1
vote
1answer
54 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
1
vote
2answers
38 views

is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
3
votes
1answer
63 views

About reparametrization of timelike curves in $\mathbb{L}^3$ (Lorentz-Minkowski space)

I think there is something wrong with the proof this text gives of Lemma $2.1.5$, in pages $19$ and $20$, for timelike curves. I used another function, and it seems to work. Either I'm wrong, or he ...
0
votes
1answer
50 views

Are these derivatives correct?

Given the map $E: \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}^3$, such that (notice that this excercise is taken from Physics(Electrodynamics)). $$E(r,t) = -\frac{1}{4 \pi \varepsilon_0} ...
0
votes
4answers
69 views

Multiple choice question on rates of change (or so I thought)

If I were to find the resistance of the component (see image below), I would either find the equation of the curve and use differentiation or I'd draw a tangent at $V_2$ and then find the reciprocal ...
1
vote
0answers
37 views

Meaning of this differentiation operators

I have been just reading this paper here: paper and was wondering how they carry out the differentiation in (4.9). In principle, this should be just the differentiation of 4.8 with the help of 4.7a. ...
0
votes
1answer
32 views

Delta distribution - integration by parts of its differentiation

Some delta distribution physicist calculus. Assume there is given $$ \int_{\mathbb{R}^3} \sum_i f(\mathbf{x}) \delta^{(3)}(\mathbf{x}-\mathbf{a}_i) \ d^3x $$ with $f$ vanishing at infinity and ...
3
votes
3answers
99 views

Compute the integral $\int_0^\pi \frac{\sin x}{x}dx$ .

How to compute precisely the integral $$\int_0^\pi \frac{\sin x}{x}dx$$ analytically? It is well-known that $$\int_0^{+\infty}\frac{\sin x}{x}dx=\frac{\pi}{2}.$$ One way to compute the above integral ...
0
votes
1answer
20 views

Deriving relative position from instanteous acceleration and time

I'm working on a mobile app that uses the accelerometer to move a cursor. Although it's technically a computer science problem, once you get past how you get the values, it's more of a math problem, ...
0
votes
1answer
18 views

When finding the frequencies of normal modes, can you have a negative frequency?

Do you simply just consider the positive solutions? I tried a google search but didn't find anything quickly. The work I am studying is Lagrangian systems.
3
votes
3answers
44 views

Planet Simulation Newtons Law - Downscaling

I would like to scale-down the original numbers of our planet's motion, as i cannot properly visualize it in Unity3D (Game-Engine). I have: 1) Initial Position (-3.5e10, 0) (km) 2) Initial ...
2
votes
2answers
139 views

Integral calculus question relating to particle motion

"A particle of mass m is attracted toward a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at distance L, from 0, find ...
0
votes
0answers
57 views

What is the reason that Veneziano uses Euler's beta function?

This is veneziano amplitude: $$B(-a(s),-a(t))$$ where the $a(s)$ and $a(t)$ are a kind of leading trajectories (regge trajectory). which is born of strings theory. What is the reason that Euler's ...
0
votes
3answers
39 views

Finding the total distanced covered (physics)

A subway train starts from rest at a station and accelerates at a rate of $1.60\frac{m}{s^2}$ for $14.0 s$ . It runs at constant speed for $70.0 s$ and slows down at a rate of $3.50\frac{m}{s^2}$ ...
3
votes
2answers
67 views

Asymptotic expansion of $\sum_{k=0}^{\infty} k^{1 - \lambda}(1 - \epsilon)^{k-1}$

I'm seeing a physics paper about percolation (http://arxiv.org/abs/cond-mat/0202259). In the paper the following asymptotic relation is used without derivation. $$ \sum_{k=0}^{\infty} k P(k) (1 - ...
0
votes
1answer
34 views

Size of square formed by soap in a cube frame

So through the work of Plateau (as I understand it), we know that soap tries to find the shortest connection between points. At least, that's what I was taught. With this in mind, I had to solve the ...
1
vote
1answer
36 views

Why are the uncertainties so different?

Here is my scenario: I am trying to calculate the uncertainty of the function $y=x^2$, that is, I want to find $\Delta y$, and I found that we can get a great difference in the $\Delta y$, depending ...
1
vote
1answer
65 views

proof of $\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial \frac{\partial f(x,y)}{\partial y}}{\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial ...
3
votes
1answer
48 views

Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} ...
2
votes
1answer
41 views

Angle that car is at after angular acceleration

A car starts from rest on a curve with a radius of $150m$ and tangential acceleration of $\displaystyle 1.5\frac{m}{s^2}$. Through what angle will the car have traveled when the magnitude of its ...
0
votes
1answer
61 views

Fourier transform of Legendre

I am trying to figure out the Fourier transform of Legendre polynomial $P_\ell [\cos(\theta-a t )]$: $Q(\omega)=\int_{-\infty}^\infty P_\ell [\sin\phi\cos(\theta-a t )] e^{i \omega t} dt,$ where ...
3
votes
0answers
64 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
2
votes
0answers
54 views

Ratio of Hankel functions

I am trying to evaluate the ratio $\frac{H_m'^{(1)}(z)}{H_m^{(1)}(z)}, m\in N$. $'$ indicates derivative with respect to $z$. For large $m$, each term in the numerator or denominator can overflow ...
2
votes
1answer
59 views

Hankel trasformation of acoustic wave equation

We consider a simplified version of acoustic wave equation \begin{equation} \frac{\partial^2 p}{\partial r^2}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^2 p}{\partial z^2}+k^2 ...
2
votes
3answers
185 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...
1
vote
2answers
116 views

Why is acceleration $\frac{1}{2}at^2$ halved when finding final height (distance)?

The final distance of an object dropped from a certain height is: $$S_f=S_0-\frac{1}{2}at^2,$$ $S_f=$ Final distance $S_0=$ Initial height from which the object was dropped $a=$ acceleration due ...
0
votes
4answers
76 views

Regression analysis for non linear function

I am trying to model a problem with damped sine wave, $f(x) = a\sin(bx)\exp(-cx)$. I want to find optimum $a,b,c$ for my data. Can anyone please shed some light on this?
0
votes
0answers
63 views

Representing real function as integral over trigonometric functions

Since one can clearly express any function g(x) as integral from 0 to infinity of A(k)cos(kx)dk + integral from 0 to infinity of B(k)sin(kx)dk, how would G(k) relate to A(k) and B(k)? In other words, ...
4
votes
1answer
194 views

How to analyze the asymptotic behaviour of this integral function?

Based on the asymptotic analysis of correlation functions at large distence in Physics, now I get a math question. Let the function $$f(x)=\int_{-1}^{1}\sqrt{1-k^2}e^{ikx}dk.$$ Without working out ...
1
vote
1answer
164 views

Operator curl and gradient

Operator curl $\nabla$ x$(\cdot)$ (x is cross product) working on a $ C^1$ vector field and operator gradient $\nabla(\cdot)$ working on scalar fields. And results of these operators is vector ...
-1
votes
2answers
991 views

can an electric field exist at a point where the electrical potential there is zero?

can an electric field exist at a point where the electrical potential there is zero? 0 v=integral of E.dl
1
vote
1answer
248 views

Find the position equation from this velocity equation

Find the position equation from this velocity equation $$\displaystyle \frac{dr}{dt} = v_{t}\sqrt{1-e^{-v_{t}t}},$$ where $t$= time and $v_t$= constant I'm wondering if there's a way to solve this ...
1
vote
1answer
69 views

Need help with boundary conditions of a differential equation.

QUESTION: A particle $A$ is moving along the $X$ axis at a constant horizontal velocity $u\hat{i}$. Another particle $B$ is moving such that its velocity vector always points towards the particle ...
1
vote
1answer
70 views

Mathematical question concerning Lagrange multipliers of a Lagrangian

In Lagrangian Mechanics I have in general holonomic constraints of the form $f(q_1,...,q_n,t)=0$ and then I am able to use the method of Lagrange multipliers, where I go from a Lagrangian $L$ to a ...
0
votes
1answer
60 views

Write the normal and vector form of the equation in $\mathbb{R}^2$

This is more of a check then anything else. Here is what I have. Need to find the normal and vector form of the equation $$-2x+3y=5$$ Normal form: $$(-2,3) \cdot [(x,y) - (-1,1)]$$ Vector form: Now ...
3
votes
0answers
97 views

Physical Significance of normalised and non-normalised function?

$\dfrac {\sin(x)}{x}= \rm{sinc}(x)$ unnormalized \rm{sinc} function And for the normalized sinc we have: $\dfrac {\sin(\pi x)}{\pi x}=\rm{sinc}(x)$ normalized \rm{sinc} function Is there any ...
3
votes
1answer
41 views

For nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, is the length of the projection of $v$ along $u$ always less than the length of $v$.

For all nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, the length of the projection of $v$ along $u$ is less than the length of $v$. This is a true/false question and when I said true it ...
2
votes
1answer
192 views

Evaluating an integral

Gaussian-profile initial condition has the solution, $$\phi (r,t)=\frac{R^{3}}{2}\frac{A}{\sqrt{\pi }}\int_{0}^{\infty }ke^{-R^{2}k^{2}/4}\frac{\sin (kr)}{r}\cos (\omega t)\ dk,$$ where A is an ...
3
votes
1answer
506 views

integrals with error function

Can anyone help me to compute these integrals? \begin{equation} \int_0^t\frac{1}{x}\exp\left(-\frac{a^2}{x}\right) \operatorname{erf}\left(\frac{b}{\sqrt{x}}\right)\,dx \end{equation} here ...
2
votes
2answers
123 views

Electric field of a symetrically charged ball surface

I've been trying to solve this for some time, to no avail I must say. I am to calculate function of intensity of electric field on $z$ axis. The problem is: We have a charged ball surface with radius ...
1
vote
1answer
58 views

On one representation of Green's function

The Green's function for heat equation on finite interval is well known (with Dirichlet conditions): $$ G(x,x', t) = \frac{2}{l}\sum\limits_{n=1}^{\infty} ...
2
votes
1answer
59 views

Acceleration of series convergence

everyone! I am currently struggling with following problem: compute the series $$ \sum\limits_{m=1}^{+\infty}\frac{1}{m}\sin(m\alpha)(\cos(m\beta_1) - \cos(m\beta_2)) $$ and $$ ...
1
vote
1answer
71 views

Is the following differentiating under the integral sign correct?

Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial ...
1
vote
1answer
300 views

How to solve a tensor differential equation?

Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$ where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor. The original Problem How does ...
2
votes
1answer
144 views

Trouble understanding a common vector calculus example

I have difficulty understanding the following vector calculus example. Text can be found here. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a ...
0
votes
1answer
319 views

Straight Line Motion w/ Acceleration and Deceleration Rates?

PROBLEM: A subway train travels 400ft between two stations. It starts from rest and accelerates at the rate of 8ft/sec^2 until it's velocity reaches 20ft/sec. It then moves at this constant velocity ...
1
vote
1answer
98 views

Calculus of Variations-question on rotating curve of max volume

My calc of variations is still rusty. I'm assuming implementation of arclength revolution formula is necessary, but how to find y(1/2a)?
2
votes
1answer
187 views

Determine the Fourier Transform and Fourier Series of the function

$$ f(t)=\frac{\sin(at)}{t} $$ Since the term is parameterized, it's easy to see that if I take the first derivative with respect to 'a', then the function becomes considerably easier. I do this to ...
4
votes
3answers
168 views

Physics notation justified

Sometimes in physics they do things like this one: If $dq=f\left(x\right)\cdot dr$ then $\frac{dq}{dt}=f\left(x\right)\cdot \frac{dr}{dt}$ Which mathematically is a wrong deduction. Is there any ...
1
vote
1answer
103 views

Can any dynamical system be written as a hamiltonian system?

Can I always find a Hamiltonian for any given Dynamical System such that the Hamiltons' equations are satisfied? The hamiltonian may be an extremely complicated function (Possibly containing complex ...