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Questions tagged [calculus]

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3answers
395 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
55 views

An object moves with constant acceleration. What would be the derivative of the unit velocity vector of that object with respect to time? [closed]

there was a question saying that: "An object is moving with constant acceleration a. which of the following are also constant?" (a) $$\frac{\text d\vert\vec v\vert}{\text dt}$$ (b) $$\vert\frac{\...
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2answers
24 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
17 views

Isomorphism between functions [closed]

Is there an intuitively achievable isomorphism between the functions of a component in $3N$ variables (for example, in a configuaration space of $N$ particles in a three-dimensional physical space) ...
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0answers
48 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
57 views

How to derive kinematics equations using calculus? [closed]

I read derivation of kinematics equations using calculus: $$a=\frac{\text dv}{\text dt}$$ $$\implies \text dv=a\text dt$$ $$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$ $$\implies v-v_0=at$$ $$\...
1
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1answer
36 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
171 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
36 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
77 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
65 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 ...
1
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2answers
108 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
23 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
24 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
42 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
33 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
106 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
41 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
48 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
33 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 ...
0
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1answer
20 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
43 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
154 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
40 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
116 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
286 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 ...
1
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2answers
66 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 ...
1
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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
193 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
103 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 ...
3
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2answers
167 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
83 views

Dispersion Relations in Particle Physics [closed]

Please tell me how to get the identity(2) in this image
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1answer
40 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
25 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 ...
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1answer
28 views

Acceleration and Velocity Zero at Same Time [closed]

Since velocity is the antiderivative of acceleration ∫a(t)dt, how would one take a starting velocity and acceleration and determine what constant change in acceleration would be needed for a and v to ...
0
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1answer
21 views

Scalar field and 2 types of line integrals

Consider the line integral, $\int _ c$f(x,y)$\vec dr$ , where $f(x,y)$ is a scalar field, and it is evaluvated on a curve $c $. After integration we get a vector let it be $\vec I$ . $\int _ c$f(x,...
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1answer
44 views

What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?

In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as: $a(t) = a_0$ $v(t) = a_0t + v_0$ $x(t) = \frac{1}{2}...
1
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1answer
229 views

Electric field at any point due to a continuous charge distribution

I am reading Purcell and Morin's Electricity and Magnetism 3rd Edition. Equation ($1.22$): $$\vec{E}(x,y,z)=\dfrac{1}{4 \pi \epsilon_0} \int \dfrac{ρ\ (x^\prime, y^\prime, z^\prime)\ \hat{r}\ dx^\...
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1answer
77 views

Calculate launch angle of object moving away from view

I'm writing image processing software and my goal here is to take an image of a projectile moving away from the camera and determine the launch angle. What I already know is: The actual size of the ...
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1answer
42 views

Calculating the distance between two masses with respect to gravitational force [duplicate]

Call them $m_1,m_2$. They are compressed to their center of masses, if you wish. If the initial distance at $t=0$ is $d$, is there a formula or an efficient way to calculate the distance between them ...
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1answer
89 views

Two-level laser rate equation

I am stuck on what I assume is a very basic rearranging of terms in Siegman's Lasers, Page 204. Here, the saturation of a laser medium is introduced. The change of the populations of two energy levels ...
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1answer
57 views

No clue about a term [closed]

$\int_S\int \vec{A}\cdot\hat{n}dS= \int_S\int A cos(\theta)dS= \int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$ I have no clue about the term $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$ ...
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1answer
55 views

Dot product in cylidrical coordinates

I'm given the vector: $$\vec{V}{(r,θ,z)}=\frac{1}{r}\hat{e_r} + (r\cosθ)\hat{e_θ}+\frac{z^2}{r^2}\hat{e_z}$$ I want the scalar product ${\vec{\nabla}}\cdot{\vec{V}}$ We know that in cylindrical ...
1
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1answer
82 views

Abuse of Calculus [duplicate]

I'm following Professor R. Shankar's Fundamentals of Physics course on YouTube. There I saw him doing manipulations of Calculus I never saw before. Here it goes, $$\newcommand\deriv[2]{\frac{\mathrm ...
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2answers
38 views

Is the work in line integrals equivalent to the work as an area under the curve?

A little explanation is needed here. Let me use two dimensions. In a line integral a curve is given and usually one parametrizes yet another curve and then substitutes this into the original equation ...