Variables Definition and 1000 Threads

Greetings all. I just got confused by the following. Consider volume integral, for simplicity in 1D. $$ V(A) = \int_{A} dz. $$ If ##z## can be written as an invertible function of ##x##, i.e. ##z=f(x)##, we know the change of variables formula $$ V(A)=\int_{A} dz= \int_{z^{-1}(A)} |z'(x)|dx...

Hi PF Given the following def f1(var1, var2): var3 = var1 + var2 return var3 def f2(var1, var2, var4) var4 = 10 var5 = f1(var1, var2)*var4 return var5 it is obvious function f2 does not explicitly need variables var1 and var2. However, it needs the result of f1, which...

Is the answer given above correct?

Find the volume V inside both the sphere $x^2 + y^2 + z^2 =1$ and cone $z = \sqrt{x^2 + y^2}$ My attempt: I graphed the cone inside the sphere as follows. But I don't understand how to use the change of variables technique here to find the required volume. My answer without using integrals is...

After plotting the above (not shown) I believe one way (the hard way) to solve this problem is to compute the following integral where ##f(x) = e^{-x^2/2}/\sqrt{2\pi}##: $$\frac{\int_0^\infty \int_{3X}^\infty f(X)f(Y)\, dydx + \int_{-\infty}^0 \int_0^\infty f(X)f(Y)\...

Summary: Find the volume V of the solid inside both ## x^2 + y^2 + z^2 =4## and ## x^2 +y^2 =1## My attempt to answer this question: given ## x^2 + y^2 +z^2 =4; x^2 + y^2 =1 \therefore z^2 =3 \Rightarrow z=\sqrt{3}## ## \displaystyle\iiint\limits_R 1dV =...

Hello! I have the following initial value problem: \[ x' = x + 2y + 3z \] \[ y' = 4y + 5z \] \[ z' = 6z \] All I'm looking to do is find the general solution to this system, and as long as I'm doing this correctly I have these answers: \[ y(t) = K_2e^{4t} + \tfrac{5K_1}{2}e^{6t} \] \[...

it's come to my attention that there are arguments that loop quantum gravity is simply wrong, and specifically in the way it method of quantization of Ashtekar variables. So what are more promising ways to canonically and nonperturbatively quantize Ashtekar variables Ashtekar variables are...

Prove directly that the transformation $$Q_{1} = q_{1}, P_{1} = p_{1} − 2p_{2}$$ $$Q_{2} = p_{2}, P_{2} = −2q_{1} − q_{2}$$ is canonical and find a generating functionSo the first part is easy and can be skipped here. I have some difficults regarding the second part, namely, the one that ask for...

Hi there. Let's say I have the following relationship: x = a + b*z + c*y z is distributed normally y is distributed according to a different distribution, say exponential Is there a way to figure out what is the distribution of x? Thanks!

Hello! I need to perform a fit with several variables and 2 of them are very correlated (above 0.99). The functional form of these 2 variables is something like: ##(p+q)x+qf(x)##, where ##f(x)## contains polynomials and some square roots of x, but the coefficients appearing in ##f(x)## are much...

Hi. I have found the answer to a and c (I don't know whether it is correct) but I do not know what I should find in question b. Is my method correct and how should I solve part b? Thank you for your help!

I calculated the mean which is 78.4 And the Standard deviation is 5.6 I thought the answer would be (90^(-78.4)/78.4!)*e^-90 But looking back having a decimal factorial doesn't make sense I have the numerical answers for c)= 0.019226 and d)=0.022750 but I my solution was wrong. Any help on...

Suppose I have 2 variables q and t (time), where q is some reparameterization of x (position) : ##x \to q = x f(t)##. Suppose I have a partial differential equation : $$\frac{\partial u(q,t)}{\partial t} = k \frac{\partial u(q,t)}{\partial q}$$ where k = constant Then I do a separation of...

Computer languages handle the scope of variables in a precise way so that if one symbol, such as "k" is used in different contexts, the program keeps these separate. When sophisticated human beings re-use symbols in writing mathematics, they can keep things straight, but I don't think they...

I do not know the solution.

Problem statement : Let me copy and paste the problem statement from the text : Attempt at solution : I could not solve the problem reducing the L.H.S into the R.H.S. However, I could solve the problem by expanding the R.H.S. into the L.H.S., though it is less than satisfactory. Below is my...

Problem : Let me copy and paste the problem statement as it appears in the text, as shown above. Attempt : I can sense there is an "elegant" way of doing this, but I don't know how. I show below my attempt using ##\text{Autodesk Sketchbook}##. I hope am not violating anything. Ok so I have...

$$f_{XY}=1$$ $$dzdy=2xdxdy⇒\frac{1}{2\sqrt{z}}dzdy=dxdy$$ $$f_{ZY}=\frac{1}{2\sqrt{z}}\quad \text{on some region S}$$ $$F_{ZY}=\int^y_{g}\int^x_{h}\frac{1}{2\sqrt{z}}dzdy\quad\text{for some}\quad g(x,y),h(x,y)$$ im learning how to find the region S using a change-of variables technique

I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...

(a) the agrea of the triangleses is 1, so γ one. (b) I'm not sure how to prove. i feel like ##X## and ##Y## are dependent because ##E(Y|X=0)=\frac{1}{2}## and ##E(Y|X=1)=0## so ##Y## seems dependent on ##X##. ##f_X=1-x## for ##x>0## ane ##f_X=1+x## for ##x<0## so X seems independent on Y.

I am refreshing on the pde's, and i am trying to understand how the textbook was addressing change of variables, i find it a bit confusing. I will share the textbook approach, then later share my own understanding on change of variables approach. Here is the textbook approach; My approach on...

Hey! :giggle: We have the table of distribution of $X$, $Y$ and their joint random variable : with $$(c,d)\in \left \{(c,d)\in \mathbb{R}^2\mid 0\leq c\leq \frac{1}{4}, \ 0\leq d\leq \frac{1}{2}, \ \frac{1}{4}\leq c+d\leq \frac{1}{2}\right \}$$ I want to calculate the values of $c$ and $d$...

This is a spring problem From this, it says I need to answer in terms of kinematic friction which to me doesn't make much sense. I also looked at similar questions online to the "in terms of" problems and they don't use all four variables in their derived equation. Do I not need to use all...

Good day all:smile:! I've been researching on 'Atmospheric water harvester' (AWH) devices. As I researched many functioning mechanisms on such devices, the most workable mechanism to me seems to be the one illustrated in the picture attached to this thread. (i.e running hot...

Hi, a basic doubt about thermodynamic functions and state variables. Take for instance transformations I and II in the following ##(p,V)## plane. As far as I can tell, just because the transformations are drawn as continuous lines they are reversible by definition. Namely we can transform...

I'm using Visual Studio Code as my text editor. My ultimate goal is to use PythonTex in a Latex file so that I can generate multiple versions of the same worksheet for my class. As a test, I defined a variable myvar = "hi" but I can't get it to render correctly in the pdf when I use \py{myvar}...

I have a question about changing variables in the context of thermodynamics, but I suppose this would extend to any set of variables that have defined and nonzero partial derivatives on a given set of points. First I should define the variables. ##T## is temperature, ##U## is internal energy...

#2 $ y'= \dfrac{x^2}{y(1+x^3)}$ Separate y dy =\dfrac{x^2}{(1+x^3)... ok i tried to get the book ans but someahere derailed why is =c in the answer

#24 in the spoiler ov!347.21 nmh{987} $\dfrac{dy}{dx}=\dfrac{(2¬e^x) }{(3+2y)}\quad y(0)=0$ $\begin{array}{lll} separate & (3+2y)dy=(2¬e^x)dx \\ integrate &3y + y^2=2x-e^x +c \\ complete the square &y^2+3y+(3/2)^2 =2x-e^x +(3/2)^2+c\\ and &(y+(3/2))^2\\ from...

Hi all, simple question but I cannot understand what is going wrong. I have a CLASS.hpp file of the form class CLASS { private: double varA = 5.; //some numerical value public: double func(double z); }; and then in the CLASS.cpp file I have a call to this function func()...

Hi, I just found this problem and was wondering how I might go about approaching the solution. Question: Given three random variables ## X##, ##Y##, and ## Z ## such that ##\text{corr}(X, Y) = \text{corr}(Y, Z) = \text{corr}(Z, X) = r ##, provide an upper and lower bound on ##r## Attempt: I...

Kindly see the attached problem below (i find the topic to be easy and straightforward). My concern is only on the highlighted part: In my understanding, to define the type of correlation i have always approached a straightforward approach. For value ##1## perfect positive correlation and...

I got (a) and (b) but I'm still working on (c). The solutions can be found here for your reference: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/lecture-9/MIT6_041SCF13_assn05_sol.pdf. But...

Hi, This is as part of a larger probability change of variables question, but it was this part that was giving me problems. Question: If we have ## 0 < x_1 < \infty ## and ## 0 < x_2 < \infty ## and the transformations: ## y_1 = x_1 - x_2 ## and ## y_2 = x_1 + x_2 ##, find inequalities for...

We have two different integrals, the first one being ∫∫erdrdθ where -1≤r≤1 and 0≤θ≤π which equals approximately 7 and ∫∫erdrdθ where 0≤r≤1 and 0≤θ≤2π which equals approximately 11. Why do these integrals have different values and do not go against the change of variables theorem? I'm having...

So first thing I tried was to separate the variables then differentiate by parts, setting u = E and v = 1/ln(E) (and the other way around) but I couldn't do the integral it gave. Then I tried to reason that because dx was constants then dE/dx is equal to E/x but I was told that's not the case...

Hi, I was attempting the problem above and got stuck along the way. Problem: Suppose that ## Y_1 ## and ## Y_2 ## are random variables with joint pdf: f_{y_1, y_2} (y_1, y_2) = 8y_1 y_2 for ## 0 < y_1 < y_2 < 1 ## and 0 otherwise. Let ## U_1 = Y_1/Y_2 ##. Find the probability distribution ##...

Hello, I'm carrying out an assignment on the simulation of correlated (random) variables. Can you suggest to me equations pertaining to the field of engineering (or science) that involve correlated variables? My question may sound redundant, because all equations create dependencies between...

I am looking for books that have sections or even chapters devoted to complex random variables, or random variables that can take on complex values (NOT probabilities that are valued in the complex range, in this regard). On the other hand, if someone does know any books that contain material on...

The answer calculates the integral with ##du## before ##dv## as shown below. However I decided to compute it in the opposite order with different bounds. Here is my work: According to the definitions, $$\begin{cases} u=x+y\\ v=2x-3y \end{cases}$$ First we need to convert the boundaries in xy...

(I know how to prove it). Prove that a finite sum of of independent normal random variables is normal. I suspect that independence may not be necessary.

Using the related equations, I have come up with an equation that I think would answer this question (#4). However, I was just wondering does the 'L' from equation #1 and 'l' in equation #3 cancels each other out when the substitution is made to give out equation #4 (since they are both the...

Given a probability density distribution ##P(\vec{x})##, for what named distributions is the following true: \begin{equation} \begin{split} P(\vec{x}) &= P_1(x_1) P_2(x_2) ... P_n(x_n) \end{split} \end{equation}

If individual atoms are indistinguishable from one another, then how can you tell if atom A will experience radioactive decay before identical atom B? ISTM there would have to be some underlying structure beyond electrons and quarks and unique to each atom / particle to be able to do this...

I've been trying to get change of variables in PDEs down (I don't particularly like my textbook or professor's approach to it), and I want to ask here if I am getting this right. Let ##\vec{x}=(x_1,x_2,...,x_n)^T## and ##\partial_\vec{x}=(\partial_{x_1},\partial_{x_2},...,\partial_{x_n})^T##. I...

The image above is the problem and the image below is the solution I have tried but failed.

Each different boundary condition means a different charge configuration, how can this problem be solved using superposition?

For a thought experiment, if you defined: An electron's "state" can be in 16 states measured as increments of 45 "degrees" starting at 22.5 (modulus 360) An entangled electron (e') simply gets aligned 180 degrees away from (e) A "measurement" M(n) at n degrees simply adds n degrees "Spin" is...

I have a problem where I am given the density of states for a Fermion gas in terms of momentum: ##D(p)dp##. I need to express it in terms of the energy of the energy levels, ##D(\varepsilon)d\varepsilon##, knowing that the gas is relativistic and thus ##\varepsilon=cp##. Replacing ##p## by...