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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3 votes
2 answers
169 views

Converting differential to gradient

Landau & Lifschitz's fluid mechanics book proposes the following statement for an isentropic proccess: $$dH=vdp \Rightarrow \nabla H=v\nabla p$$ What's the rigorous way to get this result (...
2 votes
1 answer
62 views

Leibniz's Theorem [closed]

I'm not familiar with Leibniz's Theorem, and by the time I added my substitutions, I got lost in the variables and how they are suppose to transform. Please help?
0 votes
3 answers
51 views

Thermodynamic adiabatic process, question regarding mathematical operations

I have a question regarding mathematical operations often seen in physics books: In an adiabatic process the heat is 0, so by the first law of thermodynamics we have that $E = W$, and an infinitesimal ...
0 votes
2 answers
35 views

Circular motion equivalent in three dimensions [closed]

Are there equations or even a concept of circular motion/tangential acceleration/centripetal acceleration in three dimensions? Maybe something called "spherical acceleration"? or am I just ...
3 votes
1 answer
149 views

Prove that Gauss's Law Holds Under Translations

We know that Gauss's Law says $\nabla \cdot E(\textbf{x}) = \frac{\rho(\textbf{x})}{\varepsilon}$, and we also know that this relationship should be true regardless of where you're located in three ...
-1 votes
1 answer
41 views

Derivative of distance [duplicate]

I know that $speed = |\frac{\vec{dr}}{dt}|$ and first derivative of distance with respect time will be $\frac{d\vec{|r|}}{dt}|$ These 2 expressions don't seem to represent the same thing. But when I ...
1 vote
1 answer
36 views

Limit definition of scalar curvature for flat vs curved space in 2D, 3D and so on in Zee

In Zee's book, Einstein Gravity in a Nutshell, p. 6 + p. 77, he says that \begin{equation} R = \text{lim}_{\text{radius} \rightarrow 0} \frac{6}{(\text{radius})^2} \left(1 - \frac{\text{...
0 votes
2 answers
27 views

Finding the distance under a position-time graph

I have to find the distance travelled by a particle between t=0 and t=4 secs for x(t)= 4t^2 - 2t^3. I tried differentiating the equation to find v(t) then integrating it within t=0 and t=4, but that ...
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0 votes
1 answer
23 views

Clarification for derivatives under a change of variables

In Special Relativity and Classical Field Theory by Susskind, he says that we can imagine a function of $(x+ct)$, then he says that we can consider its derivatives and easily see that $$\frac{\...
2 votes
0 answers
51 views

Dirac delta function at singularities in spherical coordinates [migrated]

Background Information Let $\delta^3(\vec x-\vec a)$ represent a point density at $\vec a$. It satisfies $$ f(\vec a)=\int \delta^3(\vec x-\vec a)f(\vec x)|J(\vec x)|\mathrm d^3\vec x, $$ where $f(\...
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0 votes
1 answer
37 views

What are the relationships between the motion-time graphs?

I was wondering if someone could explain the relationships between the three motion graphs (Position-Time, Velocity-Time, and Acceleration-Time). I believe that the slope of the P-T is Velocity and ...
0 votes
1 answer
61 views

Does the arc length constant of the sine function interval occur anywhere in physics?

π occurs in many formulae, but does the arc length of a single sine function interval have any meaning in physics? Does it occur in any formula? According to What is the length of a sine wave from 0 ...
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1 vote
2 answers
51 views

Question regarding eliminating volume term from Gauss Law

Gauss law is given by $$\oint_{\partial S}\vec E\cdot d\vec {A}=\dfrac{q_\text{enclosed}}{ε_0}.$$ $$q_\text{enclosed}=\iiint \rho\ dV.$$ For a closed surface $$\oint_{\partial S}\vec E\cdot d\vec{A}=\...
1 vote
0 answers
33 views

Is $n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} =\frac{60}{2\pi}\sqrt{g\frac{\int y_idx}{\int y_i^2dx}}\quad ?$

I have a question about this formula used to calculate the first critical speed of a drive shaft. $$ n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} \tag {1} \quad .$$ It is the ...
-2 votes
1 answer
100 views

What is the General formula of gradient of $r^n$? [closed]

so, the question is that r is the separation vector from a fixed point $(x',y',z')$ to the point $(x,y,z)$ and let $r$ be its length. the answer to the question of what is the general formula of $$\...
0 votes
1 answer
79 views

How do I reconcile these two definitions of acceleration?

How do I reconcile these two definitions of acceleration? $$a=\frac{d\bar{v}}{dt}=(\frac{dv^k}{dt}+v^i v^j \Gamma^k_{ij})\bar{e}_k \tag{1}$$ and $$a^k=v^{\small\beta} \nabla_{\small\beta} v^k.\tag{2}$$...
0 votes
2 answers
58 views

How to calculate $\nabla\cdot (|L_{\perp}| \vec{v})$?

Let's assume I know $\nabla\cdot (|L_{x}| \vec{v})$ and $\nabla\cdot (|L_{y}| \vec{v})$, where $\vec{v}$ is the velocity field and $L_x$ and $L_y$ represent classical angular momentum components in $\...
1 vote
1 answer
82 views

Proof that $\nabla \times E = 0$ using Stoke's theorem [closed]

One way that Jackson proves that $\nabla \times E = 0$ is the following: $$ F = q E $$ $$ W = - \int_A^B F \cdot dl = - q \int_A^B E \cdot dl = q \int_A^B \nabla \phi \cdot dl = q \int_A^B d \phi = ...
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1 vote
3 answers
117 views

Is the acceleration vector half of the gradient of velocity squared?

Consider the differentiation of speed squared with respect to time: $$\frac{d(v^2)}{dt}=\frac{d(\mathbf v\cdot\mathbf v)}{dt}$$ $$=2\mathbf v\cdot\frac{d\mathbf v}{dt}$$ $$=2\mathbf v\cdot\mathbf a$$ $...
0 votes
1 answer
32 views

Spherical and Cartesian forms of divergence [closed]

Suppose the electric field found in some region is $$\overrightarrow{E} = ar^3\vec{e}_r$$ in coordinates spherical (a is a constant). What is the charge density? So, using the spherical form of ...
3 votes
2 answers
118 views

Acceleration in terms of displacement

I am having problems understanding the derivation of acceleration in terms of displacement. The first step is fine: $$a(x) = \frac{\mathrm dv(x)}{\mathrm dt} = \frac{\mathrm dv(x)}{\mathrm dx} \frac{\...
0 votes
1 answer
36 views

Two questions concerning dirac delta function and Hamiltonian

I'm trying to compute to quantities with Hamiltonian and Dirac delta function but I don't how to do it properly. I'm stuck calculating the following quantity $$ \frac{d}{dE} \left[ \theta(E-H(x,p;V)) ...
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0 votes
1 answer
30 views

Why should I find value of constant of integration instead of using definite integration?

$v = 2t$, find $s$ (displacement) at 3 seconds if at $t=0$ body is 2m behind origin. My teacher said that I cannot use definite integration as we do not know the boundaries for time, so I have to ...
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1 vote
2 answers
71 views

What does instantaneous displacement really mean?

I've read and (I think) understood that instantaneous velocity is the velocity of an object for an "infinitesimally" small time interval: $$ \lim_{\Delta t\to0} \frac{\Delta s}{\Delta t} = \...
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0 votes
1 answer
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What is the justification for the "physics approach" for integration?

When I say the "physics approach" to integration, I mean considering tiny slices and taking the limit, such as when you find the total charge in a rod by considering tiny slices of length $...
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5 votes
5 answers
227 views

Is $dx$ always positive?

When we refer to change in a quantity, we define it to be (final-initial). If it is positive it indicates an increase from the initial value and negative indicates a decrease. But when we take this to ...
1 vote
0 answers
12 views

Shockley-Queisser limit calculations

I am working through the paper, Detailed Balance Limit of Efficiency of p-n Junction Solar Cells by Shockley & Queisser, for my own research with perovskite solar cells. I am able to follow their ...
0 votes
3 answers
72 views

Why does the integral of $E\psi(x)dx$ go to zero around the the delta function? [closed]

My lecturer writes: Firstly, I assume the term with a second derivative is, well, exactly that - a second derivative and therefore intended to be $\frac{d^2\psi(x)}{dx^2}$ and not $\frac{d^2\psi(x)}{...
2 votes
1 answer
46 views

When exactly does velocity increase or decrease on an acceleration time graph? [closed]

How does the acceleration time graph show if and object is speeding up or slowing down? Is it possible to find the answer without any deep calculations? If yes then how? Like how can I find the ...
2 votes
1 answer
33 views

Does the Continuity Equation Imply it's Zero for Incompressible Flow?

This might be an incredibly simple question, but I haven't been able to figure it out on my own. I'm an engineer by trade, so please forgive my unfamiliarity with vector calculus. I'm interested in ...
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0 votes
2 answers
52 views

Trying to understand gravitational force equation

I don't understand the red underlined equations but I understand gravitational force equation in simpler form. Can anyone please explain the equations? Source An Introduction to Celestial Mechanics by ...
0 votes
0 answers
24 views

Minimum force to hold a large number of repelling particles in a volume

I need a way to find the minimum amount of force that would need to be exerted to keep a large number of identical point particles which repel each other contained in some region, assuming no other ...
0 votes
2 answers
32 views

Upper and lower limits of integration in the derivation of kinematic equations?

I have already read part of this answer, but I need further clarification. In the derivation of kinematic equations, the following two lines are present: Acceleration is constant. $\text dv=a\text dt$ ...
2 votes
2 answers
67 views

How to show that a radially symmetric central force is conservative?

Let $U\subseteq \mathbb{R}^3$ be open and $f:U\to\mathbb{R}^3$ be a radially symmetric central force, that is, a force field such that $$f(p) = -g(r)u_r$$ where $r=|p|$ and $u_r$ is the unit vector ...
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0 votes
3 answers
248 views

Goldstein: derivation of work-energy theorem

I am reading "Classical Mechanics-Third Edition; Herbert Goldstein, Charles P. Poole, John L. Safko" and in the first chapter I came across the work-energy theorem (paraphrased) as follows: ...
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0 votes
2 answers
93 views

What problems would arise in physics by treating infinitesimals as ~1 (in units much smaller than the measurement precision) rather than ~0?

I have a hopefully simple/ignorant question. The difference quotient, where $h$ is an "infinitesimally" small value: $$f'(x) = \frac{f(x + h) - f(x)}{h}$$ What problems (if any) would arise ...
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0 votes
0 answers
33 views

Wavefunctions and Hamilton-Jacobi equation [duplicate]

I was reading this paper, Wavefunctions and Hamilton-Jacobi Equation. The author started with the Hamiltonian formalism and then came up with a connection to the Schrodinger equation. There was a ...
1 vote
0 answers
62 views

Wavefunction from the Hamilton-Jacobi formalism [closed]

I was reading this paper, Wavefunctions and Hamilton-Jacobi Equation by Sabrina Pasterski. The author started with the Hamiltonian formalism and then came up with a connection to the Schrodinger ...
2 votes
1 answer
105 views

How does this scalar transformation law make sense?

A scalar field doesn't transform under a change of co-ordinates. Therefore, a scalar field $\phi(x)$ transforms to $\phi'(y)$ under the coordinate transformation $y^{\mu} = x^{\mu} + \epsilon^{\mu}(x)$...
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-5 votes
1 answer
86 views

Differentiation [closed]

Why is $$\frac{d}{dt}v^2=2v\frac{dv}{dt},$$ When: $$\frac{d}{dx}x^2=2x,$$ where $v$ is velocity? I don't understand why the variable $x^2$ has the derivative of $2x$, whereas the variable velocity has ...
1 vote
5 answers
112 views

The value of $g$ in free fall motion on earth [closed]

When we release a heavy body from a height to earth. We get the value of $g=9.8 \ ms^{-2}$. Now, I'm confused about what it means. For example, does it mean that the body's speed increases to $9.8$ ...
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0 votes
1 answer
93 views

How to attain the period of this nonlinear differential equation/system?

Lately, I've been trying to find the period of an angle included in the following differential equations, but only could with the basic model: Basic or original: $$\mathrm{For}\ (\Phi (0), \Omega (0))=...
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6 votes
7 answers
141 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 ...
1 vote
3 answers
135 views

Issue expanding $\sin \theta$ about $\theta_{eq}$

Quoting a textbook: $$(m_1 + 2m_2\sin^2\theta)\ddot\theta = m_1\Omega^2\sin\theta\cos\theta - \frac g L (m_1 + m_2)\sin\theta.\tag{10}$$ We can simplify this expression a bit by relating $\frac g L (...
0 votes
2 answers
96 views

Solving an inexact differential

So we have something called an inexact differential which is when a function is path dependent meaning I can't just subtract the initial and final states to get an answer. Take for example work which ...
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0 votes
6 answers
95 views

Deriving Work-Kinetic Energy Theorem

I am currently reading Physics for Scientists and Engineers (Ninth Edition) by Serway and Jewett and in Chapter 7.5, a derivation of the work-kinetic energy theorem was shown. To give context, ...
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0 votes
1 answer
53 views

Advection term for a matrix equation

How can I calculate a quantity like $(\vec{v} \cdot \nabla) M$ where $\vec{v}$ is the velocity vector, and $M$ is some 3x3 matrix? (if one wants, assume $M$ is a tensor) This would be the advective ...
0 votes
1 answer
44 views

Determine the meaning of a gradient of a graph [closed]

How do you determine the gradient of a graph in physics, such as how with a velocity-time graph, the gradient is acceleration. I want to know the general method for figuring out what the differential ...
0 votes
1 answer
68 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}{\...
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1 vote
2 answers
69 views

Why can we change $dt$ with $(dt/dp)_s dp$?

In my homework assignment there's the following question: A general thermodynamic system is being compressed isentropically from pressure $P_i$ to $P_f$ while keeping the number of particles constant....

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