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Questions tagged [calculus]

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0answers
9 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 ...
0
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0answers
19 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
22 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 ...
0
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0answers
40 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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0answers
17 views

How to get the derivative of $F(a(x,y,t),b(x,y,t))$ wrt $t$ displayed as $f(x,y,t)(dF/dx) + g(x,y,t)(dF/dx)$? [migrated]

I have a function $F(a(x,y,t),b(x,y,t))$ whose derivative wrt t I would like to write as $$\frac{\partial F}{\partial t} = f(x,y,t)\frac{\partial F}{\partial x} + g(x,y,t)\frac{\partial F}{\partial y}...
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1answer
25 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:
4
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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
67 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
23 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
46 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
27 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
19 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
38 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
28 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 ...
0
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1answer
76 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 ...
2
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0answers
38 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
113 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: $$\...
0
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1answer
165 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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1answer
67 views

Distance Travelled by a Projectile [closed]

I was trying to find the distance travelled by a Projectile during its time of flight, $t=\dfrac{2u\sin \theta}{g}$. I used the Equation of the Path i.e. Trajectory and built up the Arc Length ...
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2answers
44 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
35 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 ...
1
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0answers
178 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 ...
0
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1answer
65 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
163 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
81 views

Dispersion Relations in Particle Physics [closed]

Please tell me how to get the identity(2) in this image
0
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1answer
39 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 ...
0
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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 ...
-1
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1answer
27 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
20 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
167 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
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1answer
72 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
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1answer
38 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
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1answer
60 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
56 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
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1answer
54 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
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1answer
72 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
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2answers
35 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
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1answer
87 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, ...
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3answers
51 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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1answer
30 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
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2answers
120 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
65 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
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1answer
79 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
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0answers
31 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
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2answers
65 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
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3answers
120 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
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1answer
42 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
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1answer
473 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-...