Function Definition and 1000 Threads

Hello, To first clarify what I want to know : I read the answer proposed from the solution manual and I understand it. What I want to understand is how they came up with the solution, and if there is a way to get better at this. I have to show that, given a vector field ##F## such that ## F ...

Let $X$ be a topological space and ##Y## a set. A function ##f: X \to Y## is said to be locally constant if, for every ##x \in X##, there is an open set ##U## containing ##x## so that the restriction ##f|_U: U \to Y## is constant. Prove that if ##X## is connected, a locally constant function on...

Hi, I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$ Since I have 2 singularities which are double poles. I'm using this formula $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$ then, $$\lim_{z\to\ \pm ia}...

If ##\hat{T} = -\frac{\hbar}{2m}\frac{\mathrm{d^2} }{\mathrm{d} x^2}##, then the expectation value of the kinetic energy should be given as: $$\begin{align*} \left \langle T \right \rangle &= \int_{0}^{L} \sqrt{\frac{2}{L}} \sin{\left(\frac{\pi x}{L}\right)}...

is the magnetic field B→. a state function and exact differential? I argued that it's a state function, what do you guys think

This is a textbook problem: now for part a) no issue here, the range of the function is ##-1≤f(x)≤299## now for part b) i got ##x≥-1## but the textbook indicates the solution as ##x≥0## hmmmmm i think, that's not correct...

I would like to find a function such that for $$a(x) \rightarrow 1~\text{for}~(x \gg x_c)$$ $$a(x) \rightarrow f(x)~\text{for}~(x \ll x_c)$$ What could be the ##a(x)## ? I have tried some simple functions but could not figure it out. Maybe I am just blind to see the correct result.

We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...

Hi, I was trying to solve the attached problem which shows its solution as well. I cannot understand how and where they are getting the equations 3.69 and 3.69A from. Are they substituting the values of θ₁ and θ₂ into Expression 1 after performing the differentiation to get equations 3.70 and...

Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem. The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in...

The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ## I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ## I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've...

Sketch by hand the function determined as f(x) = 1/28 (7√−16x^2 + 16x + 5 + 12) and then From the sketch, determine the domain and range of f in interval notation. Hint: Interpret f as part of a circle. You must include in your solutions the inputs and outputs you used to help you sketch the...

Hi, I cannot figure out how they got Table 2.1. For example, how come when x=1, F_X(x)=1/2? Could you please help me with it? Hi-resolution copy of the image: https://imagizer.imageshack.com/img923/2951/w9yTCQ.jpg

Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...

Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is: $$ u(x, t) = \int \phi(x-y, t)f(y)dy $$ where ##\phi(x, t)## is the heat kernel. The integral looks a lot similar to using Green's function to solve differential...

While studying the solution to a integral problem I found online I ran across a special function I am unfamiliar with. The integral is $$ \int_0^{\infty}\frac{t^{\frac{m+1}{n}-1}}{1+t}dt=\mathcal{B}(\frac{m+1}{n},1-\frac{m+1}{n}) $$ This certainly isn't the normal beta function. What is it...

Let ##S_n## denote ##\{1,\ldots,n\}##, where ##n\in\mathbb{N}##. Recall that the ##\textrm{sgn}## function maps a permutation of ##S_n## to an element in ##\{1,0,-1\}##. We want to rework the definition of ##\textrm{sgn}## because it is not sufficient for some proofs about determinants. For...

Attempt at a solution: \begin{aligned}Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }\\ \Rightarrow Z=e^{\beta NJ}\sum ^{N}_{r=0}C\left( N,r\right) e^{-2\beta rJ}\end{aligned} Let ##e^{-2\beta J}=x##. Then ##e^{-2\beta rJ}=x^{r}##. \begin{aligned}\therefore Z=e^{\beta...

I have an experimantally obtained time series: n_test(t) with about 5500 data points. Now I assume that this n_test(t) should follow the following equation: n(t) = n_max - (n_max - n_start)*exp(-t/tau). How can I find the values for n_start, n_max and tau so as to find the best fit to the...

I would say that what this matrix does is rotate e.g. a vector by ##\pi/2## clockwise. Am I right? I would like to check my understanding.

GRAPH WITH VALUES: Sorry I have a small dilema, I don't know if this is a exponential or polynomial function. I'd think its exponential but it doesn't have same change of factors.

I live close to the (landing) flight path to a major airport. The city has been working with the airport and the FAA to deal with complaints about the noise the jets make as they come in for a landing. I recently attended a meeting where various solutions were discussed, such as changing the...

Suppose a program computes or reads-in a symbolic mathematical expression like ##2x^2 - xy + y##. What's an effective way to cause the program to implement the expression as a function (e.g. implement ## f(x,y) = 2x^2 - xy + y##) when the programmer doesn't know in advance what the expression...

Hi, I am 16 year old and I am very interested in Physics. This summer I solved Schrödinger equation using griffiths' introduction to quantum physics and other sources. I achieved to get an exact solution of the wave function but I would like to plot it in a programm in order to get the 3d...

I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...

I would like to check my understanding of this problem. There are the following possibilities: a. Isolated points where the gradient is 0. b. The level curves of height 0 c. The level curves of height 1. d. The level curves of height -1. e. None of the above. I would choose a, c, d. Where...

Tried this, but not sure how am I supposed to square the whole equation and then square root it since this will inevitably give me imaginary values. Am I supposed to ignore the imaginary values? Also, how can I find out the phase in this case? Usually, it's taking the exponents but in this case...

this had ahttps://mathhelpboards.com/threads/2-2-21-ivp.27772/ but wanted to add tikz graph orifinally authored by Klazs van Aarsen \begin{tikzpicture}%[scale=.6] [declare function = {radius(\phi)=sqrt((3*sin(\phi)+cos(\phi)) / (sin(\phi)^3 -cos(\phi)^3)); },] % \draw[help lines] (-3,-2) grid...

Hello, I have a few questions and I'd appreciate if you can please help me. 1. If I want to say "for every ## i \in \Bbb N ## and ## 0 \leq j \leq i ## define ## A_{i,j} := i ## and ## B_{i,j} := i \cdot j ## ", then is the logical formula used for this is as such?: ## \forall i \in \Bbb N...

##A=(-1,1)## ##B=[0,1)## Define ##f:A\longrightarrow B## by ##f(x)=x^2## Set ##X=A##. ##f(X)=\{f(x):x\in X\}=\{x^2:x\in(-1,1)\}=B## ##f^{-1}(f(X))=\{x\in A:f(x)\in f(X)\}=\{x\in (-1,1):x^2\in B\}=X## Now choose a non-zero point ##y\in f(X)##. There are two pre-images of this point: ##x,-x\in...

ok I don't think MHB will process a newcommand but I don't know how to put this in the after \begin{tikzpicture} line the problem with posting pic here is eventually they get remove and OP is useless... this tikz code renders in overleaf but I also have many newcommands in preamble

Summary:: According to Yale’s University PHYS: 200: v*(dv/dt) = d(v^2/2)/dt Could someone explain how has he reached that conclusion? He claims to be some standard derivation rules, but I can’t find anything about it. As much as I can tell: (dv/dt)* v = v’ * v = a* v thanks! [Moderator's...

Last week I encountered a problem in my graduate project. The project was about designing an autonomous and mobile robot that picks up 9 glass tiles from a stack point and place them into a 3x3 matrix with minimum tolerance. I am using a DC motor with an infinite turn potentiometer for closed...

What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?

Please help me I am struggle with this question Thank you in advance

Can you please help me how to do it ? I am really struggle with this question. Thank you in advance

I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept...

Can you help me with this two questions I am really struggle on how to do it Please help me Thank you in advance

I'm trying to see if I can calculate the peak draw weight of my bow based on the draw length and the velocity of the arrow and a known shape of a curve, but I'm not quite sure how to make such a function, if there even is such a way. This is the shape of the draw weight plotted against...

Hi All I am currently doing Master in data science. I came across the function PDF probability density function which is used to find cumulative probability(range) of a continuous random variable. The PDF probability density function is plotted against probability density in y-axis and...

My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!

I have calculated $Z$s as $$ \begin{aligned} Z_1 & = 1 + \frac{3g^2}{16\pi^2} \left[ 2 C(R) - \frac12 T(A) \right] \frac1{\epsilon} + \cdots, \\ Z_2 &= 1 + \frac{3g^2}{8\pi^2} C(R) \frac1{\epsilon} + \cdots, \\ Z_3 &= 1 + \frac{g^2}{24\pi^2} \left[ 5 T(A) - T(R) \right] \frac1{\epsilon} +...

Hello! Let's say we have a wave function. Maybe it's in a potential well, maybe not, I think it's arbitrary here. This wave function is one-dimensional for now to keep things simple. Then, we use a device, maybe a photon emitter and detector system where the photon crosses paths with the wave...

The ac signal is converted to DC signal which is connected to a capacitor to filter the DC signal. The filter DC signal is step down for 12volt to 5 volt using a voltage regulator. The regulated DC signal is connected to a crystal oscillator that converts the DC signal to a square wave signal...

{\catcode`\^^M=12 \endlinechar=`\^^J \catcode`\^^J=5 This is an M: This is a middle line. This is a "J": } As I see it, the TeX processor would first need to feed this to the input processor to be transformed into a token list. The input processor would see the code I posted above with their...

Does anyone knows the name of the probability density function f(y) =aeby-cy2

I'm used to calculating Jacobians with several functions, so my only question would be how do I approach solving this one with only one function but three variables? I think our function becomes (s^2+sin(rt)-3)/since we are looking for J(f/s). So then would our Jacobian simply be J=[∂f/∂s...

Since z=0, the only variable that counts is x. So the solution would be: $$\frac {f \left(a + \Delta\ x, b \right) - f(a,b)} {\left( \Delta\ x\right)}$$

E.g. all real numbers could be a domain but not necessarily, etc. Am I right?

Hello, I don't know to solve this exercise: Let $\mathcal{B}_\mathbb{R}$ the $\sigma-algebra$ Borel in $\mathbb{R}$ and let $\mu : \mathcal{B}_\mathbb{R} \rightarrow{} \mathbb{R}_{+}$ a finite measure. For each $x \in \mathbb{R}$ define $$f_{\mu} := \mu((- \infty,x]) $$ Prove that: a)...