Integration Definition and 1000 Threads

Hello! In Nakahara's Geometry, Topology and Physics in chapter 6.4.5 second edition, he proves at a point that on a simply connected manifold, the first de Rham cohomology group is trivial. In the proof he defines ##\alpha : I=[0,1] \to M##, homotopic to a point. Now, by the rules of integration...

Hi. I think I jumped too eagerly into my latest module, as I am positive it makes many assumptions on previous experience. I am obviously lacking. I have created a question in Word, but the equations don;t appear to be copying into this thread. So I have uploaded question instead. If this is...

Homework Statement I'm trying to figure out the other parameters to solve the problem Homework Equations I know sqrt(x^2+y^2+z^2) = p The Attempt at a Solution I changed the integrand to p^3 sin(theta) since p * p^2 sin(theta) Then for the first integration sign, I know how to get the...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding the first example on page 63 ... ... The the first...

Homework Statement I am supposed to evaluate the contour integral of the positive branch of ##z^{-1/2}## over the following contour: I believe the answer should be 0, by Cauchy's theorem (loop encloses no poles), but my methods of parameterization have led to non-zero answers. Homework...

I'm solving a first order linear ODE system numerically three different ways: implicit Euler, explicit Euler, and RK4. Attached are the plots of the numerical solutions (line and stream of small closely-connected dots) and the exact solution (big dots). Also I plot the maximum error (right...

Homework Statement Hi. I ve got a problem, where I have to show torque by integrating the weight of the rod over the whole it's length. Homework Equations [/B] Result, what I am suppose to get is: ## \tau_{rod} = \frac{mgb}2 ##The Attempt at a Solution [/B] When I try to integrate, I am...

Homework Statement So this is a problem for my numerical methods class. We are doing integration and derivation. The part that I am stuck on is why does the midpoint method and trapezoidal method both have error of order ##h^2##, when the first one is one point integration method while the...

Homework Statement A solid is bounded by the surface z=x^2-y^2, the xy-plane, and the plane x=1 and x=3. Compute, by double integration, the volume of the the solid. Homework EquationsThe Attempt at a Solution I know that the height is the z function and the lower limit of is x= 1 and upper...

Is there a book that can teach me every single method of integration (except Ryzhik) and Special Functions (Bessel,Hermite,Legendre ,Spherical Harmonics). since I'll be use it in QFT ,It will be very useful to have strong in Bessel Functions and Others Functions and Integrals.

I'm following a derivation in my lecture notes of total average particle number in an ideal classical gas (statistical physics approach). I follow it to the line (though the specific terms don't matter): \left<N\right> = e^{\mu/\tau} \frac{\pi}{2} \int_0^\infty \left(n \,dn \,e^{- \frac{\hbar^2...

Let ##P## be a ##U(1)## principal bundle over base space ##M##. In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase ##\gamma = \oint_C A ## where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of...

Homework Statement [/B] Determine the value that A (assumed real) must have if the wavefunction is to be correctly normalised, i.e. the volume integral of |Ψ|2 over all space is equal to unity. Homework Equations Integration by parts (I think?) The Attempt at a Solution So, I've managed...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 7: The Riemann Integral ... I need help in fully understanding an aspect of the proof of Theorem 7.2.9 ...Theorem 7.2.9 and its proof ... ... read as follows...

I have learned differentiation (till multiple order differentiation). But I don't just understand integrals.

Homework Statement [/B] $$\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx$$ or http://www.HostMath.com/Show.aspx?Code=\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx Homework Equations the sinh identity, which is (e^x-e^-x)/2 The Attempt at a Solution Tried...

Hello all, I am trying to understand the rational behind the visualization of integration by parts, however I struggle with it a wee bit. I was trying to read about it in Wiki, this is what I found...

Homework Statement Suppose an infinitely long wire carrying current ##I=sin_0(\omega t)## is a distance ##a## away from a equilateral triangular circuit with resistance ##R## in the same plane as shown in the figure. Each side of the circuit is length ##b##. I need to find the induced voltage...

$\textsf{a. Sketch the region of integration and evaluate.}\\$ \begin{align*}\displaystyle \int_{6}^{\frac{\pi}{6}} \int_{0}^{\sec{\theta}} 6r^{3} drd\theta\\ W|A=\frac{5}{3\sqrt{3}} \end{align*} OK I really didn't know how to do this in desmos

Hello! I am reading how to integrate on an orientable manifold. So we have ##f:M \to R## and an m-form (m is the dimension of M): ##\omega = h(p)dx^1 \wedge ... \wedge dx^m##, where ##h(p)## is another function on the manifold which is always positive as the manifold is orientable. The way...

Hi. I would like a source code to integrate the ikonal equation. I would like to compute the ray path. Of course I am able to compute the phase refractive index n(x,y). Cartesian system is preferred. Can anybody give me a suggestion? Bye, Carlos

Hello all! I usually don't like to ask for help... But this is the first week of courses and I'm already stumped on a homework question... 1. Homework Statement So the question states: Find the work by the force F = x^i + xy^j. If the object starts from the origin (0,0), moves along the...

$\textsf{Reverse the order of integration in the following integral }$ \begin{align*}\displaystyle I&=\int_0^1 \int_2^{2e^x}f(x,y) dydx \end{align*} ok tried to follow some examples but 😰 \begin{align*}\displaystyle I&=\int_0^1 \int_?^{?}f(x,y) dxdy \end{align*}

solve by reversing the reversing the order of integration this was given: \begin{align*}\displaystyle I&=\int_0^8 \int_{\sqrt[3]{x}}^2 \left[\frac{x}{y^7+1}\right]dy \, dx\\ \end{align*} ok I put this in an dbl int calculor but it turned the order around to \begin{align*}\displaystyle...

Hi, Simple question, sort of: I see that according to the internet the mathematical description of a triangular wave is rather complex, so I'll try to stay as far away from that as I can, because I'm a bit rusty. I understand that if you integrate a square wave you get a triangular wave on the...

Homework Statement A CD case slides along a floor in the positive direction of an x-axis while an applied force Fa acts on the case. The force is directed along the x-axis and has the x component Fax = 7.0x – 2.0x^2, with x in meters and Fax in Newtons. The case starts at rest at the position x...

<Moderator's note: Moved from a technical forum and thus no template.> Calculate flux of the vector field $$F=(-y, x, z^2)$$ through the tetraeder $$T(ABCD)$$ with the corner points $$A= (\frac{3}{2}, 0, 0), B= (0, \frac{\sqrt 3}{2},0), C = (0, -\frac{\sqrt 3}{2},0), D = (\frac{1}{2},0 , \sqrt...

I would like to solve an equation below using MATLAB: All the parameters except p are known, so I only need to solve for p. However since I need to consider the sign of the integrand and there is an absolute value sign in it I don't know how to solve it. Could anyone please help? Thank you.

I can obviously do the chain rule and see how the final expression of the derivative is related to the original function but I can't seem to figure out the substitution Rule as an intuitive way of solving the indefinite integral of functions... bear with me if I'm too verbose, I've attached an...

$\tiny{w.8.8.19}$ $\textsf{Integration of $2$ to $\infty$}$ $$\displaystyle I_{19} = \int_{2}^{\infty} \frac{\cos\left({\pi/x}\right)}{{x}^{2}}\,dx $$ $\textit{Not sure how to break this down but it appears the denominator will advance faster than the numerator}$

Use integration methods to establish the formula A = π r^2 for the area of a disc ofradius r. so the equation of the circle is x^2 +y^2 =r^2 . i will try and find the area of one quadrant using integration. so it will be ∫ r,0 y dx so ∫ r,0 √(r^2 -x^2) dx so from here i am trying to integrate...

I'll cut the long story short. What on Earth happened here: I seem to be unable to do the integration by parts of the first term. I end up with a lot of dx's.

Hello, In your opinion, why the integration with dx in the denominator is undefined ?? (as in the attached picture)

A force F = -K(yi + xj) (K is a positive constant) acts on a particle moving in the x-y plane. Starting from the origin, the particle is taken along the positive x-axis to the point (a, 0) and then parallel to the y-axis to the point (a, a). What is the total work done by the force F on the...

Homework Statement ##\displaystyle \int \frac{\log (x)}{x}~ dx## Homework EquationsThe Attempt at a Solution I am a little confused about the first part. We know that the ##\displaystyle \int \frac{1}{x}~ dx = \log |x|##. So how can we proceed with integration by parts if one of the logs has...

Homework Statement If ##a>0## and ##b≠0##, solve the following stating the maximal domain for which the solution is valid: ##\sqrt{a^2-x^2}\cdot \frac{dy}{dx}+b=0,\ y\left(0\right)=0## Homework Equations [/B] ##\int _{ }^{ }\frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \left(\frac{x}{a}\right)+c,\ a>0##...

Homework Statement I am struggling with solving this integral that seems to look easy on the surface but integration isn't my strong suit and so I'm not 100% sure on how to go about solving this: ∫ (120(e-15.24t - e-39984.75t))2dt where the integration is from 0 to 0.3 Homework EquationsThe...

Homework Statement Simplify $$\int_{-1}^1\left( (1-x^2)P_i''-2xP'_i+2P_i\right)P_j\,dx$$ where ##P_i## is the ##i^{th}## Legendre Polynomial, a function of ##x##. Homework Equations The Attempt at a Solution Integration by parts is likely useful?? Also I know the Legendre Polynomials are...

I am beginning to learn about differential equations and I saw in an explanatory video a solution to this separable differential equation: ##\frac {dy} {dx} = \frac {-x} {y{e^{x^2}}}## from there through simple steps the equation changed to ##y⋅dy=-xe^{-x^2}⋅dx ##. Then the video did an...

How is this done? My textbook only specifies integrating polar graphs with respect to theta.

Really excited about this blog post by yours truly. Wineman Technology is going places!

Homework Statement I need to integrate this expression : P(k, w) = A * δ(w-k*v) * f(k, w) A is constant and δ, Dirac Delta.Homework Equations [/B] There is double integration : I = ∫0∞ dk ∫0∞ P(k,w) dw = A ∫∫0∞ δ(w-k*v) * f(k, w) dw dk The Attempt at a Solution [/B] I'm confused with...

Hey everyone, first, let me say I understand the complement rule. Where I am confused is over the integration. My professor said that suppose you have a continuous cumulative distribution function F(x) = 1-e-x/10, if x > 0 (0, otherwise). And suppose you want to find P(X>12) you can use the...

I have a question about integration and I hope I can word it correctly. When we integrate over something (i.e. a line, a surface, or volume), what exactly do I integrate over? Is it the domain of that "thing"? I feel like it's something I've been told but I just can't remember. If this is the...

Hi everybody, I'm trying to calculate this: $$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$ where ##P_{l}## are the Legendre polynomials, ##\Omega## is the surface of a sphere of radius ##R##, and $$ \cos{\gamma} = \cos{\theta'}...

Homework Statement Integrate: (sinx + cosx)/sqrt(1+sin2x) Homework Equations Simple trigo formulae: cos2x=cos^2(x)--sin^2(x) sin^2(x)+cos^2(x)=1 sin2x=2.sinx.cosx The Attempt at a Solution I tried to rationalize the given term,multiplying both numerator and denominator with: 1st time:cosx-sinx...

Homework Statement Integrate: 4cos(x/2).cos(x).sin(21x/2) Homework Equations ▪sin(A+B)=sinAcosB+sinBcosA ▪2sinAcosA=sin2A And obviously, ▪Integration of sinx is (-cosx) ▪Integration of cosx is (sinx) The Attempt at a Solution ○I multiplied the numerator and denominator with sin(x/2) ○The...

Homework Statement Integrate: [(x^4+x^2+1)/2(1+x^2)]dx Homework Equations [/B] Integral of x^n=x^(n+1)/(n+1) Integral of 1/(1+x^2)=arctanx The Attempt at a Solution I have attached my solution.All the steps seem to be correct,but the answer isn't matching,don't know why.

Homework Statement Solve the following integral: [(x^2+3)/(x^8+x^6)] dx [/B] Homework Equations The question has also said to integrate by substitution(though other methods are welcome) That would mean substituting an expression in x with a variable,say, 't' such that the integral comes of...

Hello, I want to integrate this expression : ∫ (x5 + ax4 + bx3 + cx2 + dx)-1 between xmin>0 and xmax>0 a is positive but b, c and d can be positive or negative. I have no idea to integrate this expression... Do you have methods to do this ? Thanks in advance !