want to ask about infinitesimals and non-standard analysis. In physics we always use $\mathrm dx,~\mathrm dv,~\mathrm dt$ etc. as infinitesimal quantities. When we deduce equations in physics, when ...
- 41
3
votes
1
answer
76
views
Derivative with respect to a difference of independent variables
I am dealing with an equation from nonlinear acoustics (Khokhlova-Zabolotskaya-Kuznetsov equation) where a strange term (for me as a mathematician) is used.
The equation looks like this
$$ \frac{\...
- 241
2
votes
4
answers
733
views
Can a particle have no instantaneous velocity at all points of the path taken but a finite average velocity?
I have a question on kinematics.
Say the path traced by a particle is given by a Koch curve or Koch snowflake.
Now consider the particle starts from some arbitrary point $A$ on the curve and ...
- 4,984
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
5
votes
1
answer
5k
views
Second derivative of Dirac delta expression
I have come across the expression
$$ \int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$
where the prime represents the derivative.
Usually with derivatives of the Dirac delta distribution I'd partially ...
- 9,197
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
7
votes
6
answers
8k
views
How is gradient the maximum rate of change of a function?
Recently I read a book which described about gradient. It says
$${\rm d}T~=~ \nabla T \cdot {\rm d}{\bf r},$$
and suddenly they concluded that $\nabla T$ is the maximum rate of change of $f(T)$ ...
- 872