All Questions
Tagged with calculus differentiation
318 questions
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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 ...
-3
votes
1
answer
84
views
Math explanation needed [closed]
Can someone explain to me how did they come from equation $(23)$ to $(24)$? What does the straight-line denote mean?
and the change in energy due to the addition of matter as
$$\mathrm dE_{\text{...
- 1
0
votes
3
answers
114
views
What and how are you measuring with $\frac{dy}{dx}$, $dx$, $\mathbf{\nabla} \cdot$ and $\mathbf{\nabla} \times$?
In geometry, the gap between the mathematical model, be it axiomatic or algebraic, and the measurable real world is often non-existent. We have the intuition going from the mathematical model to world ...
- 229
0
votes
1
answer
129
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Why intuitively is the tangent vector the derivative of velocity of position with respect to their modulus?
When trying to find the tangential velocity, many textbooks define the tangent direction as one of the following:
or
Intuitively, why is the tangent vector the derivative of the position with ...
- 849
0
votes
1
answer
933
views
Change with time
I was going through a silly doubt, but aren't able to find its answer.
If we say something is changing with time, what do we mean by that?
Should we multiply the quantity with time or divide it? Like ...
- 49
1
vote
2
answers
126
views
Partial derivative in hydrostatic equilibrium in star
In a simple model, a gaseous, non-rotating star consists of many thin, concentric spherical shells with radius $r$ and mass $\text{d}m$. The total mass of the shells within radius $r$ is $m$. The ...
- 1,169
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votes
1
answer
113
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Partial differentiation of tensors
We know that $∂x^ρ /∂x^μ = δ^ρ_μ$
Τhen, $∂x_ρ /∂x^μ = η_{ρμ}$
Should be correct, right?
Similarly,
$\frac{∂x_ρ} {∂x_μ} = δ^μ_ρ$
Also, if
$x'^μ = e^α x^μ $, then
$∂'_μ$ should be $e^α ∂_μ$
I am new to ...
- 358
1
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2
answers
319
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What is the time derivative of the linear velocity vector $\vec{v}\,(t)$?
If $\vec{v}\,(t)$ denotes linear velocity, we can then write $\vec{v}\,(t)$ as $|v(t)|\hat{v}$. My question is what is $\displaystyle\frac{d\vec{v}\,(t)}{dt}?$
The answer I have seen to this question ...
- 39
1
vote
1
answer
113
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Calculating the variation of an operator in two different ways
Let
$$
H_{T}=\dot{x}^{I}\frac{\partial}{\partial \psi^{I}}T(x,\psi)
$$
and consider the transformation:
$$
x^{I}\mapsto x^{I}+i\epsilon\psi^{I}
\\
\psi^{I}\mapsto\psi^{I}-2\epsilon\dot{x}^{I}
$$
where ...
- 587
1
vote
1
answer
265
views
Maximum value of a quantity
While solving problems based on finding the maximum value of quantities such as maximum force or maximum power dissipated, I was told to differentiate the obtained expression and equate it to zero as ...
0
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1
answer
91
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Discrete quantities and calculus
Why we can apply calculus in cases where discrete quantities take place?
Suppose we have a box that has two partitions, namely A and B (look at the figure below). Suppose we know the rate that ...
- 281
0
votes
2
answers
204
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Physical meaning of the exterior derivative of the first law of thermodynamics
We know, $$ dU = d \overline{q} - d \overline{W}.$$ suppose we took the exterior derivative on both sides, then:
$$ 0= d( d \overline{q}) - d( d \overline{W})$$
This means, $$ d^2 \overline{q} = d^2 \...
- 8,040
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1
answer
101
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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
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1
answer
42
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Is such a situation realistically possible where $v$-$t$ graph is continuous but $a$-$t$ graph is not?
Taking for example $v = \cos(t-1)$ from $t \in [0,1]$ and $v = e^{t-1}$ from $t \in (1,\infty)$ and $t \ge 0$. At $t = 1$, the function shifts from cosine to exponential, but remains continuous since ...
0
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1
answer
435
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Find the distance travelled between $t=0$ and $t=5$ [closed]
The position vector of a particle is given as $\vec r = \frac43 t^{3/2}\hat i - \frac{1}{2} t^2\hat j + 2 \hat k$, $t$ is in seconds. Find the distance travelled between $t = 0$ and $t = 5$ seconds.
...
- 231
1
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3
answers
307
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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
80
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Confusion in derivation of Formula of divergence
This is from the book Mathematical Methods (Arfken),
Can someone explain how did one del(x) and one dx came here,?
0
votes
1
answer
55
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Which is the differential $\text{d} p_i$ of a generalized momentum?
I want to get a partition function, but I introduce a generalized momentum, my doubt is about, when I define a differential respect to $p$, it means $\text{d} p$, which is the correct form to get it?
...
- 13
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2
answers
145
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Spin coherent state path integral derivation
I'm trying to follow the exposition of spin coherent state path integral presented in Condensed Matter Field Theory by Altland and Simons (section 3.3, Page 134-142), and I have a problem with the ...
- 3
0
votes
1
answer
34
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Help decipher the notation said to denote a common pattern in various branches of science in Prelude to Mathematics by W. W. Sawyer
In Section 1.2 - Nature's Favorite Pattern? (excerpted below) of Prelude to Mathematics by W. W. Sawyer (1982), he said mathematicians used the notation $\nabla^2 V$ to denote a pattern that occurs &...
- 109
1
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1
answer
73
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Finding the maximum electric field strength above a ring with a hole in the middle
I'm doing a problem (not homework, by the way) which asks for the electric field strength on the axis of symmetry a distance $x$ above the centre of a circular disc, which has uniform surface charge ...
- 221
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votes
2
answers
326
views
Is there any difference in superscript and subscript notation in finite difference method
Is there any difference in superscript and subscript notation in the finite difference method?
I see the same paper use (superscript for $x$ and superscript for $y$ notation) and (subscript for x and ...
- 3
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:
$$(1)
\left\{
\begin{array}{c}
dU=T dS-pdV \\
dU=\frac{\partial U}{\partial S}dS+ \frac{\partial U}{\partial V} dV
\end{array}
\right.
\implies
\...
- 321
1
vote
2
answers
73
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How do you differentiate this differential equation? [closed]
I have to differentiate this equation (Gravitational force between N-Bodies)
$\begin{align}
\frac{d^2}{dt^2}\vec{r_i}(t)=G
\sum_{k=1}^{n}
\frac
{m_k(\vec{r}_k(t)-\vec{r}_i(t))}
{\lvert\...
1
vote
3
answers
647
views
Derivative as a fraction in deriving the Lorentz transformation for velocity
Consider a frame $S$ and $S'$ which is coincides at $t=0$ and then $S'$ starts moving with velocity $v$ in $+x$ direction.
By Lorentz transformation equation,
\begin{align}
x'&=\gamma(x-vt) \\
...
- 446
9
votes
2
answers
959
views
Do partial derivatives of different coordinate systems commute?
Consider an arbitrary set of coordinates $x^\mu$ and another set of coordinates $y^{\mu}$, which is a (lorentzian) transformation from $x^\mu$ given by $y^\mu = f(x^\mu)$.
So I want to know whether $\...
3
votes
3
answers
296
views
If the displacement of an object is not differentiable at some point, say $x(t)=t\sin(1/t)$ at $t=0$, how is its instant $v$ defined? [closed]
If instant velocity at any given time $t_0$ is defined as the derivative of $x(t)$ at $t_0$, what if the derivative does not exist? How are we supposed to deal with $x(t)=|t|$ at $t=0$, or for more ...
- 39
3
votes
2
answers
233
views
Generalization of straight line motion under constant acceleration
My question is that, we all know the three equations of straight line motion under constant acceleration,
\begin{align}
x & =x_{\rm o}+v_{\rm o}\,t+\tfrac12 \mathrm a\,t^2
\tag{1d-a}\label{1d-a}\\
...
1
vote
1
answer
459
views
Expressing acceleration in terms of velocity and derivative of velocity with respect to position
we know that
$$a = \dfrac{dv}{dt}$$
dividing numerator and denominator by $dx$, we get $$a=v\dfrac{dv}{dx}$$ provided that $dx$ is not equal to zero or instantaneous velocity not equal to zero
when I ...
- 892
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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 ...
2
votes
3
answers
131
views
Is motion in infinitesimal interval is linear?
As a kind of thought experiment I tried to think if a motion (including circular motion), when divided into infinitesimal time intervals is always linear motion (whether each interval of the motion is ...
- 121
2
votes
1
answer
87
views
How to express the elementary work definition as an explicit functional expression [duplicate]
My assumption here is that in the definition of elementary work :
$dW = F ds$
symbol $d$ represents a differential.
But a differential implies a function :
$dy =_{df} d[f(x)] = f'(x) \Delta x = f'(...
- 373
-4
votes
1
answer
71
views
Given that $m \dot v \cdot v = 0$ , how is it equal to $m \frac{d}{dt} (v \cdot v)/2$? [closed]
While studying about scalar triple product in vector algebra, I stumbled upon the following question with the solution.
I want know how is $m \dot v \cdot v $ = $m \frac{d}{dt} (v \cdot v)/2$?
- 1
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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
196
views
Derivative of a complex potential for the $\lambda \Phi^{4}$-model
A charged scalar particle is described by a complex field $\Phi(x) = \phi_{1}(x)+i\phi_{2}(x)$. Consider a Lagrangian of the $\lambda \Phi^{4}$-model whose potential in the Euclidean action is given ...
- 75
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votes
1
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305
views
What does an "elementary value $\delta$ of a quantity" mean?
In page-11 of I.E irodov Fundamental laws of mechanics, some notation used in the book is introduced. There, it is said that $\delta$ denotes the elementary value of a quantity but what exactly does ...
- 8,040
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votes
1
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78
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Particle paths - the distance moved by a particle in a velocity field
This question is is context to particle paths.
Particle paths are trajectories of a given particle in the velocity field:
$$\boldsymbol{u}(\boldsymbol{x},t)$$
A particle location at position $\...
- 187
0
votes
2
answers
173
views
Taylor Series Expansion of unknown, fraction function
I am learning about deformation, and the deformed state between two points can be defined as
$$E(x) = \frac{(f(x+dx) - f(x))^2 - (dx)^2}{2(dx)^2}$$
My textbook says
When $dx \to 0$ we can use a ...
- 1
0
votes
0
answers
273
views
Best Calculus one book [duplicate]
I’m currently in my senior year of high-school. I’m planning to major in physics. I really enjoyed basic calculus but I really want to start studying it for real. I know university courses include ...
2
votes
1
answer
292
views
Is the relation "slope=velocity" mathematically valid?
$\text{Slope= tan(angle with respect to positive X-axis)= scalar output}$
$\text{velocity= a vector }$
Source: Hugh D Young_ Roger A Freedman - University Physics with Modern Physics In SI Units (...
- 439
2
votes
1
answer
267
views
Is there a difference between instantaneous speed and the magnitude of instantaneous velocity?
Consider a particle that moves around the coordinate grid. After $t$ seconds, it has the position
$$
S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, .
$$
The particle traces a quarter arc of ...
- 131
8
votes
2
answers
1k
views
What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?
I'm trying to find out
$$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$
I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
- 12.1k
3
votes
2
answers
133
views
Is this notation inconsistent? If not, can some explain why not?
Im working through a textbook section on particle kinematics. An example given is relating vertical velocity to horizontal velocity and states:
$y$ has a constant velocity of $10 \ \rm [m/s]$
$y=(0....
- 31
-1
votes
1
answer
4k
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Lennard-Jones potential, distance $r$ for minimum energy
I'm sorry if the question seems stupid. I found (wikipedia) that the Lennard-Jones potential has it's minimum at a distance of
$$r = 2^{\frac{1}{6}}\sigma.$$
If $U(r)_{min} = -\epsilon$
$$U(r) = 4\...
- 137
1
vote
1
answer
101
views
What does the $d$ mean in metric tensor calculations?
In many metric calculations, like the Schwartzschild metric, we see formulas like $d^2X / dt^2$ and many other formulas with a $d$ in them. You'd be surprised that I've been looking for months to ...
- 4,855
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
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
2
answers
78
views
Translation of coordinates to generalised coordinates
The translation form $r_i$ to $q_j$ language start forms the transformation equation:
$r_i=r_i (q_1,q_2,…,q_n,t)$ (assuming $n$ independent coordinates)
Since it is carried out by means of the ...
- 187
-2
votes
1
answer
97
views
Can we write scalar potential associated with a Vector Function? [closed]
If we are given a vector function, can we directly write its associated scalar potential? Should there be some other "cross" terms too?
Let's take a Vector function $$\vec A(x, y, z) = (4xy-...
- 187
0
votes
0
answers
147
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How to calculate the derivative of the angular momentum vector $ d\vec L = d(\hat I \vec \omega)?$
My last question, but also the most important one How to calculate the derivative of the angular momentum vector?
$$ d\vec L = d(\hat I \vec \omega)$$
I'm especially interested in derivative tensor to ...
