Questions tagged [calculus]
Calculus is the branch of mathematics which deals with the study of rate of change of quantities. This is usually divided into differential calculus and integral calculus which are concerned with derivatives and integrals respectively. DO NOT USE THIS TAG just because your question makes use of calculus.
1,160 questions
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Why are independent variables treated differently in kinetic energy calculations across problems?
In two different problems involving Lagrangian mechanics, I am confused about how independent variables are treated in the kinetic energy calculations. Specifically, in one case, an independent ...
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Continuity of Electric Potential in the Vicinity of a Charged soap bubble
This is not completely a homework question, I have relatively deeper questions at the end.
My Solution:
I assumed that say, a charge Q is distributed over the bubble and thus the electric potential ...
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Limits of the integral for the calculation of work
To calculate the total displacement for a time dependent velocity, one can start from an infinitesimal displacement and integrate as follows
$$dx=vdt$$
$$ \int_{x_i}^{x_f}dx= \int_{t_i}^{t_f}vdt $$
$$ ...
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Connection between gravitational potential function and Gaussian distribution function in 3 dimensions
I was reading Kai Lai Chung's book Green, Brown, and Probability. Consider the Gaussian distribution function in 3 dimensions:-
Now, this is a function of y, mean is x, and variance is t, which is ...
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Integral equation of the convolution type with singular kernel [migrated]
I have stumbled upon an integral equation whose kernel involves a quotient of Gamma functions:
$$ x(2-x)f(x)+g(x)=-\int_{0}^{x}dy\;f(y)\;g(x-y), $$
where
$$g(x)=\frac{\Gamma(1-x)}{(1+x)\Gamma(x)}.$$
...
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Use of $dv/ds$ in defining acceleration [duplicate]
We can write acceleration as either
$dv/dt$ or $v dv/ds$.
And surprisingly the work-energy theorem arrives from the second definition.
I feel it would be fundamentally understanding towards work ...
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Second derivative of unit vector
We know that the second derivative of unit vector (the vector from a point toward the source) is proportional to the Electric field caused by the source in a particular point.
If we imagine that our ...
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Can something be at rest if it has a non-zero acceleration?
I think I have a decent grasp on the physics - I understand that something can be accelerating while stationary. That's the basis of my question. I just wanted to clarify some of the language used.
We ...
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Dot product under parallel transport
I was wondering about simple justification that parallel transport preserves dot product value. I came up with an idea like that:
$u^i$ - 1st vector in covariant basis
$v_i$ - 2nd vector in ...
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Explaining the equations of motion in kinematics [closed]
Hi I Have these equations and I don't know and understand what each of them means.
Can someone help me?
$$\tag 1 v=v_0+at$$
$$\tag 2 x=v_0t+\frac{at^{2}}{2}$$
$$\tag 3 v^{2}=v_0^{2}+2ax$$
What does ...
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What happens when a car starts moving? The last moment the car is at rest versus the first moment the car moves
Imagine a car that's at rest and then it starts moving. Consider these two moments:
The last moment the car is at rest.
The first moment the car moves.
The question is: what happens between these 2 ...
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Doubt in Verlet's Algorithm
In studying the temporal evolution of a system according to the deterministic model, we begin by considering a Taylor series expansion for the displacement $r$. First, we consider a positive variation ...
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Why take the derivative of variables such as area, mass, and radius?
I'm taking a module on stars and the solar system; I've attached notes from our first lecture- hydrostatic equilibrium. I'm confused about the notation $\mathrm{d}$ for $\mathrm{d}A, \, \mathrm{d}r$, ...
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Hausdorff formula for Dilatation operator
Using the Hausdorff formula I can find the dilation operator at all position:
$$
e^{i x^\rho P_\rho} D e^{-i x^\rho P_\rho}=D+\left[D,-i x^\rho P_\rho\right]+\frac{1}{2}\left[\left[D,-i x^\rho P_\rho\...
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Perfect gas relation in differential form [closed]
I have a problem to understand the transformation of the perfect gas relation:
$$ \rho\cdot R\cdot T = P $$
into its differential form:
$$\frac {dp}{p} = \frac {d{\rho}}{\rho} + \frac {d{T}}{T}$$
How ...
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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 ...
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Confusion with classic kinematics and functions: weird results
Let think I am studying some simple system (this is a thought experiment), where I have two classic objects, and the position versus time plots follows two curves (I will left out physics constants, ...
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Does centripetal acceleration affects the magnitude of velocity?
A vector has no component in the perpendicular direction, obviously.
But let us consider a situation where a ball is moving in a uniform circular motion tied to a string, then $a = \dfrac {v_1^2} r$.
...
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Where did $1/2$ of this come from? [duplicate]
Work done by an external force $F$ upon a particle displacing from point 1 to point 2 is defined as
$$
W_{12} = \int_1^2 F \cdot dr
\, .$$
Kinetic energy and work-energy theorem: According to Newton's ...
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Force in thermodynamic configuration space
Consider a thermodynamic system whose internal energy $U$ may not be conserved in general. It's a direct consequence of the First Principle that the variations in internal energy do not depend on the ...
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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 ...
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Reduced mass in a Harmonic Oscillator [closed]
I recently came across the harmonic oscillator and the concept of reduced mass, i.e
$$
\mu = \frac{m_1m_2}{m_1 + m_2}
$$
To begin, I understand the derivation from the point of view of sitting on one ...
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Why the normal vector addition does not seem to work in centripetal acceleration? [duplicate]
It is known that centripetal acceleration acts at an angle of 90 degrees to the tangential velocity. This acceleration vector then causes an increment $\mathbf{a} \Delta t$ to be added to the original ...
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Difference between the infinitesimal elements of hemispherical shell and solid hemisphere [duplicate]
Question:
While calculating the center of mass of a uniform solid hemisphere, we consider the differential element:
$$\mathrm{d}m = \rho \pi r^2 R \, \mathrm{d}\theta \, \cos\theta$$
where $r = R \cos ...
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Derivative of the product of a scalar function and a vector valued function
According to Berkeley Physics Course, Volume 1 Mechanics,
The time derivative of a vector valued function can be derived from the formula:
$$
\mathbf{r}(t) = r(t)\mathbf{\hat{r}}(t)
$$
where the ...
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References on a variant of Geometric Calculus
Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism.
I have read a bit about fractional calculus and time-scale calculus, both very ...
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McVittie in Schwarzschild coordinates
I am trying to get McVittie's metric to Schwarzschild's as a homework. In flat geometry ($k=0$) it is given by:
$$ds^2 = -\left(\frac{\mu_-}{\mu_+}\right)^2c^2dt^2 + \left(\mu_+\right)^4a(t)^2\left(dr^...
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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 ...
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Why is linear charge density $dq/dl$ and not $q/l$?
If linear charge density is charge per unit length then shouldn't it be $q/l$. Why is it $dq/dl$ instead? Wouldn't that mean it is only being calculated for a small element and not the whole length?
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How to "rectify" this wave-equation derivation for Longitudinal waves?
To derive the differential equation for longitudinal waves, my professor proceeded like this:
We are using the concept of $N$-coupled oscillators. Consider a slab of length $l$ and cross sectional ...
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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 ...
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Integrating density function that is the inverse square law? [closed]
If I want to calculate the mass of a sphere, and I know the density function is the inverse square law, then it is just a matter of using a volume integral to calculate the mass, that's no problem.
$M=...
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Adding equations of motion
Consider an object of (constant) mass $m$ subject to forces $F_i$ where $i=1,\ldots,n$. Now assume $s_i, v_i, a_i$ are the corresponding equations of motion (position $s$, velocity $v$, acceleration $...
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Fermi poles expansion
I want to prove the following formula:
$$\frac{e^{-tE}}{1+e^{-\beta E}} = \frac{1}{\beta}\sum_{k \in \frac{2\pi}{\beta}\mathbb{Z}}\frac{e^{-ikt}}{-ik+E},$$
for $\beta > t > 0$. I know the trick ...
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Does angular absement exist?
Probably a dumb question. I'm a highschool student, and I don't even know if it is even possible to integrate an angle (for reference, I haven't even learnt integration yet at school, my calculus ...
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Conservation of energy in a mechanical system with a discontinuous potential function
I've started reading Landau-Lifshitz Mechanics, and I'm having trouble with the problem at the end of section 7.
A particle of mass $m$ moving with velocity $\mathbf{v}_1$ leaves a half-space in ...
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How to calculate jerk in uniform circular motion?
We can calculate the centripetal acceleration in circular motion by the equation v^2/r. But how do we calculate the jerk (which is acceleration over time)?
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Series Expansion of logarithm of integral by Landau and Lifshitz in Statistical Physics (First Part)
Let $$E(p,q) = E_0(p,q) + V(p,q)$$ where $V$ represents the small terms. To calculate the free energy of the body we put
$$
e^\frac{-F}{T} = \int' e^{-\frac{E_0(p,q) + V(p,q)}{T}} d\Gamma \approx \int'...
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Need help understanding lagrangian applied to mass on spring problem [closed]
I am interested in learning physics, and I was trying to learn about the Lagrangian method for classical mechanics through https://scholar.harvard.edu/files/david-morin/files/cmchap6.pdf, but I ...
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How acceleration affects velocity?
I understood that the acceleration changes the velocity and the velocity changes the position.
So I tried to calculate the position of a falling object, where $y_{acc} = 9.81$ and the initial values ...
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How to linearise on Lagrangian level?
Consider a Lagrangian density
$$\mathcal{L}(\phi, \nabla \phi) = \frac{1}{2} \, g^{\mu \nu} \, \partial_{\mu} \phi \; \partial_{\nu} \phi + V(\phi) \tag{1}$$
The equation of motion (EOM), i.e. the ...
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How to Derive the Time Evolution Equation for Quantum Phase?
In quantum mechanics, the wavefunction $\psi(x,t)$ outputs a complex number that describes the probability amplitude of finding a particle in a particular place and time. The complex number can be ...
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What is $F$ and what is $P$ in the sentence "$F × dP$ is a total differential"?
The physicist Emilio Segrè, as a student, attended lessons of Calculus given by Francesco Severi and of Analytical Mechanics given by Tullio Levi-Civita. Segrè wrote in his autobiography1
For many ...
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How do force and mass work with all derivatives of position?
I think if $F(t) = kt^0$ then $$x(t) = x_0 + v_0t + \frac{k}{m}\frac{t^2}{2!},$$ and if $F(t) = kt^1$ then $$x(t) = x_0 + v_0t + \frac{k}{m} \frac{t^2}{2!} + \frac{k}{m} \frac{t^3}{3!},$$ and so on, ...
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Is the answer given in the option wrong? [closed]
The question is
"An Object moves along a straight line. The graph illustrates how the acceleration of the object changes with time. The direction of the motion of the object changed only once, ...
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Derivation of Schrödinger equation in Feynman-Hibbs
I am going through the derivation in chapter 4-1 of "Quantum Mechanics and Path Integrals. Emended Edition" by Feynman and Hibbs. The chapter starts with a proof of the equivalence of the ...
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Precise relation between temperature change and physical quantities [duplicate]
I've learnt that many physical quantities like length or volume etc depend on the change in temperature and some proportionality constant as: $\Delta{L}=l\alpha\Delta{\theta}$. In our physics class, ...
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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 ...
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2
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Physics Kinematics Equation $\Delta x = v_f\Delta t - \frac 12 a \Delta t ^2$ Derivation Using Calculus [closed]
I was wondering how you can derive the physics kinematics equation $\Delta x = v_f\Delta t - \frac 12 a \Delta t ^2$ algebraically. I understand where this equation comes from geometrically (when a=...
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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 ...
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