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.

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25 views

Degenerate perturbation with the projection operators

I'm studying degenerate perturbation theory in graphene superlattice. The problem is \begin{align} H=H_{0}+\lambda V \end{align} with \begin{align} \mathcal{H}_{0}=&D\begin{pmatrix} 0 &a &...
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Does the number of field lines crossing an area depend upon angle between them?

Consider Electric Field Lines crossing a square area (for simplicity) such that all field lines are parallel and make an angle say $\alpha$ with the area vector of the square. Let us vary the angle $\...
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47 views

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 ...
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1answer
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Why should we integrate to find displacement instead of directly plugging in $t$ for displacement function?

context:preparing for jee, chapter kinematics. text book problem given $v = 2t^2$ find $x$ at $t=2s$ if at $t = 1s$, $x= 3m$ My solution integrating v to find displacement fucntion $$ x(t) = \int2t^...
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2answers
75 views

Solution to differential equation

If I have a differential equations of the form $$\frac {d^2y}{dt^2}=\alpha^2y$$ Assuming the roots of the characteristic equation is complex the solution to the differential equation is: $$y=C_1e^{j\...
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1answer
66 views

Show identity about divergence theorem

I'm trying to show the following integral equality, but I really can't come up with a proof. The context here is the one of an introductive book to continuum mechanics, so everything is smooth and ...
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2answers
64 views

A clarification on acceleration and velocity

This is one of those questions which require an answer that does not take practical limitations into account. It is a theoretical physics question, perhaps. If there are any loopholes used, please ...
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1answer
76 views

Maximum height of a projectile when $g$ is not constant [closed]

How can I calculate the maximum height of a projectile that is launched from the surface of the earth with a given initial velocity? (ignoring air resistance in the atmosphere) I understand how to ...
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Singular Integrand vs Diverging integral

I am reading Jackson's Electrodynamics and came across this part that I'm not sure I understand. Specifically what is Jackson referring to when he says "it turns out that the resulting integrand ...
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1answer
27 views

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 &...
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Changing dummy integration variables for Lorentz measures

Let's say we have a double integral in spacetime, \begin{equation} \int d^4 x_1 d^4 x_2 f(x_1, x_2)= \int d^3 \vec{x}_1 d^3 \vec{x}_2 \int d x_1^0 d x_2^0\,\, f(x^0_1, x^0_2,\vec{x}_1, \vec{x}_2) \end{...
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Explaining how we cannot account for changing acceleration questions without calculus

For context, I am a high school physics teacher. I am teaching students about the basics of electromagnetic force between two point charges. The equation we use is $F=\frac{kq_1q_2}{r^2}$. This gives ...
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For $\int_{x}^{x+dx} p(x) \,dx = P(x)$ which one is the correct interval, $(x,x+dx)$ or $[x,x+dx]$? [migrated]

In Physics (both in Statistical & Quantum Mechanics) when we describe the probability function of finding a particle between $x$ and $x+dx$, we write $\int_{x}^{x+dx} p(x) \,dx = P(x)$. Here in ...
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1answer
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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 ...
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2answers
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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 ...
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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 \...
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1answer
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Mysterious path integral divergence after Hubbard-Stratonovich transformation

Let us consider common gaussian path integral over some complex random field $\displaystyle \Psi (\mathbf{r})$: \begin{equation*} N=\int D\Psi ^{*} D\Psi \ \exp\left( -\int d^{n} r\ \Psi ^{*}\hat{K} \...
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Real world application of L'Hôpital's [closed]

Can you name and describe some of the real world applications of L'Hôpital's rule where one needs the limit of a model/function at certain point where limit is not defined? I'm looking for examples in ...
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Expressing this result in different coordinates [migrated]

Is there a neat way to express this in Cylindrical and Spherical coordinate systems? $(\vec{A}.\nabla)\vec{B}$ Reference: this occurs quite frequently in Electrodynamics books including Griffiths. ...
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1answer
86 views

Can position be derived from acceleration in practice?

We know that acceleration is the derivative of velocity, and velocity is the derivative of position. But does that mean that we can find position from acceleration in practice (as opposed to in theory ...
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1answer
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Area under any Physics graph

What all aspects of a graph determine or are used to determine the area under a graph? How can a person tell exactly what the area under any graph in this world represents?
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2answers
64 views

How is Green's function used in converting differential equations into integral one

I was reading this section of W. C. Gibson, The Method of Moments in Electromagnetics, Second. Chapman and Hall/CRC, 2014, and I was confused on how they got from 2.25 to 2.26 It seems the integrand ...
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2answers
69 views

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\...
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3answers
95 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) \\ ...
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1answer
18 views

Issue with a derivation in Marion's Dynamics [closed]

I was solving problem 2-14 in Marion's "Classical dynamics of particles and systems" edition 5. In this problem we calculate the range of a trajectory to be $d=\frac{2{v_0}^2\cos{\alpha}\sin{...
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2answers
364 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 $\...
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1answer
43 views

Am I applying this vector-based equation correctly to my dataset? [closed]

A coworker and I need to perform what is probably a simple calculation with our data so that we can calculate something else. However, our vector calculus skills are rusty and we don't know if we ...
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3answers
60 views

Problem in finding the divergence at a point [duplicate]

I am solving a problem given as Divergence of $\frac{\textbf{r}}{r^3}$ is a) zero at the origin b) zero everywhere c) zero everywhere except the origin d) nonzero everywhere The answer is given as (...
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3answers
104 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 ...
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1answer
45 views

Proving that acceleration perpendicular to velocity only changes it's direction [duplicate]

In a recent class, I learned about centripetal acceleration and that if a body moves in uniform circular motion the direction of velocity continuously changes implying presence of an acceleration. My ...
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2answers
154 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}\\ ...
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2answers
72 views

Logarithm of Grassmann numbers

What is $\log \theta$ where $\theta$ is a Grassmann number such that $\theta^2 = 0$. How does one then look at its logarithm? Does it even make sense?
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1answer
45 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 ...
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0answers
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Finding the time required for an object to move from point A to B when the Force exerted is a function of distance? (Proportional to the inverse sqr) [duplicate]

I know we will have to use double integration. But I need some help in making and solving the equation. The example of these scenarios is the time taken for an object falling towards Earth from ...
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1answer
28 views

How did the result of the following derivations come about? [closed]

Can someone please explain how the exponent was included in the equation? How did they convert the equation into constant x exponent form? I think I'm missing some rules of how to convert into ...
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2answers
61 views

Integration with complex Grassmann numbers

I have a question about a convention from Peskin & Schroeder, namely that $$\int d\theta^{*}\, d\theta \, (\theta \theta^*) = 1,$$ where $\theta$ and $\theta^*$ are independent Grassmann numbers. ...
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1answer
118 views

Proof of empirical temperature from Zeroth Law given in Kardar's book

I am studying Mehran Kardar's Statistical Physics of Particles. In chapter-1, they derive the existence of empirical temperature using Zeroth Law. It is given as Definition- If two systems, A and B, ...
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0answers
10 views

Voltage fluctuations

I simulate a neuron by compartments, where each compartment is represented by an electrical circuit. Myelinated nerves have reduced capacitance - so for example, if $n$ denotes the membrane layers, ...
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1answer
58 views

Doubt in deriving a relation for internal energy as a function of $T$ and $P$

I am solving a problem given in Dittman and Zemansky's Heat and thermodynamics. The problem is given below- Regarding internal energy of a hydrostatic system to be a function of $T$ and $P$, derive ...
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3answers
73 views

Problem with derivation of formula for change in entropy for latent heat

We know, $dS=\frac{Q}{T}$[when $Q\to 0$] $\implies \Delta S=\int\frac{Q}{T}$ $\implies \Delta S=\int \frac{ms}{T}$[m= mass, s= latent heat] $\implies \Delta S=\frac{ms}{T}\int$ This is where I get ...
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2answers
94 views

What is distance multiplied by time?

From grade 6th we have been taught that if we find the area of the velocity-time graph is distance covered or in other words if velocity is constant then distance= velocity* time as velocity = ...
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2answers
29 views

Contact surface between circle and curved floor?

First, consider an inelastic circle on a hard, flat floor, the shared contact surface between the two is some infinitesimally small length dx. Consider a second case, where if the floor had some ...
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48 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 ...
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3answers
68 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 ...
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6answers
1k views

Can acceleration depend linearly on velocity?

Is it possible that acceleration may vary linearly with velocity. Is it practically possible, if so is there a practical example of it? By integration I was able to verify that for the above case to ...
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3answers
52 views

Motion in a plane situation

There is something weird I find about the following situation. Suppose a particle has the $X$-coordinate $= 2+2t+4t²$ and $Y$-coordinate $= 4t+8t²$. So it's velocity in $X$ is $2+8t$ and velocity in $...
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33 views

How can I calculate two gases approaching equilibrium over time?

There are two well insulated, rigid tanks (a and b) with known volumes. Each contains a gas with different temperatures, mass, density, specific heat, etc. Gas in tank 'a' has higher pressure than gas ...
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0answers
35 views

Linear harmonic oscillator in quantum mechanics, why we expand the potential into Taylor series?

We have started our first problem in QM by solving Schrödinger equation for the linear harmonic oscillator potential. I noticed that the first thing we did was to expand the potential near the stable ...
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1answer
49 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'(...
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1answer
27 views

Barometric formula with variable gravity

Recently, I have wondered about what would be the atmospheric pressure as a function of altitude in a planet that only consists of gas. The equation that does just that is the barometric formula (see ...

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