All Questions
37 questions
0
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33
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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
1
vote
6
answers
539
views
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 ...
- 177
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 $dr$...
- 45
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
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
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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
-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
133
views
Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
- 535
1
vote
1
answer
170
views
What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
- 1,254
1
vote
2
answers
70
views
Expressing infinitesimal physical quantities
In physics class, my teacher demonstrated that in polar coordinates, an infinitesimal area involving radial length dr and infinitesimal angle dθ is equal to rdr dθ, since the area is roughly a square ...
- 303
1
vote
1
answer
65
views
"To order $n$ of" arguments
Often one finds in physics textbooks that arguments will be made "to order $n$". I am not sure on what the procedure or argument ought to be when we have some denominator dependence though. ...
- 1,271
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
9
votes
1
answer
598
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
0
votes
0
answers
50
views
Why area under a curve equals the sum of values of function/quantities taken elementally? [duplicate]
Background:
I was taught basic formulas of differentiation and integration when I started learning physics. However, it wasn't taught through intuitions and concepts. It was like : " Hey these ...
- 325
0
votes
1
answer
57
views
What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
- 85
0
votes
2
answers
68
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Why isn't $C_v=\left( \frac{\partial U}{\partial T}\right) _v$ equivalent to $C_v=\left( \frac{\Delta U}{\Delta T}\right) $?
This might be a naive question but I just saw
$$c_v=\frac{1}{n}\left( \frac{\partial U}{\partial T}\right)_V \approx \frac{1}{n}\left( \frac{\Delta U}{\Delta T}\right)$$
Refearing to the LHS as the ...
- 237
0
votes
1
answer
60
views
Discussion about Taylor expansion
I have a function like
$$\mathcal{F}(\phi_{e}(r),\phi_{c}(r),\phi_{a}(r)) = \exp\left({-\beta\left(\phi_{e}(r) + \phi_{c}(r)\cos{\theta}+\phi_{a}(r)\sin{\theta}\right)}\right).$$
If we assume $\beta\...
user316761
6
votes
7
answers
255
views
Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
0
votes
1
answer
110
views
Taking the second time derivative of a scalar field
Given some scalar field $\phi(x,y,x,t)$, taking its first total derivative we get:
$$\frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\...
- 613
0
votes
4
answers
127
views
Standing Wave Equation: Why does assuming a small slope $du/dx$ not make $d^2u/dx^2$ negligible as well?
Referencing the above image, just change the label for $y$-axis to $u$-axis.^
Following the derivation of the standing wave equation: https://www.youtube.com/watch?v=IAut5Y-Ns7g&t=1324s
So if ...
- 719
0
votes
3
answers
79
views
Function Values Surrounding Stationary Points
Taylor, in his widely read book "Classical Mechanics," writes on page 218 that
When $df/dx = 0$ at a point $x_0$, but we don't know which of the 3 possibilities obtains, we say that $x_0$ ...
- 463
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. $$ \frac{\...
- 128
0
votes
1
answer
101
views
When should we differentiate an equation? [closed]
If we think of a rectangle and differentiate the whole are that means I am just taking small piece of area of that rectangle. But, in physics we find a new equation by differentiating. Even, in ...
user286470
1
vote
2
answers
153
views
Two total differentials with equal variable differentials. Why coefficients in front of differentials are equal?
Could you prove that inference like that is valid:
$$(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
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
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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
157
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
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
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
0
votes
3
answers
2k
views
How to derive kinematics equations using calculus? [closed]
I read derivation of kinematics equations using calculus:
$$a=\frac{\text dv}{\text dt}$$
$$\implies \text dv=a\text dt$$
$$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$
$$\implies v-v_0=at$$
$$\...
- 259
2
votes
1
answer
2k
views
Derivative of tensor product of quantum states
Recently I asked a question over at the math stack exchange:
https://math.stackexchange.com/q/3210375/.
However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
- 569
1
vote
0
answers
55
views
Relation between computation of curl and divergence and their formal definitions
both divergence and curl of a vector field have a formal definition, however, we don't use these definitions when we compute the divergence or curl.
so can we just derive the computations from the ...
- 29
-1
votes
3
answers
2k
views
What is the significance of the second derivative of a function? [duplicate]
Basically, I just want to know the significance of the 2nd derivative of a function, or what does it tell us.
- 411
0
votes
1
answer
2k
views
Use of infinitesimals in physics [duplicate]
I want to ask about infinitesimals and non-standard analysis. In physics we always use $\mathrm dx,~\mathrm dv,~\mathrm dt$ etc. as infinitesimal quantities. When we deduce equations in physics, when ...
- 41
14
votes
4
answers
22k
views
How do you do an integral involving the derivative of a delta function?
I got an integral in solving Schrodinger equation with delta function potential. It looks like
$$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$
I'm trying to solve this by ...
- 325