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Questions tagged [calculus]

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How do I show that the two definitions of the curl of a vector field equal each other? [migrated]

The curl of a 3D vector field is a 3D vector itself and has two definitions - one in integral form and one in differential form. Definition 1: $$ \operatorname{curl}\vec{F}(x,y,z) \, \cdot \, \hat{n} ...
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1answer
45 views

Why $\int dx \partial_\mu\neq \partial_\mu \int dx$ but $\int dp \partial_\mu=\partial_\mu\int dp$?

It's well known that $\int dx \partial_\mu\neq \partial_\mu \int dx$. But I have a hard time understanding $\int dp \partial_\mu=\partial_\mu\int dp$, because $[p,x]\neq0$ do not commute. However, ...
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49 views

The two definitions of the divergence of a vector field? [migrated]

Now, I am aware that the divergence of a vector field, $\vec{F}$, can be defined in two ways. What I don't understand is why do these equal each other formally? Definition 1: $$\text{div}\vec{F} = \...
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51 views

Why is this equation correct?

This problem seems to be solved the exact same way in multiple solution books, so I'm certain that the way it is done is correct and that I'm just rusty when it comes to calculus due to multiple years ...
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1answer
44 views

How do you derive $v_f^2 = 2ad + v_i^2$ from $v_f = v_i + at$ and $d = v_it + (a/2)t^2$?

I'm trying to use substitution to find the equation Vf^2 = 2ad + Vi^2 from Vf = Vi + at and d = Vit + a/2(t^2), but I get stuck understanding part of the math. I've attached a picture, and you can see ...
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1answer
64 views

Find $v(t)$ and $x(t)$, How do I treat $δt$? [closed]

We apply a force to a particle with a mass $m$ and inicial velocity $v_0$: $$ F(t) = \left \{ \begin{matrix} 0 & \mbox{ $t<t_0$} \\ \frac{p_0}{\delta t} & \mbox{ $t_0<t<t_0 +\...
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1answer
48 views

(Physics version of) Taylor expansion. In the the context of deriving a Lie groups generators (a Lie algebra from a Lie group)

Statement which I'm confused about: "Consider some n-dimensional Lie group whose elements depend on a set of parameters $\alpha = (\alpha_1 ... \alpha_n)$, such that $g(0) = e$ with e as the identity,...
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1answer
60 views

Divergence of a vector multiplied by dot product [closed]

If I am correct, then $\operatorname{div} [(\vec A\cdot \vec B)\vec C] = (\vec A \cdot \vec B) \operatorname{div} \vec C + \vec C \cdot \nabla (\vec A\cdot\vec B)= (\vec A \cdot \vec B) \operatorname{...
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0answers
46 views

Electric field of an infinite sheet of charge calculated without Gauss law

Problem Given an infinite sheet with a surface charge density $\sigma$ in the x-y plane. The charge density is given by $\rho(z)=\sigma\operatorname{\delta}(z)$. Calculate the electric field without ...
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“Euclid’s test” , Negative Pressure and Measure Theory

I don't understand what does it mean for 'Euclid's Test' when they talk about negative pressure Using Euclid’s test for a hypothesis of examining its implications, one finds that negative pressure ...
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2answers
2k views

How does instantaneous velocity or acceleration have any other numerical value than 0? [duplicate]

Instantaneous velocity is defined as the limit of average velocity as the time interval ∆t becomes infinitesimally small. Average velocity is defined as the change in position divided by the time ...
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0answers
15 views

Born approximation of scattering field

So I am trying to solve the Born approximation of the scattering field for an incoming plane wave $E_0 e^{i K x}$ for a Kramers-Kronig permittivity profile which is given as $$\alpha (x)=\frac{i-x}{...
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1answer
45 views

Gauss's law for magnetism : double integral

Gauss's law for magnetism is stated as followed with the beautiful closed surface double integral (by wikipidia): $$ \mathop{\vcenter{ \huge\unicode{x222F}\, }}_{S} \mathbf{B} \cdot \text d\...
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6answers
1k views

Question about derivation of kinematics equations

Apologies if this has been asked before, but I browsed the sub and couldn't find something specific. I understand the derivation for one of the equations as follows: \begin{gather} \frac{dv}{dt} = a ...
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1answer
65 views

Derivation: Maximum entropy implies minimum energy

Callen claims in his book (chapter 5 in my copy at least) that the condition of minimum energy for fixed entropy is exactly equivalent to the condition of maximum entropy for fixed energy. I have seen ...
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2answers
133 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
65 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
52 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
26 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
73 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
56 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
73 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
57 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
34 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
53 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
408 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
43 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
83 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
101 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
51 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
179 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
46 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 ...
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2answers
97 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
69 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
123 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
11 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
28 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
48 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
53 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
116 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
113 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
63 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
45 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
23 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
46 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....