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0answers
42 views

Why is the angular average of the direction of the momentum squared a third instead of one?

In the article http://www.nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf (in going from eq. (6.13) to eq. (6.14)) it is stated that $$\langle\hat{p}\cdot\hat{p}\rangle_{\hat{p}}=\dfrac{1}{3}...
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2answers
156 views

Work done against gravity comes out positive

Given two points in 3-space, $A = (9, 1, 5)$ and $B=(2,8,7)$, the work done by the gravitational field $\bf{F}$ when an object is moved from point $A$ to point $B$ comes out positive, even though it ...
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2answers
51 views

Faraday's law and potential along a closed curve

$$\int(\nabla \times \vec{E}) \cdot d\vec{S} = \oint \vec{E}\cdot d\vec{l} = -\frac{\mathrm{d} \phi_b}{\mathrm{d} t}$$ The second expression is the potential difference along a closed curve. Isn't ...
1
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0answers
83 views

Is $d^3r$ the same as $dV$, the volume element?

I've seen the term $d^3r$ being used instead of $dV$. Are they exactly the same? Do they have a different connotation?
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1answer
56 views

Derive an equation related to magnetism [closed]

Solve the equations for $v_x$ and $v_y$ : $$m\frac{d({v_x)}}{dt} = qv_yB \qquad m\frac{d{(v_y)}}{dt} = -qv_xB$$ by differentiating them with respect to time to obtain two equations of the form: $$...
0
votes
1answer
28 views

Electric field uniform circle $R$ direction cancel out

I am doing a physics problem involving a uniform circle with a total charge of X, and am attempting to find the electric field on a point charge on the axis of the circle a distance of Z away. I ...
0
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1answer
61 views

Hermitian Conjugate in Spherical Coordinate System [duplicate]

Help me find $\hat{B^\dagger}$, when we know that $$\hat{B}=i\frac{d}{dr}$$ with the condition that $\hat{B}$ is defined in spherical coordinates. My approach: $$ \langle\psi|\hat{B}\psi\rangle=\int_{...
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2answers
51 views

Find $\oint xdx+2ydy+3zdz$ on the line segment formed by points $A(1,0,-1)$,$B(2,5,-1)$ [closed]

Find $\oint xdx+2ydy+3zdz$ on the line segment formed by points $A(1,0,-1)$,$B(2,5,-1)$ My approach: First I find the line segment formed by $A,B$ which is $\vec l(t)=\vec{OA}+t\vec{AB}=(1,0,-1)+t(1,...
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votes
2answers
58 views

Do I need to differentiate this equation or not?

I've been given this equation: $$\mathrm{velocity}, v = 2\,\mathrm{cm/s^3} \,t^2 + 5\,\rm cm/s.$$ If someone now asks me to tell the velocity at, say 4 sec, then before proceeding with the equation, ...
1
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1answer
146 views

Derivation of Torricelli's equation

I am reading "Fundamentals of Physics" by Shanker and he is deriving Torricelli's equation. First he says $\displaystyle a = \frac{dv}{dt}$. Next, he multiplies both sides by velocity to get $\...
0
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2answers
87 views

Derivation of a kinematical equation [duplicate]

My question is, we can calculate distance of free falling objects with the equation of $d = \frac{1}{2}gt^2$ where $d$ is distance in meter, $t$ is time in seconds, but I wonder how we achieved this ...
0
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0answers
55 views

Little problem with surface integral

I've read a problem and I think there's a mistake in it. The problem is as follows. Note: The surface is the surface of a sphere. \begin{equation} \oint\frac{d\vec{a}}{d} \end{equation} Where $d=|\...
1
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2answers
88 views

Units of the derivative of a function

I have a function $\phi(\mu, \sigma)$. $\mu$ and $\sigma$ are voltages (in mV in my case), so $\phi$ is a function of two voltages. $\phi$ itself, however, is in units of time (ms in my case). (...
1
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2answers
90 views

Would condensed matter physicists need more than three dimensional calculus? [closed]

The difference between "multivariable" and "vector" calculus, as stated on Yale's website, is that multivariable would only go through 2 or 3 dimensions, and so would rely heavily on geometric ...
0
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0answers
90 views

Expanding Electric Field, Magnetic Field in post-Newtonian Gravitational Potential

I've cross-posted this question from Mathematics since Physics is probably better suited for the nature of the question. I've ...
0
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0answers
11 views

Calculating Trajectory in non-Uniform electric field? [duplicate]

Above is the equation of the electric field. Where k is the electric constant, q is the charge on the particle and m is the mass of particle. The particle q initial position is (0,10) and starts off ...
0
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1answer
50 views

Electric potential of a sphere at a point on its surface

I'm struggling to figure out how to determine the infinitesimal area of the sphere given in the question. More specifically, in part (b) of the question. For part (a) it's the volume so $$V = \frac{...
1
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2answers
116 views

Does calculus work on the quantum scale [duplicate]

I have read somewhere that Leibniz championed the idea that the world is continuous as this was needed for his (or maybe Newtons) new invention (or discovery?) of calculus. But if I am not mistaken a ...
0
votes
1answer
57 views

Current Electricity

If $$ \frac{dQ}{dt} = I $$ and if an accelerated current produces E.M. waves (radiation), does that mean $d^2Q/dt^2$ (second derivative of a charge w.r.t. time) will give me the magnitude of the wave ...
1
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2answers
171 views

Composing integrals in physics?

OK... so this problem isn't really specific... it's more of a conceptual puzzle. I've recently started using integrals while solving problems in physics (specifically Newtonian Mechanics and other ...
0
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0answers
56 views

Compute distance travelled based on a yaw-rate

Assume that a rigid body is traveling with constant velocity $v$, and (this rigid body) is rotating with a constant yaw rate of $\dot{\theta}$. Find the distance travelled in one time step, $\Delta t$....
0
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2answers
169 views

Physical meaning of divergence

While reading the section on Hamiltonian mechanics in Taylor's Classical mechanics, I realized that I didn't fully understand what he was saying when he was explaining why $$\nabla\cdot\vec{F(\vec{x_0}...
0
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3answers
287 views

How do we calculate instantaneous velocity in 2D?

Suppose a body is moving with a constant speed of $10~\mathrm{ms^{-1}}$ in negative $x$ direction in $x$-$y$ plane. Let $\vec r$ be the position vector. Then what will be the instantaneous velocity ...
2
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2answers
298 views

Falling rain drop problem

I've been trying to solve this physics problem today "Spherical raindrop falls through a cloud with uniform density. After falling 1km it has a radius of 5mm. What volume fraction of the cloud is made ...
1
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1answer
81 views

Why is the elemental volume of a sphere equal to $4 \pi r^2dr$?

I was doing this question on calculating the electric field at a certain point in a sphere (length $r$ away from the centre), where the charge density is given by an equation. When I checked the ...
2
votes
3answers
558 views

Basic question about acceleration [duplicate]

Very basic question. Please show where I'm wrong in the following reasoning. The movement of an object in function of time could be described as $$ x(t) = v t + x_{i} $$ if velocity is constant. If ...
5
votes
1answer
223 views

Optimization of Bottle Rocket Water Level

My (entry-level) physics class is building bottle rockets, and we are competing to build the longest-flying bottle rocket. The rockets are filled partway (we get to decide how much to fill them) with ...
1
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2answers
435 views

Derivation of Jefimenko's Equation in Jackson's EMT book

I have been trying to understand the derivation of Jefimenko's equation in Jackson on p.246-247 which can be seen in the photographs attached. First of all I did not fully comprehend the ...
2
votes
2answers
193 views

Differential operators in curvilinear coordinates

In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
0
votes
1answer
60 views

Simplifying a Vector Integral

This question has (long) remained unanswered on MSE. While reading the book - Theory and Applications of Boltzmann Transport Equation by Cercignani, I found this integral which I am unable to ...
3
votes
1answer
101 views

What is the physical cause that circulation on a closed surface is zero?

This is quoted from Feynman's Lectures: We would like to see what happens when the loop shrinks down to a point, so that surface boundary disappears - the surface becomes closed. Now, if the ...
0
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1answer
53 views

Confusion regarding area from graph

This might be a trivial question but is illustrated below. Why is the area 'below' the graph always taken for a velocity-time graph when finding the displacement? I mean why is the area with the time ...
0
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1answer
27 views

Flux from cube: Relation between the vector field to that of the opposite side

This is an exerpt from Feynman's lecture: . . .We wish to find the flux of a vector field $C$ through the surface of the cube. . . First consider the face having edges $\Delta y \quad \& \...
1
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1answer
219 views

magnetic field due to current in a wire using Biot-Savart's law

I am learning about Biot-Savart's law to calculate the magnetic field due to the electric current in a wire. $\displaystyle\mathbf{B}=\frac{\mu_0I}{4\pi}\int\frac{d\mathbf{s}\times\mathbf{\hat{r}}}{r^...
0
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1answer
99 views

Why is $\int_{s} \mathbf{h}\cdot \mathbf{n} da = - \dfrac{dQ}{dt}$ & not $\int_{s} \mathbf{h} \cdot\mathbf{n} da = - \dfrac{dQ}{dt} .{dt}$?

I was reading the Lectures of Feynman about surface integral where a situation in which heat is conserved has been dealt. Let there be $Q$ heat energy present inside a body. Now, if there is net heat ...
2
votes
1answer
231 views

Insight into Torricelli's Equation ($v^2=u^2+2as$) [closed]

Torricelli's Equation ($v^2=u^2+2as$) is usually presented as the particular formulation of the SUVAT system which doesn't involve t. It is derived from the others using some (perhaps well-motivated) ...
0
votes
1answer
81 views

Using differentials to optimize a function [closed]

I've read in a paper by Tevian Dray an alternative way to solve optimization problems manipulating "differentials". Here is an example of how it works (next I quote the paper). Consider the ...
3
votes
1answer
412 views

$v^2 = 2ax$ or $v^2 = ax$?

As far as I am aware, $v^2 = 2ax$ is the formula to find the velocity in various questions. If kinetic energy = work, $$\frac{1}{2}mv^2=Fx$$ $$mv^2=2max$$ $$v^2=2ax$$ We use this formula to solve ...
0
votes
0answers
66 views

Solve unsteady-state pressure-falloff axial gas flow equation for permeability

Equation 17 in Section 6.2.1.2 of American Petroleum Institute Recommended Practices 40 for Core Analysis gives a permeability equation of: $$ \frac{(P_1−P_2)(P_m+f_{Fo}b)z_r}{P_rq_rz_m}=A_1+A_2\left(...
1
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0answers
17 views

Find center of mass and moment applied on beam structure. [duplicate]

I have a simple mathematical problem to solve but it is giving me a slightly difficult time to figure out. The problem: I have a beam structure with same cross section. It consists of three beam. ...
-1
votes
3answers
168 views

What is physical interpretation gives integration?

It is my understanding that the integration is the inverse process of differentiation and its meaning is a fine sum (in fact, so is its symbol) but what physical interpretation do we get from this? At ...
1
vote
1answer
119 views

Can moment of inertia be defined as function of mass?

For continuous bodies, moment of inertia is found as $$ \int dI = \int_{m_i}^{m_f} r^2(m) .dm$$ . Now, $$\int dx = \int_{u_i}^{u_f} f(u).du \implies X_f(u) = \int_{u_i} ^{u_f} f(u) .du + X_i(u)$$ , ...
0
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2answers
135 views

Is the Fundamental Theorem of Calculus really applicable to the definition of work?

When the force $F$ on an object is not constant, then the work it performs is defined as $$W = \int_{x_0}^{x} F(X)dX.$$ Now, the Fundamental Theorem of Calculus states that $$\text{If}\,\,\, f(x) ...
1
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3answers
84 views

Problem in understanding the process of calculating the rotational inertia

As we know, rotational inertia is the mass-equivalent in rotation. For a discrete body, it is measured as $$I = \sum m_i{r_i}^2 $$ . But when a continuous body comes, $$I = \int r^2 .dm$$ which ...
1
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3answers
476 views

Problem in deducing the equations of motion using indefinite integral

As we know, antiderivative or indefinite integral is the function the derivative of which gives the actual function. Let $F(x)$ be the derivative of $f(x)$ ie. the instantaneous rate of change of $f(...
3
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2answers
4k views

Why and when do we differentiate or integrate equations in physics? [closed]

I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like: The object is moving in a positive ...
2
votes
1answer
207 views

Integration by parts [closed]

Zee in his book, Quantum field in a nutshell mentioned the following, p. 22: Equation (14): $$Z=\int{D\phi}e^{i\int d^4x{ {\frac{1}{2}[(\partial \phi)^2-m^2\phi^2]+J\phi}} }$$ He said then, ...
1
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5answers
1k views

What is divergence?

What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
4
votes
1answer
128 views

How is taking the average of an integral over an interval justified?

I have been studying classical mechanics. Often when going through a worked problem, I see a step where there is an integral from 0 to 2$\pi$ of $\sin^{2} \theta \ d\theta$. Instead of using the ...
4
votes
2answers
599 views

A basic math identity often used in integrals [closed]

I'm just wondering about why $y_i=A_{ij}x_j$ implies $$d^Ny=|\det A|d^Nx.$$ I see that $\det A$ is the product of the eigenvalues of a diagonal matrix but still don't exactly see how. Please help.