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Questions tagged [calculus]

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4
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1answer
29 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}...
0
votes
1answer
67 views

Elementary calculus in physics [closed]

Consider charge flowing through a given area governed by the equation $q(t) =t^2+t$.Now we are supposed to answer the current and charge flowing at time 2 and between 1 to 2(where $t$ is time and $q$ ...
1
vote
1answer
85 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^\...
0
votes
1answer
53 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 ...
0
votes
1answer
34 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 ...
1
vote
1answer
22 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 ...
-1
votes
0answers
27 views

Computing definite integrals (with force) [migrated]

Say we wished to compute the change in momentum of a body, which undergoes a time-variable force. Then: $p(t)|^{t_f}_{t_i}=\int_{t_i}^{t_f}F(t)dt$. Could you please explain why $\int_{t_i}^{t_f}F(t)...
-4
votes
1answer
55 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 $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$ ...
2
votes
1answer
40 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 ...
0
votes
1answer
69 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 ...
0
votes
2answers
30 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 ...
0
votes
1answer
23 views

How to find the net electric force exerted on a uniformly charged rod by another, same rod on the x-axis (they don't touch)? [duplicate]

How to find the net electric force exerted on rod 2 by rod 1, both being on the x-axis, both having the same length and constant linear charge density, being some distance apart? More specifically, ...
-1
votes
3answers
45 views

What is the significance of the second derivative of a function? [duplicate]

Basically, I just want to know the significance of the 2nd derivative of a function, or what does it tell us.
0
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0answers
24 views

Best multivariable calculus books that covers partial derivative and multiple integrals …etc [duplicate]

all that I want is some books that cover partial derivative and multiple integrals, gradient, curl and divergence in a way that doesn't leave anything without saying why are we doing it in this way ...
0
votes
1answer
26 views

calculate the time elapsed for a robot to pass certain distance with a load [closed]

For a robotics project I wanted to find the optimal gear ratio for my robot to travel 10 meters. Unfortunately. the acceleration is nonconstant, and that proved to make my life much harder. I think I ...
0
votes
2answers
73 views

Derivation of generalized velocities in Lagrangian mechanics

So I know that: $$r_i = r_i(q_1, q_2,q_3,...., q_n, t)$$ Where $r_i$ represent the position of the $i$th part of a dynamical system and the $q_n$ represent the dynamical variables of the system ($n$ =...
2
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1answer
53 views

Real Lagrangian with complex variable

I have a general question concerning real valued Lagrangians that take complex arguments. I have seen in many works of physicists and lecture books where extremal problems are discussed in Lagrangians ...
0
votes
1answer
58 views

Why is velocity gradient not called a velocity Jacobian?

I started thinking about the rate of deformation of fluid in the boundary layer. but here we consider only one of the components of the velocity vector (which is a scalar). But what about just general ...
3
votes
0answers
24 views

Why does the RG group flow's linearization provide an eigenbasis at fixed points?

I'm reading Conformal Field Theory by David Sénéchal, Philippe Di Francesco, and Pierre Mathieu. Let $T$ be the map that generates the renormalization (semi-)group by taking couplings $J$ to $J'$ (...
1
vote
2answers
49 views

Determining charge distribution from electric field (Griffiths 4th)

I am trying to teach myself Electrodynamics by following Griffiths' book. This is probably what's considered a "homework question", but as I don't have an instructor to ask for help, I'm hoping ...
1
vote
3answers
103 views

How to mathematically prove that point charge and infinitesimal volume charge are same?

In electrostatics, while deriving certain elementary equations, I have seen all the books just assuming that point charge and infinitesimal volume charge are same. How can we rigorously, ...
1
vote
1answer
38 views

Finding Terminal Velocity given non uniform acceleration

So I was doing a question which involved non-uniform acceleration. It went something a bit like this. If we have a particle that starts from rest and has initial acceleration $a_0$ and it varies with ...
0
votes
1answer
226 views

Why is displacement equal to the area of velocity-time graph? [duplicate]

why is the distance of a body equal to the area of its speed-time graph? the general formula of speed(v) is v=distance(s)÷time taken(t) so the formula of distance(s) should be s=v×t so if the speed-...
0
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0answers
21 views

Hamiltonian formalism, polynomial of a certain rank and velocity-independent potential

In page 88 of Shankar's Principles of Quantum Mechanics, we have the following lines: $$\mathcal{H}(q,p) = \sum_{i=1}^{n}p_i \dot{q_i} - \mathcal{L}(q, \dot{q}) \tag{2.5.8 } $$ where the $\dot{q}$'...
1
vote
0answers
23 views

Shifting the derivative outside the integral [closed]

In page 62 of Shankar's Principles of Quantum Mechanics, the author conveys the following: $$\int \delta'( x-x') f(x') dx' = \int \frac{d\delta(x-x')}{dx}f(x')dx'= \frac{d}{dx} \int \delta(x-x') f(x') ...
1
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0answers
56 views

Treating a differential as a difference of two quantities---legal or no? [duplicate]

In some textbooks, I have seen the authors treat differentials such as $dx$ as a difference, so $dx$ = $x_2$ - $x_1$ My question is, how legal is this? I realize the differential is a quantity ...
-1
votes
1answer
29 views

Dynamics - Find the height of the bridge

Working : $$ \text{Let the origin be where the paintbrush hits the ground} $$ $$ \text{Let time = 0 when the paintbrush is dropped} $$ $$ \text{For the car:} $$ $$ v_c(t)=v $$ $$ d_c(t)=vt+c, v(0)=-...
2
votes
1answer
95 views

Trying to understand the difference between $\Delta t$ and $dt$ [duplicate]

I'm trying to gain a more conceptual understanding of derivatives and would appreciate your feedback on this. Say I have a quantity, $x$, at time $t$. Now $x$ moves to a different location $x'$ in ...
1
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0answers
46 views

Problem in understanding Differential form of Gauss's Law

I am well aware of the integral form of Gauss's Law and the mathematical deduction through which it is reduced to the differential form. But I think I have a flaw in my understanding of divergence. ...
1
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0answers
42 views

Why are distributions not dependent on higher order time-derivatives?

When considering equations such as the Vlasov equation, the particle distribution, $f$, is taken to be a function of time, position, and velocity. But why are higher-order time-derivatives such as ...
2
votes
2answers
71 views

Dimensions of a distance time relation

Recently I came accross a question which was:- Suppose the velocity of a moving particle varies with time as $$v=50t^2.$$ And we have to find out the acceleration at $t = 10s.$ I know that I can use ...
0
votes
1answer
19 views

How to predict when a storm will hit given current & historical barometric data?

As a storm approaches, the barometric pressure gets lower. Suppose we have a barometer, we know a storm's approaching within the next 24 hours. We take measurements every second or two and have a ...
0
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3answers
30 views

The force used when calculating work done

When calculating the work done over a distance due to a variable force, you're supposed to use $$\int^{b}_{a} F(x)dx$$ Would the force $F(x)$ be an equation representing the resultant force acting on ...
1
vote
2answers
83 views

Cylindrical coordinate $\theta$ when $r=0$

When we use the cylindrical coordinate system $(r, \theta, z)$ where $r$ is the distance from the point in the $xy$-plane, $\theta$ is the angle with the $x$ axis and $z$ is the height. As can been ...
0
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0answers
50 views

Get rid of the derivatives and relativistic mass in Feynman lectures

i have a problem with get rid of the derivatives in Feynman lectures (chapter 15, Equivalence of mass and energy). The problem: we have $\frac {d(mc^2)}{dt} = v\cdot \frac {d(mv)}{dt}$, then we ...
1
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0answers
55 views

Can I use calculus to calculate how fast a the cracker of a whip is moving when it cracks?

So I know that when a whip cracks it is breaking the sound barrier and makes a sonic boom. I understand that this is Because of conservation of momentum as the whip tapers in thickness and hairpins at ...
2
votes
1answer
214 views

How is dot or cross product possible using the del operator?

Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
2
votes
1answer
47 views

How do you expand $\langle x'-\Delta x'\rvert \alpha\rangle$?

In my textbook (Sakurai) the following identity is often used: $$ \left< x'-\Delta x' \, \middle| \, \alpha\right>~=~\left< x' \, \middle| \, \alpha \right> - \Delta x'\frac{\partial}{\...
0
votes
2answers
139 views

Gravitational Force of hemispherical shell

My approach: $dF= \dfrac{Gm(dM)}{R^2} $ $ \int dF = \int \dfrac{Gm}{r^2} dM$ $ dM=\sigma dS $ $ dS=R^2sin \phi d\theta d\phi $ $F= \int_{0}^{2\pi} \int_{\frac{pi}{2}}^{0} \dfrac{Gm\sigma R^2}{R^2}...
1
vote
1answer
112 views

How can I find the deflation rate of a balloon?

I'm trying to model the deflation of a balloon. Assuming that deflation occurs through a small opening and shape of the balloon remains spherical during deflation, we may start with, $$\frac {dV}{dt}=\...
18
votes
7answers
4k views

What's the difference between average velocity and instantaneous velocity?

Suppose the distance $x$ varies with time as: $$x = 490t^2.$$ We have to calculate the velocity at $t = 10\ \mathrm s$. My question is that why can't we just put $t = 10$ in the equation $$x = 490t^2$...
0
votes
2answers
40 views

Is there anything wrong with my Euler's method equations for a pendulum outside of small angles?

I'm trying to write a program to calculate the angle, angular speed and energy of a pendulum at different times using Euler's method. The equation I started with was:$${\rm d}^2θ/{\rm d}t^2 = - g\sin(...
0
votes
1answer
51 views

How to use Newton's second law to derive conservation of momentum and how to use derive conservation of momentum to derive the second law?

I know if taking the integral of $F=ma$, then I can get $p=mv$. I'm weak in calculus, so I wondered how to do this exactly. Is there anything wrong in my logic below? \begin{align}\int F\left(t\...
2
votes
1answer
32 views

Thickness of a $3N$ dimensional spherical shell (entropy of classical gas)

I'm brushing up on statistical mechanics and calculating the entropy of a classical gas (i.e. particles in a box). Working through the calculation, we end up with an integral of the form: $$ \int^{'} ...
-1
votes
1answer
83 views

What is difference between $d\vec{l}$ and $\vec{dl}$? [closed]

What is difference between $d\vec{l}$ and $\vec{dl}$? $d$ means differential as usual.
-3
votes
1answer
70 views

What's the result of this integral? [closed]

$$\int_{|\vec k|<k_F} \frac{d^3k}{(2\pi)^3} e^{i\vec k\cdot \vec r} $$ it's not a Fourier transformation since the integrand is not infinite.
1
vote
1answer
45 views

Integral of Ion Distribution Function in Derivation of Dreicer Field

Disclaimer: I am including a fair bit of the physics background, but I believe this can be solved by someone with a strong calculus understanding I am following the derivation of the critical ...
0
votes
3answers
84 views

Issue with deriving the work-energy theorem

I'm a little confused regarding the way Total work = Change in kinetic energy is derived using calculus. My issue can be seen at 3:26 of this video: https://youtu.be/2dqO4sy4Njg?t=3m20s Why can the ...
0
votes
2answers
68 views

What is the significance of the temperature derivative of the isentropic bulk modulus?

I've been investigating various properties of the Gruneisen parameter and in my calculations the temperature derivative of the isentropic bulk modulus keeps coming up, i.e. $$\left( \frac{\partial ...
0
votes
0answers
48 views

What is the Jacobean this paper is talking about?

I am looking at this paper: https://arxiv.org/abs/1102.3938 The system is as follows: There are two states, A and B. Here we try to find the transition rates between the two, one represented by $k_+...