Inverse Definition and 1000 Threads

I noticed the scan was cut off on the second image at the bottom right, but I came up with x= 31/5 My first test in Calc I begins tomorrow and I want to know that I'm headed in the right direction. I think I understand to some extent how logarithms can be expanded and condensed though I'm...

Given the Lorentz matrix Λuv its transpose is Λvu but what is its transpose ? I have seen ΛuaΛub = δb a which implies an inverse. This seems to be some sort of swapping rows and columns but to get the inverse you also need to replace v with -v ? Also In the LT matrix is it the 1st slot...

Hello all, Can anyone give me a pointer on how to start this proof?: f:E\rightarrow F we consider f^{-1} as a function from P(F) to P(E). Show f^(-1) is injective iff f is surjective.

I am currently confused with the concept of the blackbody radiation and the inverse square law. Planck's function for the radiation of a blackbody is in ##W sr^{-1} m^{-3} ##, is this somehow a form of intensity (because of the watts per square meter unit)? If it does, doesn't intensity...

I almost understand how the inverse square law is derived from the area of sphere equation, 4πr2, but I'm not quite clear on what happens to the 4π. I found one equation that seemed to say that the intensity is equal to the area of the sphere of the source point times the amount of whatever...

I have a set of data points that I must fit to an inverse function like this: y(x) = a/(x+b) + c My problem is that least-squares fitting using this equation is extremely unstable and heavily dependent on initial guess. No matter how accurate parameters I start with, the algorithm often goes...

Is there a way to formally prove that if ##f## and ##g## are multiplicative inverses of each other, then ##f^{-1} (x) = g^{-1} (\frac{1}{x})##?

Is this normal? it doesn't seem correct. The equation for the portion of circle with radius 1 unit in the 1st Quadrant is: ## y = f(x) = \sqrt{1-x^2} ## Domain is 0<x But when I calculate f'(x) I also get ## f'(x) = \sqrt{1-x^2} ## I thought inverse functions always reflect about y=x. Please...

What is the inverse function of ##x+sin x## ?

Homework Statement A spectator is standing 50 ft from the freight elevator shaft of a building which is under construction. The elevator is ascending at a constant rate of 20 ft/sec. How fast is the angle of elevation of the spectator's line of sight to the elevator increasing when the elevator...

Does anyone perhaps have a good way for me to get a lasting 'intuition' about what inverse hyperbolics are? I look at, for example, the well known sin x; it is periodic. Then, it seems, sinh x is a reflection of sin x about the line y=x. (I found an example at 7. The Inverse Trigonometric...

Hi MHB, The following problem has been a really vexing problem (for me) because I believe there would be a tricky way of approaching it but I could not solve it after working with it on and off for two days, it is an Olympiad math competition problem, and so far no one that I know of has solved...

Hi, I know that the inverse function of y= f(x) =2x+1 is y-1=2x x=(y-1)/2 and then we just replace x with f-1(y) and then when we plug in any value of y it gives us a corresponding x value. Now my question is this: If we want to find a line or function that is perpendicular to another...

I am familiar with the importance of the following inverse circular/hyperbolic functions: ##\sin^{-1}##, ##\cos^{-1}##, ##\tan^{-1}##, ##\sinh^{-1}##, ##\cosh^{-1}##, ##\tanh^{-1}##. However, I don't really get the point of ##\csc^{-1}##, ##\coth^{-1}##, and so on. Given any equation of the form...

To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...

Hello all, I have this equation \sum_{k=\lceil \frac{n}{2}\rceil}^n{n\choose k}P^k\left(1-P\right)^{n-k}=\epsilon and I want to find P as a function of epsilon and n. Can I do that? If so, then how? Note: \epsilon<10^{-3} if it helps for any possible approximation. Thanks

Hello, When evaluating the step response of a circuit, the resulting Laplace representation is: $\frac{I_{pd}}{s^2 C1 R1}$ If I look this up on a table of Laplace Transforms, this results in $\frac{I_{pd}*t}{C1 R1}$. However, I'm struggling to solve this via partial fraction expansion--is...

Homework Statement Show that ##n\times n ## complex matrices such that ##\forall 1\le i \le n,\quad \sum_{k\neq i} |a_{ik}| < |a_{ii}|##, are invertible Homework EquationsThe Attempt at a Solution If I show that the column vectors are linearly independent, then the matrix has rank ##n## and...

I've been looking at the torus parametrization \begin{equation} \phi(u,v)=((r\cos u+a)\cos v, (r\cos u +a)\sin v, r\sin u) \end{equation} with \begin{equation}a>0, r\in(0,a)\end{equation}. I want to invert this map to get a chart map for the torus. Can anyone give me a hand with this? Thanks!

Hi, I have an idea which when tested looks like its clearly flawed. I am hoping someone can tell me where my procedure is flawed, or point me to some other theory that has already done something similar. The first two are the laplace transform. The third line is the Fourier Transform. The...

Homework Statement ##f : A \rightarrow B## if and only if ##\exists g : B \rightarrow A## with the property ##(g \circ f)(a) = a##, for all ##a \in A## (In other words, ##g## is the left inverse of ##f##) Homework EquationsThe Attempt at a Solution I have already prove the one direction. Now I...

In other words, why does a function have to be onto (or surjective, i.e. Range=Codomain) for its inverse to exist?

Running into a little trouble when doing this integral by hand: \int arccsc(x) dx u = arccsc(x) \implies du = -\frac{1}{x\sqrt{x^{2}-1}} dx dv = dx \implies v = x \int u dv = uv - \int vdu \int arccsc(x) dx = xarccsc(x) - \int x\cdot -\frac{1}{x\sqrt{x^{2}-1}} dx \int...

Hi! Here is my task: Find inverse z transform of $$X(z)=\frac{1}{2-3z}$$, if $$|z|>\frac{2}{3}$$ using definition formula. I found that $$x(n)$$ is $$\frac{1}{3}(\frac{2}{3})^{n-1}u(n-1)$$ (using other method). But how can I find it using definition formula, $$x(n)=\frac{1}{2\pi j}\oint_{C}^{ }...

Homework Statement Solve the following: $$\mathscr{L}_s^{-1} \left\{ \frac{s}{s^2-s+\frac{17}{4}} \right\}$$ Homework Equations Table of Laplace Transforms.The Attempt at a Solution The solution is $$f(t) = (1/4 )e^{t/2} (\sin(2 t)+4 \cos(2 t))$$ I know I need to break up ##F(s)## into...

Hi! My task is to find discrete signal x(n), if z transform of that signal is $$X(z)=\frac{5}{(z-2)^{2}}$$. It is known that signal is causal. Here is what I have done. Since signal x(n) is causal, convergence of z transform of that signal will be outside of circle with radius r: We have in...

if we consider a separable measurement operator $$(A+A')\otimes (B-B')$$ then quantum mechanics predict the result is in [-2;2] Whereas going to classical results would give in [-4;4] This could indicate that going from measurement operator in the quantum realm to measurement results is maybe...

Homework Statement Find ## \sum_1^{23} tan^{-1}(\frac{1}{n^2+n+1}) ## Homework Equations ## tan^{-1}x + tan^{-1}y = tan^{-1}(\frac{x+y}{1-xy} )##The Attempt at a Solution I think we have to split the question in a form of relevant equation given above. First thing what should I do?

Homework Statement Homework Equations H(z) = Y(z)/X(z) The Attempt at a Solution I realized this wasn't in partial fraction form because the 1+z-1+0.5z-2 has non-real roots. I multiplied the 1st fraction part by z1 and the 2nd fraction by z2, then I combined them into one fraction and I...

Seems exist some relationship between the inverse of a matrix with the inertia tensor, looks: This relationship really exist?

Be the matrix A defined by [aijk] (a matrix 2x2x2), do you know how to compute a inverse this cubic matrix?

Homework Statement How can I take the Inverse Laplace Transform of $F(s) = \frac{d}{ds}\left(\frac{1-e^{5s}}{s}\right)$? I have tried going with inverse of the derivative and convolution (even tried evaluating the derivative and go from there) but although I can get to some results none of them...

Homework Statement Show that f(x) = \sqrt{5x+2} is one-to-one. Homework Equations If f' >0 or f' < 0 everywhere on f's domain, f is one-to-one. The Attempt at a Solution f^\prime = \frac{5}{2} \frac{1}{\sqrt{5x +2}} = \frac{5}{2f} f' is positive on (-2/5, infinity) but is undefined...

I have a problem I am trying to solve but no example provided can clarify the steps. I find this to be the problem with math examples, they provide problems but utilize same numbers so people who are not as math savvy such as I cannot figure the steps out as easily as they are provided. Anyway...

Homework Statement I'm looking for how to calculate inverse of the 4th order tensor. That is, [FONT=Times New Roman]A:A-1=A-1:A=I(4) If I know a fourth order tensor [FONT=Times New Roman]A, then how can I calculate [FONT=Times New Roman]A-1 ? Let's just say it is inversible. Homework...

Homework Statement If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x). Homework Equations (None I believe.) The Attempt at a Solution I know that for a function to be invertible, it must be both one-one and onto...

Homework Statement Piezoelectric crystals will change about 0.1% of their static dimension when an external electric field is applied to the material. Homework Equations 1. What happens with the static dimension if the external electric field applied is a constant? 2. Will the static dimension...

Hi, I'd like to get some help with my homework. Piezoelectric crystals will change about 0.1% of their static dimension when an external electric field is applied to the material. 1. What happens with the static dimension if the external electric field applied is a constant? 2. Will the static...

I am reading Munkres book, "Topology" (Second Edition). I need help with an aspect of Theorem 18.2 Part (f) concerning the inverse image of a set under the restriction of a function ... Theorem 18.2 Part f reads as follows: In the above text we read: " ... ... Let $$V$$ be an open set in...

In Munkres book "Topology" (Second Edition), Munkres proves that a function F is a homeomorphism ... I need help in determining how to find the inverse of $$F$$ ... so that I feel I have a full understanding of all aspects of the example ... Example 5 reads as follows:Wishing to understand...

Homework Statement I will post a picture of the problem and then the second picture will be my work. The problems are the first two. Homework EquationsThe Attempt at a Solution I didn't know how to do this at first so I don't know if I am doing it correctly now. Also I don't know the correct...

Homework Statement From an old edition of the Anton calculus text, I am asked to prove cos-1(-x) + cos-1(x) = π or equivalently cos-1(-x) = π - cos-1(x) Homework EquationsThe Attempt at a Solution Earlier I proved that sin-1(x) was odd by noting sin-1(sin(x)) = -sin-1(sin(-x)), so I...

Is it possible to produce the inverse of a natural radioactive decay? If yes, what would happen? Release energy?

Could I by any way find out average thermal retention of a blackbody that is irradiated with x rays? Since wavelength is inversely propotional to temp, in order to emit and absorb x rays it must be at a fairly high temperature I guess.

Why is the solubility of some solids lower when the solution is heated? I read that it is because the process is exothermic (heat from breakdown is greater than the heat needed for breakdown). But why would having extra heat from outside sources inhibit the dissolving process, wouldn't it only...

And also: y`+2y=2(1-e^-2t) Y(0)=0 y¨-2y`+y = t+e^t y(0)=1 and y`(0)=0 Please help me out here folks ;)

Hi. I`m new here and I need some help with Inverse Laplace Transform: f(t)=5+3t+e^3t g(t)=(t+1)u(t-2) g(t)=(t^2-9t+20)u(t-5) and Laplace Transform: F(s)=1/(s+2)^5 F(s)= 2s^2+10/s(s^2+2s+10) G(S)=2s/s^2+4e^-sso if anywone can please help me:)

Homework Statement In the inverse compton scattering, find the formula of the recoiled electron as a function of the incoming electron Homework Equations The energy of the incoming photon and electron are known. The Attempt at a Solution Starting from this...

Hello everyone, This is probably going to come off as a very silly question. However, I have not had calculus in several years. I was angry that my physics textbook did not have a derivation of Electric Potential Energy. So, I finally came across it, and I see that the integration of the...

Homework Statement 1.\int{\frac{sinx}{1+cos^{2}x}} \, dx 2.\int{\frac{1}{13-4x+x^2}} \, dx Homework Equations Inverse trig identities. The Attempt at a Solution For the first one, I'm not too sure about what to do with the sinx on the numerator and i have tried u-substitution to no avail...