Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

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How to derive the weak form of the PDE?

I have some difficulties solving the weak form of the PDE: The proof of the preceding statement is elementary. The weak form of the PDE $\nabla \cdot (A(x) \nabla u) + \omega^2 q(x) u = 0$ for all ...
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79 views

Counter exchanging limit and integral

Background I came across this answer on Math SE which claimed it made a lot of sense to switch limit and integral. In response I came up with the following counter-examples: $\lim_{w \to 0} ...
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53 views

Why is the potential function defined differently in physics and calculus?

I am very familiar with the concept of a potential function, and potential energy, from calculus-based physics. For instance, if we have the familiar force field $\mathbf{F} = -mg \,\mathbf{j}$, then ...
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112 views

Trigonometric identities — a parallel RLC circuit connected to an AC-supply [closed]

An RLC-circuit is connected to an AC-supply as in the figure below. $I_{tot}(t)=I_0sin(\omega t+\phi)$ (denoted as $I_{ges} ( t)$ in the picture), $\phi$ is the phase angle between ...
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47 views

Can I calculate the scalar potential from the electric field using: $\phi = \int \nabla \phi = - \int \vec{E}$

If I have a relation between the electric field and the radius, can I calculate the relation between the scalar potential and the radius using: $\phi = \int \nabla \phi = - \int \vec{E}$? $$\nabla ...
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1answer
36 views

Failing to calculate earth's standard gravitational parameter

I am using this equations from Wikipedia: $$\frac{4\pi^2 a^3}{T^2}=\mu$$ Where: $\mu$ = standard gravitational parameter ($\mbox{km}^3 \mbox{s}^{−2}$) $a$ = the orbiting body's semimajor axis (AU) ...
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1answer
17 views

Hamiltonian Constant on integral curves

Let $H \in C^{2}(\mathbb{R}^2)$ and let $(x(t),y(t))$ be a solution to the equations $$\frac{dx}{dt} = \frac{\partial}{\partial y} H(x(t),y(t))$$ $$\frac{dy}{dt} = -\frac{\partial}{\partial x} ...
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43 views

Determining movement time with uniform acceleration/deceleration

Consider some movement along a path segment $s$ with constant acceleration/deceleration (see figure below). The initial speed is $v_0$ and the final speed is $v_1$. The constant acceleration is $a$ ...
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20 views

When is the density matrix not diagonal?

The density matrix (density operator) in quantum mechanics is defined as $$\hat{\rho} = \sum_{i} p_i |\psi_i\rangle \langle \psi_i|\, ,$$ where the $|\psi_i\rangle$ are a full orthonormal system and ...
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36 views

Derivation of the momentum operator

\begin{align*} \langle p \rangle &= \int_{-\infty}^\infty \frac{d p}{2\pi \hbar}\, \phi(p, t)^\ast \, p \, \phi(p, t) = \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} \int_{-\infty}^\infty dx' ...
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1answer
44 views

Mass of Ocean to Atmosphere [closed]

This is a bizarre question, but here goes... If all of the water in the oceans were boiled into steam by the newly forming molten earth, could the atmosphere retain the steam? In other words, ...
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1answer
33 views

Fourier transform of a Green's function

I was studying for an exam and I found this question which has caused me a bit of trouble: Given the Green's function that satisfies the equation $$\Box ...
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1answer
30 views

Bezier curve and deceleration

I have a question regarding calculation of a cubic Bezier curve. I'm programming an app where in there's continuous straight line motion of a vehicle at a constant speed. (Let's call it $u$). When the ...
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34 views

Decomposition of acceleration into normal and tangential components

If the velocity $v=\|\mathbf{v}\|$ of a point having position $\mathbf{x}(t)$ at time $t$ is never null, then acceleration $\mathbf{a}:=\frac{d^2\mathbf{x}}{dt^2}$ can be written ...
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37 views

Rewrite the system in the form $\dot x = Ax + bu.$

$Ml\ddot\theta = (M + m)\dot x + mL\ddot\theta = u$ $M\ddot x = u - mg\theta$ Using the variables $x_1 = \theta, x_2 = \dot\theta, x_3 = x, x_4 = \dot x$ Rewrite the system in the form $\dot x = Ax ...
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3answers
562 views

Trying to understand the true meaning of integral and Derivative in calculus [duplicate]

I'm solving a physics question, and i just encountered some question i had no idea how to start, i just got the right answer and inside it it has something in math i never thought possible, I know ...
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55 views

Why we have to take the vector $\overrightarrow{e}$ ?

The differential equation of the balance of the momentum is $$\rho \frac{\partial{\overrightarrow{u}}}{\partial{t}}=-\rho (\overrightarrow{u} \cdot \nabla )\overrightarrow{u}-\nabla p+\rho ...
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25 views

Law of hydrostatic pressure

For a calm fluid of uniform density $\rho_0$, that occupies the space $W \subset \mathbb{R}^3$, and is subject to massive forces (per unit of mass) $\overrightarrow{b}(\overrightarrow{x})$, write the ...
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1answer
17 views

Null velocity and piecewise smooth path

On texts of multivariable calculus and real analysis I have always seen the work made by $\mathbf{F}$ along the path $\gamma$ defined as the ...
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1answer
33 views

Stress vector - Stress tensor

Is the definition of the stress vector the following? The stress vector is the force per unit surface. The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the ...
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24 views

Find the streamlines of the velocity field

I have to find the streamlines of the following velocity fields: $$u=x(1+2t), v=y$$ $$u=xy, v=0$$ I have done the following: $$\frac{dx}{u}=\frac{dy}{v} \Rightarrow ...
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1answer
55 views

Dirac delta function with a sum as the argument

I'm reading "First steps in random walks" by Klafter and Sokolov, and I don't understand this step involving the Dirac delta function. They want to obtain the probability density of having a walker at ...
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84 views

Upper bound on the Lipschitz constant of entanglement entropy

I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full ...
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3answers
53 views

How to visualize rotation on a hyperbola?

I am studying Lorentz transform and I do not quite get what it means to use the hyperbolic matrix to rotation a point on a hyperbola, mainly it is because the hyperbola consists of two divergent ...
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42 views

Streamlines - Pathlines

Construct and draw the streamlines of the velocity field $u=az-bt, v=\frac{b}{4}z-cy, w=2(a-1)$. Calculate $c$ (as a function of the constants $a$, $b$) such that the flow field ...
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26 views

Electric motor fed with constant power.

This problem arose from a challenged to propel a model solar electric car for a distance of 20m in the quickest time. A solar panel is illuminated with constant brightness light to deliver electrical ...
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29 views

Acceleration/Position signal correction

I have a set of data for a car position, velocity and acceleration. % my data time car_x car_velocity car_acc The problem is that these arrays have error and I ...
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22 views

Show the relation $W$ is constant

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left ...
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1answer
52 views

What is $\int \omega^2 \ dt$? Integral on a centrifuge.

I was looking at a centrifuge, and I saw the following integral: $\int\omega^2\ dt$. I was wondering if this integral has any significance? Since this is a centrifuge, I would assume that $\omega$ ...
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1answer
44 views

How do we get the last relation?

I am looking at the conservation of momentum. The force at $W$ from the tensions at the boundary $\partial{W}$ is $$\overrightarrow{S}_{\partial{W}}=-\int_{\partial{W}}p \cdot ...
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Applying a constant force on pendulum and integrating for velocity

We apply a constant horizontal force $ F $ to a weight that's connected to a solid rod. I'd like to find out the velocity of the weight when it's at its horizontal position. See image below. My ...
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110 views

Is this correct or completely wrong?

I've bumped onto the problem described below and I couldn't tell if it is wrong. It looks like it could be correct but it makes no physical sense. Thank you! Marcelo! I begin with ...
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41 views

Concept of a continuum

I have to explain the concept of a continuum that is used for the description of the dynamic behaviour of the fluids, and to explain how this concept is related on the one side with the laboratory ...
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1answer
30 views

Derivitates of parameter-dependent functions of operators

This is a problem from quantum mechanics, but purely mathematical (I've set $\hbar = 1$): Let $\psi(\cdot, t) \in L_2(\mathbb{C}^3)$ for all $t\geq 0$ and $\hat{H}$ a bounded, hermitian linear ...
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2answers
51 views

Pre-calculus - Deriving position from acceleration

Suppose an object is dropped from the tenth floor of a building whose roof is 50 feet above the point of release. Derive the formula for the position of the object t seconds after its release is ...
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Angular momentum components in a given direction.

The exam question of interest is the following: Suppose the angular momentum of $\mathbf{v} = |1\ m \rangle$ ($j=1$, $|j \ m \rangle$ denoting the usual eigenstate of $J_3$ and $\mathbf{J}$) is ...
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1answer
50 views

Mathematics of contamination [closed]

I want to know the distribution of residual material (contamination) in subsequent refills. For example, suppose a cup normally used for transferring salt is used, without cleaning, for transferring ...
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1answer
51 views

Asymmetrical Center of Gravity Problems

Calculating center of gravity asymmetrical objects find the position of the center of gravity of relative to the edges AB and AC ...
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44 views

Projectile to clear a hemispherical mound

I have only just stumbled across this site a few days ago while searching for other things. This is a great resource. I have a question for nearly 30 years now when me and my friend were trying to ...
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1answer
30 views

Light attenuation through water at an angle

I know that light intensity decreases exponentially governed by \begin{equation*} \frac{dy}{dx} = -ky \end{equation*} where $y$ is the intensity and $x$ is the distance. Now what happens when light ...
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25 views

Finding the magnetic field

A mechanical pendulum is made of a metal rod, one end of which is attached to a string. The mass of the rod is $m$ and the string length is $l$. The oscillation period of the thus obtained pendulum ...
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1answer
56 views

Integration — Work done in pulling an elevator using a rope

An elevator weighing $3000$ lbs. is supported by a $12$ ft. cable that weights $14$ lbs./ft. Find the work a winch has to do by pulling the rope to lift the elevator $9$ feet. I eventually ...
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1answer
66 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
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98 views

Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
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1answer
25 views

Calculus Velocity and Acceleration

Here's the question: I know that: $a(t) = -9.8$ So I integrated the acceleration function to find the velocity: $v(t) = -9.8t + c$ And because $v(0) = -5$, I can determine that $c = -5$, thus: ...
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Example of using the Hadamard's matrix to determine the superposition

I've came across those notes for Quantum computation from John Watrous. I am having troubles understanding the last example. We have those two vectors, or if I understood correctly, from now on ...
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2answers
31 views

Freefall with terminal velocity - expressions for velocity and position

A body weighing 29 kg is dropped from a height of 30 m with an initial velocity of 3m/sec. Assume that the air resistance is proportional to the velocity of the body. if the limiting velocity is known ...
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1answer
50 views

Prooving Kepler's Second Law through vectors.

I am taking a multivariable calculus lecture online provided by MIT OpenCourseWare. ...
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1answer
37 views

Integral equation involving Planck radiation formula

I am stuck in solving the following integral equation: $$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$ where: ...
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36 views

Tension on the string of a drawn bow.

An archer's bow is drawn at its midpoint until the tension in the string is 0.881 times the force exerted by the archer. What is the angle between the two halves of the string? *Edit: some context, ...