Questions tagged [calculus]

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Minimum calculus for Lagrangian Mechanics [on hold]

I have noticed in my classical mechanical books, the development of the motions of equation for simple systems using the Lagrangian that the main complication, if doing the work by hand, is applying ...
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2answers
115 views

Are the infinitesimal lengths in integrals bounded by the Planck length? [closed]

When we integrate something say work, $\int F\cdot ds $ then we will get work but what exactly is $ds$? how much is ds? Is it the Planck length? Are we just pretending there exists some infinitesimals ...
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1answer
64 views

An electrodinamic identity: starting point [closed]

With this request, I would like to ask you kindly how you can prove this identity. I thank you for those who can help me. \begin{equation} \overline{\nabla} \times (\overline{\nabla} \times \...
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1answer
49 views

What is the gravity on a “partial” ringworld?

This was inspired by https://worldbuilding.stackexchange.com/questions/149706/life-on-the-broken-ring-an-issue-of-size. Let's say I have a part of a Ringworld (see link for specifications). ...
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1answer
23 views

Interpretation of surface integral of vector field over surface

Is it correct to interpret the surface integral of a vector function $\mathbf{v}$ over four sides of a cube as the rate of flow of fluid (in mass per unit time) that would flow out of the cube when ...
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5answers
71 views

Equation of distance and time

How is this equation derived? $$r = r_0 + ut + at²/2$$ where $r_0$ is the initial position of particle and $r$ is the position of the particle after all the motion it has undergone, $a$ and $t$ ...
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1answer
55 views

Old unsolved question on greens function

So I was looking up Kf Riley’s 3rd edition, and bump into a problem about greens function. I went online and googled and notice other people had the same problem and no one really could answer: ...
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1answer
71 views

Partial Integration and the Levi-Civita Symbol

I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized ...
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1answer
48 views

4-gradient of retarded time

To prove explicitely that the Liénard-Wiechert potentials satisfy the Lorenz gauge, one has to find the time derivative and gradient of the retarded time. In this Wikipedia article it's calculated as ...
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0answers
31 views

Any tips for memorising acceleration in non-cartesian coordinates? [closed]

I need some efficient way to pop up how acceleration in cylindrical/spherical coordinate looks like at the test. Of course I could derive it on the spot, but it takes too much time (for me) and I ...
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2answers
30 views

Intuition of Distance covered when accelerating [duplicate]

When you're moving at $5$ m/s for $1$ second, you have traveled $5$ m. When you're moving at $5$ m/s (initial velocity) and you accelerate $2$ m/s for $1$ second, you have traveled $5$ m + $1$ m (...
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2answers
30 views

Derivation of Magnetic Field from Infinite Wire

I'm trying to get my head around the derivation for the magnetic field of an infinite wire, in my notes I have the statement: Setup: Wire centred on the z axis, current has direction +z. "Biot-Savart:...
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1answer
38 views

Change of variables in gradient

Take two coordinates with $\mathbf r$ and $\mathbf r'$ and take a function $f(|\mathbf r - \mathbf{r'}|)$. In many electromagnetism derivations I see a conversion like this $$ \nabla_r f(|\mathbf r - \...
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3answers
400 views

Calculating the moment of inertia of a uniform sphere [closed]

Currently trying to calculate the moment of inertia of a uniform sphere, radius R, I know the answer is $\frac{2}{5}MR^2$ but I keep getting $\frac{1}{5}MR^2$ Setup: Assume mass per unit volume $\...
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2answers
35 views

How can differentiating a scalar like potential energy functions give a vector like field strength functions?

Just wondering if anyone had a good explanation of how differentiating a scalar can give rise to a vector quantity.
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0answers
59 views

Why don't we define time derivative of acceleration? [duplicate]

When we started the study of kinematics we defined position and its change with respect to time. After that we defined time derivative of velocity which gave us acceleration. These 3 concepts really ...
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3answers
72 views

How to derive kinematics equations using calculus? [closed]

I read derivation of kinematics equations using calculus: $$a=\frac{\text dv}{\text dt}$$ $$\implies \text dv=a\text dt$$ $$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$ $$\implies v-v_0=at$$ $$\...
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1answer
43 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 ...
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2answers
177 views

$\int (f(x+\delta x) - f(x)) dx = \int \left ( \frac{df(x)}{dx} \delta x \right) dx$

From Landau and Lifshitz's Mechanics Vol: 1 $$ \delta S= \int \limits_{t_1}^{t_2} L(q + \delta q, \dot q + \delta \dot q, t)dt - \int \limits_{t_1}^{t_2} L(q, \dot q, t)dt \tag{2.3b}$$ $$\Rightarrow ...
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0answers
44 views

A. Zee Contour Integral

In A.Zee's book I have come a cross an integral which I found difficult to solve.
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1answer
34 views

Change of variable in function

Suppose I have a function $h(\theta)$ measuring the height of a piston, with $\theta = \omega t$. I would like to know the vertical acceleration of this piston as $\omega$ changes at the point $\theta ...
3
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2answers
86 views

Can someone provide to me an intuitive explanation of the second integral of position with respect to time?

I am aware of what the first integral of position, absement means (at least to a very superficial level). However, I can find nothing regarding the physical intuitive meaning of absity, the second ...
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1answer
66 views

How would you calculate the half-life of a source with one long measurement?

I'm working on a project, and I understand that half life of a source is typically calculated with time intervals (eg 5 sec intervals for 20 minutes) and then using that data to find the decay ...
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2answers
115 views

The chain rule and velocity transformation in relativity

From elementary calculus, we have that the chain rule occurs when we differentiate a function like $f(y(x)) \equiv f(x)$: $$\frac{\mathrm{d}}{\mathrm{dx}}[f] = \frac{\mathrm{d}}{\mathrm{dx}}[f(y(x))] ...
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0answers
10 views

Need a sample of a Probability density function in Oblate Spheroidal Coordinates

I need to develop a probability density function in Oblate Spheroidal Coordinates. That is, the volume under a this function surface is equivalent to 1 . Any idea how to propose this ? In a two ...
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0answers
27 views

Why do we change signs arbitrarily while calculating RC circuits formulas?

When I have to calculate the formulas regarding RC circuits, for example the process of charging a capacitor, there is a discrepancy between my calculations and those of all the books I can find. Let'...
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0answers
27 views

Showing conservation of momentum for arbitrary pendulum trajectories

Consider an isolated system of a pendulum driven by a motor, initially at rest. Conservation of momentum and angular momentum ordains that the center of mass and orientation cannot change in the ...
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0answers
45 views

Acceleration as the second derivative of displacement function

Let $x$ be displacement as a function of time $t$ and some other physical quantity $k$ such that $ x = f(t,k) $ Now, 1) Will the acceleration $a$ be $\frac{\partial^2 x}{\partial t^2}$ or $\frac{d^...
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1answer
39 views

Need some help to show this relationship using parseval's theorem [closed]

Use Parseval’s theorem for the Fourier series and take L → ∞ to show that:
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2answers
109 views

Denoting the antiderivative of velocity

With simple Newtonian laws (and in a specific context), I learned that the speed $\vec{v}$ of an object is the derivative of the corresponding position vector $\vec{OM}$. So that means that $$\vec{v}(...
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1answer
68 views

Proving a theorem about the average value of a function over a specific region

Let's say transient phenomenon in a function. A transient phenomenon is defined as: "A transient event is a short-lived burst of energy in a system caused by a sudden change of state." So, for ...
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1answer
77 views

Energy of continious charge distribution

In the book of Griffith intro to electrodynamics, on page 94, the energy of continuous charge distribution is derived in the following way: W(total energy) = $\frac{1}{2} \int\rho V d\tau$, where $\...
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1answer
56 views

If kinetic energy is mass times the integral of velocity, isn't it just a product of mass times distance? [closed]

I'm still learning Calculus at the moment and I'm currently on integration. The moment I realized the "$1/2$" and square value in $v^2$ are just products of integration, can't one just use ...
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1answer
41 views

Electric field on the boundary of a continuous charge distribution

In Purcell and Morin's Electricity and Magnetism, 3rd Edition, the claim is made that the magnitude of the electric field on the boundary of a continuous charge distribution is finite (assuming the ...
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1answer
22 views

Units of forcing function in the inhomogeneous wave equation

The units of the d'Alembertian are distance$^{-2}$. It should be the case that the inhomogeneous wave equation describing $$\square u = f$$ should have matching units on both sides. My understanding ...
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0answers
44 views

Can I solve this electric forces question by integrating velocity with respect to displacement?

In my physics tutorial for electricity, there was one question in particular that struck me as interesting. It is as follows: One particle has a mass of $3.00\times10^{-3}$ kg and a charge of $+8....
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0answers
30 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 ...
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1answer
298 views

What does the area under the curve of a temperature-time graph represent?

I’m trying to calculate the total heat produced by a system over a period of time and I’ve gotten a regression line of y= log x to represent the best produced by the system. To calculate the total ...
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0answers
41 views

Equivalence between Maxwell's equations and vector Helmholtz equations

When are equivalent the Maxwell's harmonic equations: $$ \nabla\times\left(\nabla\times\mathbf{E}\right)=\mu\epsilon\omega^2\mathbf{E} $$ and the vector Helmholtz equations: $$ \nabla^2\mathbf{E}=\mu\...
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1answer
24 views

Kinematics Problem requiring Calculus [closed]

Let the Instantaneous Velocity of a rocket just after launching be given by v={ 3t for 0<= t <2 2t+ 3t^2 for 2<= t<=3 t^3 for t>3 Find the ...
4
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3answers
118 views

Vector calculus in classical fields

The action is defined as: $$S = \int d^2\textbf{x}\,dt \left[\left(\frac{\partial h}{\partial t}\right)^2 + (\nu \,\nabla^2h)^2\right]$$ The equation of motion is asked for, so use Euler-Lagrange: $$\...
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1answer
386 views

What is the physical meaning of divergence? [duplicate]

I want to visualize the concept of divergence of a vector field. I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of ...
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2answers
95 views

Projectile Motion with Horizontal Variable Acceleration?

I am a High School Student and I've started to learn about Projectile Motion. One of the Assumptions made in the text is that the horizontal acceleration must be $0$ for the Equations of Projectile ...
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1answer
36 views

Why did we take gradient outside the integral sign in Scalar potential derivation?

I tried to understand the reasoning given in it but I couldn't understand it. It says that "as the gradient operation involves x and not the integration variable x', it can be taken outside the ...
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0answers
217 views

Displacement current derivation

A common explanation for the reason why displacement current is needed is in the following diagram (Giancoli): I can appreciate the reason why we need displacement current, however I really don't get ...
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1answer
177 views

How to know what the area under curve represents?

Is there a way to find out what the area under the curve represents? For eg. If i gave you a graph of $v$ with respect to $t$ would you be able to tell me what the area under the curve represents ...
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2answers
174 views

Problem with loop Integral (HQET)

I have come across the Integral: $$ \int_0^{\infty}dx [x^2-ixa+c]^{n-\frac{d}{2}}e^{-bx},$$ where $n = 1,2 ; a,b,c,d \in \mathbb{R}; b,d > 0$. This integral should contain some divergences for $d ...
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2answers
84 views

Dispersion Relations in Particle Physics [closed]

Please tell me how to get the identity(2) in this image
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1answer
42 views

Using probability of camera flash interval to get the probability density equation in Griffith's Quantum Mechanics book

In Griffith's QM, example 1 chapter 1, what is the intuition behind using the probability of camera flash interval to get the probability density equation in terms of "dx". Griffith says that ...
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2answers
33 views

Moment of Inertia equation for small volume

Below is the equation of the moment of inertia for small volume elements, $\Delta m$ $$I = \lim_{\Delta m_i \to 0} \sum_{i} r^2_i \Delta m_i = \int r^2 dm$$ Can someone please explain it to me on ...