Analysis and Calculus

If I have a given function that is defined as an integral, like of something that's clearly hard to work with like the gamma function or the exponential integral or the cosine integral or the error function or etc, is there a way to define an inverse of that function in a similar form that's NOT a cop-out dy integral? Not some dy integral that forces me to switch to integrating along the y-axis for no reason that solves absolutely nothing and never will, but something that starts as f(x) = integral(g(x))dx and hopefully takes the form of f-1(x) = integral g^-1(x)dx or something similar, something that is still defined on the domain of x. In fact, you know what, let's…

Suppose that for solving for the inverse of a function that f(x) can be manipulated into the form of [math]g(f(h(x)))=x[/math]. Then, h(x) is inverted to show [math]g(f(x))=h^{-1}(x)[/math]. Afterwards, the function is reverted to show [math]h(g(f(x)))=x[/math]. Does this correctly show that h(g(x)) is in fact the inverse of f(x)? If not, what can be done in this situation to find the inverse of the original f(x)?

I set up an excel spreadsheet to integrate the following system by first order forward difference equations: 2/(k-1)*dc/dt + du/dt = 0 in direction dr/dt = u + c 2/(k-1)*dc/dt - du/dt = 0 in direction dr/dt = u - c With r=0 boundary conditions u=Asin(wt), c = c0, I got a solution that was a very good fit to u=Asin(wt)cos(wr/c0) which is the expected result for a standing wave in a cylinder. How good a fit? Well the integration yielded maximum variance values of 0.5 in both r and t trends over one wavelength, and subtracting the above expression reduced this by 99.9954% in the r trends, 99.99902% in the t trends. A pretty good fit. However, looking…

It seems to me from my recent entry into this area that a tensor is a mathematical object with direction * and magnitude* that applies to the behaviour (?) of a cell in a manifold that exists over an area of space. These calls are non pointlike regions corresponding to what ,in school level geometry would be points in the Cartesian coordinate system such as (1,0,0) or (7,5,18) just as examples. These cells (and their corresponding tensors) can model physical activity such as different(?) forces acting in particular on the cell(=local region). My question is : (1)"Do these tensors all have to use the same units?" ** (2) Also is my understanding …

Hello everybody! I am trying to solve the wave equation in 3D [latex]\frac{\partial^2 p}{\partial t^2} - c^2 \Delta p = \delta(\vec{r}-\vec{r_0}) \delta(t-t_0),[/latex] where [latex]\vec{r_0}[/latex] is the radius-vector of the point source and [latex]t_0[/latex] is the moment when the source radiates a pulse, with Dirichlet boundary conditions. I am solving the equation with finite-difference method in time domain. The steps by time are done by Runge-Kutta 4. The source is put into one node and the time profile of the source is step-like. I made it step-like (see Pic. 1) because a point source should radiate a time-derivative of the incoming signal [latex]p(\vec{r},…

Someone claims to have a whole number solution to A^n+B^n=C^n where n is greater than two. If they have the three numbers then they can form a triangle. This triangle will always have an apex (junction of A and B) which will be less than 90 degrees. Please consider whether you agree with this statement before proceeding! This triangle can be considered as a triangle of forces and from which a parallelogram of forces can be drawn. Using the parallelogram of forces both a vector diagram and a phasor diagram can be drawn and used as reliable tools for navigators and engineers of all disciplines. A phasor diagram can only be used for adding sine waves …

Dear All, I'm trying to find function J(x) shown in attached file. Could somebody help me ? It seems that I don't have the right tool to solve it Thanks and Regards. Série.pdf

I took calculus I back in college, but never needed to take calc. II or III. I'm a little interested in learning II and III, just for general knowledge and boredom, but I don't want to spend in-depth analysis with the actual math questions and such. Since I have general knowledge of calculus, is it possible to get a general idea of how calc. II and III operates with just graphical examples? ~EE

Hi All, Hope you find of some interest how to teach young students (even children) on high-order root-approximating methods (at any desired convergence rate) based on the generalized mediant which is a well-known operation for Number theorists. No derivatives, no Trial-Error checks, No Geometry, but just the Simplest Arithmetic. Kindly take a look at the introduction on the new high-order methods for children at: url deleted As a reference for talented youngs: url deleted Best Regards,

I am stuck on this problem in this calculus book I am reading. I was wondering if someone could tell me the method of finding the answer: A mass M is drawn up a straight incline of given height h by a mass m which is attached to the first mass by a string passing from it over a pulley at the top of the incline and hangs vertically. Find the angle of incline in order that the time of ascent be a minimum. Yours sincerely, Bahoz Mirweis.

It doesn't seem right, but I could have sworn I remember seeing instances in statistics where a summation was literally just swapped with an integral sign without anything else changing, except adding a dx. Can you really just swap a sum and an integral? I mean, the sum of a linear variable k is a second degree polynomail, and the integral of a continuous variable k would be a 2nd degree polynomial, so maybe there's something to it.

Dear All, I'm trying to find a primitive of f(x)=cos(-Asin(x)) where A is a constant number. I've tried Wolfram but unsuccessfully. An approximate function of this primitive would also be welcome. Thanks for your help ! Kind Regards.

Hello, As I understand, one typically derives properties of functions that satisfy certain functional equations by substituting x for various values like 0, 1, y, -x, x^2 and so on. But what I am wondering is why algebraically solving for the unknown function doesn't work, or at least what it means when you do.

First of all I want to clarify that I posted this question on many forums and Q&A websites so the chances of getting an answer will be increased. So don't be surprised if you saw my post somewhere else. Now let's get started: When it comes to definitions, I will be very strict. Most textbooks tend to define differential of a function/variable in a way like this: -------------------------------------------------------------------------------- Let [latex]f(x)[/latex] b…

I am currently reading a book on calculus and I have come across a problem which I can't solve. I do feel like the answer is something simple. Please note that I am fairly new to calculus. Thank you.

I'm trying to figure out how to describe the geometrical coordinates of a wave contacting and reflecting off the surface area of a spherical object/particle; specifically waves traveling at the speed of light. My goal is to be able to explain the torsion in a region of surface area over time making contact with a wave reflecting off it, considering how the velocity of the object/particle changes as well, and how the geometry of the reflecting wave changes too. I have a good picture in my head of how it all comes together geometrically, but I don't know how to go about writing it down with tensor geometry on paper. I'm still very new to using tensor calculus; I…

I need this limit ((x(sqrt(x+2)))/sqrt(x+1))-x to calculate the asymptote of this function: ((x(sqrt(x+2)))/sqrt(x+1)). which, according to the class notes: y=x+1/2 with a = 1, b = 1/2 However, the online math calculators say that currently no steps are supported to show for this kind of problem. I calculated this limit as x*sqrt(1)-x = 1, but apparently the correct answer is 1/2. What is my mistake? Thx in advance. r

Have you ever noticed that real-life situations rarely conform to maths learned at school and university? So here I am, Christmas morning, and instructed to do something useful. So I get the task of rolling lengths of bacon round dates, each one forming a cylinder length 4 cm and diameter 2.5 cms. I am told to put them all into a baking dish which is elliptical, a=10cm, b=6cm (half-major axes) and get as many as possible in, on their sides. So the mathematical task is to fit rectangles 4 x 2.5 into that ellipse, and get the maximum number in. There of course is the inevitable "discussion" that a different way of arranging them would fit more in, but I claim there is no a…

Imagine that you have a rope which is 24 cm long. If you make a perfect square from this rope (each side is 6 cm) then the area becomes 6 x 6 = 36 cm^2. If you make a perfect triangle (each side is 8 cm) from the same rope, then the area becomes (768)^0.5 which is approx. 28 cm^2 So the length of your border is always the same, but this same border length surrounds different amount of area for different shapes. The border length is the same, but the area size is different. I am not asking mathematical explanation, but I am searching for the logical explanation. What is the logic behind this? Difficult to understand the logic.

Hi, Can anyone check for me if I'm doing correctly calculations based on 2 equations? Especially an integral because I don't really know how should I compute it. I'm calculating it in microsoft excel (I've got those equations from scientific paper, and I don't really know if there is a mistake or maybe I'm doing something wrong, because I should get other results than I'm actually getting) (I've put both equations and excel file in attachments). From the beginning, I've got a model created in Abaqus CAE and I am taking temperatures of elements from there (temperature depeneds on time, because it it connected with laser irradiation, laser position is changing with time).…

Hi, I want to know why is the Riemann hypothesis still unsolved? I really want to know the reasons due to which it is still unsolved.

Background - System of points a, b, c and q - a, b, and c exert a push on q moving it to the end of a vector that is 2 times the vector from a, b, or c to q. - a, b, and c do this at the same time, after each cycle the x and y values of q are the average of the end points of the vectors. - This happens n times First derivation The final x value is the sum of the averages of all the vectors endpoints divided by the number of iterations so n. Same goes for y I further thought why keep it at intervals when it can be a constant force. From this I derived The integral from 1 to n of the average of the averages of the vector endpoints in respect to the x valu…

I would appreciate very much your help or direction to solve the problem. ​ Looks like equation Q=SUM(ABS(B-Ai)) has got min and max values with the assumptions 0<B<1 0<=Ai<=1 SUM(A1:AK)=1 and i is a natural number from 1 to K Looks like the Qmax and Qmin depend on B and K only. I failed to find general solution for Qmax and Qmin. I attach the pdf with the problem written in Word Equation instead of Excel manner Q.pdf

Find integral from 0 to 2 of: X2 _______________ X2 - 2X + 2

when I tried to do: I came to the answer of 0 and I know this is not right. I do remember that you had to split the problem up and then integrate those or something like that. Could someone tell me the answer with their method.