Inverse Definition and 1000 Threads

According to the author of this article, this integration formula is well-known but rarely taught. Do you know it? $$\int f(x)dx = xf(x)-\int_{x_{0}}^{f(x)}f^{-1}(t)dt $$ where x0 is a constant and f-1(x) is the inverse function of f(x).

Hello, This is my first post. For background, I have asked Ask The Physicist who has posted our correspondences and answers here: https://www.askthephysicist.com/ask_phys_q&a.html Search "My question is with regard to an apparent mathematical disagreement" or something like that to find the...

Is there a systematic way to do it? In particular, I have the coordinates ##x=au \sin v \cos w##, ##y=bu\sin v\sin w##, ##z=cu\cos v##, where a, b, c are constants, and I want to find ##u(x,y,z)##, ##v(x,y,z)##, ##w(x,y,z)##. I could solve the three equations for u, v, and w and then try to...

I was able to solve with a rather longer way; there could be a more straightforward approach; My steps are along these lines; ##\sinh^{-1} x = 2 \ln (2+ \sqrt{3})## ##\sinh^{-1} x = \ln (7+ 4\sqrt{3})## ##x = \sinh[ \ln (7+ 4\sqrt{3})]## ##x = \dfrac {e^{\ln (7+ 4 \sqrt{3})} - e^{-[\ln 7+ 4...

I am trying to solve (a) and (b) of this question. (a) Attempt We know that ##\frac{2}{3} < \frac{e(t) - e(s)}{t - s} < 2## for ##t \neq s \in (c(-d), c(d))## Thus, taking the limits of both sides, then ##\lim_{t \to s} \frac{2}{3} < \lim_{t \to s} \frac{e(t) - e(s)}{t - s} < \lim_{t \to...

I can get the slope of the line, but then I dont know what the slope is (units or whatever it is) then i get lost from there.

I understand why certain inverse trig functions have two answers. Like for arcsin(0.5), it could be pi/6 or 5pi/6. I know both angles have the same sin value, that they're both on the same horizontal line on a graph of sin, I get all of that, but two questions about it: 1) In cases where...

I'm desperately trying to understand how to get from 2.7.11 to 2.7.16 and cannot find any reference online on how to find the inverse of an elastic tangent modulus (fourth_order tensor). Can someone help me or give me a reference I can check where they do a similar thing? I would really...

I have a probability distribution over the interval ##[0, \infty)## given by $$f(x) = \frac{x^2}{2\sqrt{\pi} a^3} \exp\left(- \frac{x^2}{4a^2} \right)$$From this I want to derive a formula for the inverse cumulative density function, ##F^{-1}##. The cumulative density function is a slightly...

For this, I don't understand how if ##(A - 2I_2)^{-1}## has an inverse then the next line is true. Many thanks!

For this problem, Find ##A^{-1}## given, The solution is, However, in the first image, why are we allowed to put together the submatrices in random order? In general does someone please know why we are allowed to decompose matrices like this? Many thanks!

For every instance of addition or multiplication there is an inverse, closed on the naturals. Not every instance of subtraction and division is defined, so not closed on the naturals. This looks like two kinds of inverse. Instance inverse - the inverse of instances of addition and...

Inverse square law would reduce the gravity from the parts of Earth that are farthest from our feet. It'll also reduce the gravity from Earth's center by a lesser amount, but would that be lesser enough so the gravity 20 kilometers under our feet is stronger than the core's gravity or even the...

Hello All, I am somewhat familiar with FFT and iFFT and its uses. However I have an issue when I edit in Freq domain and try to get back to time domain . I have an audio signal in time domain that I transform to frequency domain using an FFT routine in block sizes of N points. (in my case 256...

For this, Dose someone please know how ##ad - bc## and ##-cb + da## are equal to 1? Many thanks!

For this, Dose someone pleas know where they get ##C = CI## from? Also, What dose it mean when A and B commute? Many thanks!

I came across the mentioned equation aftet doing a integral for an area related problem.Doing the maclaurin series expansion for the inverse sine function,I considered the first two terms(as the latter terms involved higher power of the argument divided by factorial of higher numbers),doing so...

For this, Why are we allowed to interchange x and y? Is it because the equation will still be true? Many thanks!

The Schwarzschild solution could simply be expressed as $$ds^2=-(1-2GM/r)dt^2+(1-2GM/r)^-1dr^2+r^2d\omega^2$$. Is it possible that we could obtained a new metric into the form as $$ds^2=-(1-2GM/r)^-1dt^2+(1-2GM/r)dr^2+r^2d\omega^2$$? If possible, what are the steps and procedures that should be...

...y and Coulomb's law diverge as ##r\rightarrow##0? I mean, if a point light source emits light omnidirectionally, the intensity converges at the source, right? THIS is how I should've worded my previous post!

Like this one: ##f(x)=3x^3 -18x^2 +36x##.Is there any way to find the inverse of this function without using the general solution to cubic equation?

Existence: Ax = b has at least 1 solution x for every b if and only if the columns span Rm. I don't understand why then A has a right inverse C such that AC = I, and why this is only possible if m≤n. Uniqueness: Ax = b has at most 1 solution x for every b if and only if the columns are...

[FONT=georgia]Problem statement : I cope and paste the problem as it appears in the text below. [FONT=georgia]Attempt : Not being a math student, I try and prove the above statement using an "intuitive" way. Let us have a rational number ##b = \frac{n}{m}##. Multiplying with ##a## from the...

Newton arrived at "there is a force that drives a planet around the star by examining kepler's laws but how did he arrive to inverse square law by kepler's third law (##T^2=\frac {4\pi r^3}{GM}##)? Thank you.

It seems to me that all mathematics, to one degree or another, is devoted to solving inverse problems. Let's take five equations as an example: 1) ax+b=0 2) ax^2+bx+c=0 3) ax^3+bx^2+cx+d=0 4) ax^4+bx^3+cx^2+dx+e=0 5) ax^5+bx^4+cx^3+dx^2+ex+f=0 To solve the first problem, you need to spend one...

Hi everyone This is the solution for the problem. I don't understand how they got from To This was my attempt at a solution I can't seem to get rid of one of the y terms and am left with one on each side. Could someone explain the solution to me please? Thanks

If I have a ##n\times n## matrix $$ U= \begin{bmatrix} u_{11} & u_{12} &u_{13} & \cdots u_{1n} \\ 0 & u_{22} & u_{23} & \cdots u_{2n} \\ 0&0 &u_{33} &\cdots u_{3n}\\ \vdots & \vdots &\vdots & \cdots \vdots \\ 0 & 0 & 0 &\cdots u_{nn} \end{bmatrix} $$ Now, I don't want to use the fact that it's...

##H## is an ##n\times n## Hermitian matrix with eigenvectors ##\mathbf{e}_i## and all eigenvalues negative. It's claimed that ##G = \int_{0}^{\infty} e^{tH} dt## is such that ##G = H^{-1}##. I was looking at\begin{align*} G\mathbf{e}_i &= \int_0^{\infty} \sum_{n=1}^{\infty} \frac{t^n}{n!} H^n...

Hi, I have to verify the sifting property of ##\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} e^{-sa}e^{st} ds## which is the inverse Mellin transformation of the Dirac delta function ##f(t) = \delta(t-a) ##. let ##s = iw## and ##ds = idw## ##\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iwa}e^{iwt}...

I have a problem of understanding in the following demo : In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions...

What would the correct form of this? $$\stackrel{\leftrightarrow}{A} \cdot \vec{b} = \stackrel{\leftrightarrow}{C} : \vec{d}$$ I'd like to know which one is correct form 1.) $$ \vec{b} = (\stackrel{\leftrightarrow}{A})^{-1} ( \stackrel{\leftrightarrow}{C} : \vec{d} ) $$ 2.) $$ \vec{b} =...

Hi everyone! I need to inverse an label transform with sklearn. I found this example on web: from sklearn.preprocessing import LabelEncoder np.random.seed(1) y = np.random.randint(0, 2, (10, 7)) y = y[np.where(y.sum(axis=1) != 0)[0]] array([[1, 1, 0, 0, 1, 1, 1], [1, 1, 0, 0, 1, 0...

I recently encountered this problem on a test where the solution for the above problem was given as follows: $$F= \frac{Gm_1m_2} {r^2} $$ (1) but $$ m=\frac{4}{3}\pi R^3 $$ substituting in equation (1) $$F= \frac{{G(\frac{4}{3}\pi R^3\rho})^2 }{2R^2} $$ where r=radii of the two spheres m=mass...

Consider the case of a real function f for which f inverse exists. 1) We we are not used to having the y-axis (vertical axis) to denote the independent variable which it does in x=f-1(y). We rotate the system through positive 90 degree and reflect about the vertical to change the sense of the...

\mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]=Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=2]+Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=-2] From that I am getting f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})...

Hi, Is there a more general equation than the Compton equation that allows one to determine whether an electron will Compton scatter or inverse Compton scatter? If so, where can I find it (or what is it?) Thanks.

[FONT=times new roman]Problem Statement : [FONT=times new roman]Solve for ##x## : Attempt : If I take ##x=\tan\theta##, the L.H.S. reads $$\tan^{-1}\frac{1-\tan\theta}{1+\tan\theta}= \tan^{-1}\left[\tan\left(\frac{\pi}{4}-\theta \right) \right ]=\frac{\pi}{4}-\theta.$$ On going back to ##x##...

Is the highest frequency possible the inverse of Planck time? Separate or connected question, what's the highest frequency achievable practically today?

Hi I was working on a physics problem and it was almost solved. Only the part that is mostly mathematical remains, and no matter how hard I tried, I could not solve it. I hope you can help me. This is the equation I came up with and I wanted to prove it: $$\lim_{n \rightarrow+ \infty} {...

Im confused about a certain part of solving an equation. So I used the hyerbola formula to find the answer but I think I did the math wrong. X^2-y^2=c^2 X=1 Y= (2x^5-1)^2 I did the calculations as you can see in the picture but I know I messed up on the square root part. When you square one...

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...

Exchange traded funds like TZA and RWM attempt to get results that are the "inverse" of certain stock indexes. I understand they do this by shorting stocks. When do they cover their shorts? -before the end of each trading day?

Hey! :giggle: We have the matrix $\begin{pmatrix}2 & 1/2 & 1 \\ 1/2 & 3/2 & 1/2 \\ 1 & 1/2 & 2\end{pmatrix}$. We take as initial approximation of $\lambda_2$ the $1.2$. We want to calculate this value approximately using the inverse iteration (2 steps) using as starting vector...

##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...

(I must confess that, in spite of working through the chapter on inverse circular functions, I could barely proceed with this problem. Note what it asks to prove : ##x\sqrt{1-x^2}+\ldots## and how much is that at odds with the formula (1 above) of adding two ##sin^{-1}##'s, where you have...

My attempt at finding this was via convolution theorem, where we take F(s) = 1/s^2 and G(s) = e^(-sx^2/2). Then to use convolution we need to find the inverses of those transforms. From a table of Laplace transforms we know that f(t) = t. But I am sort of struggling with e^(-sx^2/2). My 'guess'...

Hello, I am puzzled about the following condition. Assume a matrix A with complex-valued zero-mean Gaussian entries and a matrix B with complex-valued zero-mean Gaussian entries too (which are mutually independent of the entries of matrix A). Then, how can we prove that...

I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?