Analysis and Calculus

I was wondering why/how the following statement is true. [math] \int_a^b f(x)\quad dx = \frac{b-a}{2} \int_{-1}^1 f \left( \frac{b-a}{2} x + \frac{a+b}{2} \right) [/math] I have come so far as to insert 1 and -1 instead of the x, and subtracting lower limit from upper limit. But that gives me [math] \frac{b-a}{2} F(b) - F(a) [/math] which has a strange constant in front of it. I have tried to interpret this constant as some sort of a mean-value of interval. I don't know why it's there or what is means. Can somebody shed some light on this matter? Thanks.

Hi, I am trying hard to solve the screened poisson equation in 2D for u(x,y): \nabla^2 u(x,y) - u(x,y) =f(x,y) with f(x,y)=exp(-(x^2+y^2)) and u(x,y)=0 for x,y -> infinity. Could anybody know what is the analytical solution of this equation? I would really thank who could give me any clue about it.

suppose f is a continues function on point x_0 prove that g(x)=(x-x_0)*f(x) differentiable on x_0?? calculate g'(x_0) i tried to think like this: if f(x) is continues on x_0 then lim f(x) as x->x_0 equals f(x_0) mvt says f'©=[f(a)-f(b)] cauchys mvt says f'©/g'©=[f(a)-f(b)]/[g(a)-g(b)] ??

Generally the function having zero points or poles with non-integer order such as f(z) = (z-a)^(1.5+i0.3) must be dealt with on appropriate Riemann surface. I tried to extend the argument principle for such functions on a single sheet of Riemann surface and got a formula similar to that of ordinary argument principle. Using that formula the winding number of f(z) = (z-a)^(1.5+i0.3) around the origin is expressed as 1.5+i0.3. For details, visit: http://hecoaustralia.fortunecity.com/argument/argument.htm

I have been trying to teach myself calculus and I came upon this: [math]\frac{d}{dx}e^{x}=e^{x}[/math] I have been struggling to understand why this is. From my basic knowledge I have got this far. [math]\frac{d}{dx}e^{x}=\frac{e^{x+h}-e^{x}}{(x+h)-x}[/math] [math]\frac{d}{dx}e^{x}=\stackrel{Lim}h{\rightarrow}0\frac{e^{x+h}-e^{x}}{(x+h)-x}=\frac{e^{x}-e^{x}}{(x)-x}=\frac{0}{0}[/math] So I keep coming up with this intermediate form and I cannot figure out how to get rid of it. I might be doing this completely wrong so could some one please point me in the right direction.

In the shower this morning, I suddenly got the idea of calculating the surface area of a sphere as an integral of circumferences. My approach is as follows: The integral is [math] 2\cdot \int_0^r 2\pi t dt [/math] where the "2" in front accounts for the surface area of whole sphere (equivalent to having [math]2r[/math] as upper limit in the integral - by symmetry). The result is then [math]2\cdot 2\pi [\frac{1}{2}t^2]_0^r[/math] giving [math]4\pi \frac{1}{2}r^2[/math] or [math]2\pi r^2[/math] which is wrong. Where did I go wrong?

Hi, I start with the continuity equation, for a CFD application. ∂u/∂x+∂v/∂y=0 (1) I’m interested in substituting the following (2)(3)(4)(5) into the above equation (1) to produce a non-dimensional continuity equation. u = (UKRe/ρR) (2) v = ((UK〖Re〗^(1⁄2))/ρR) (3) x=ϕR (4) y=ηδ (5) which produces; ∂/∂(ϕR) (UKRe/ρR)+∂/∂(ηδ) ((UK〖Re〗^(1⁄2))/ρR)=0 (6) ∂/∂ϕ (UKRe/ρR).∂/∂R (UKRe/ρR)+∂/∂η ((UK〖Re〗^(1⁄2))/ρR).∂/∂δ ((UK〖Re〗^(1⁄2))/ρR)=0 (7) My question is; when I do the substitution for x & y, and manipulating equation (6) would I yiel…

Hey there, First of all, I'm completely new to calculus in any form, so please excuse any completely obvious errors if there are any. I was just wondering the other day about the derivative of a Power Tower. I tried to treat it as [math]ax^n[/math], and came up with [math]f(a, b) = a \uparrow \uparrow b[/math] [math]f'(a, b) = (a \uparrow \uparrow b) a^{(a \uparrow \uparrow [b-1]) -1}[/math] Is this correct? Cheers, Gabe

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

How to compute the solution of a partial differential equation problem of the type : a) [math] \frac{\partial^2 f}{\partial x^2}(x,t)=D\frac{\partial f}{\partial t}(x,t)[/math] D is a non vanishing constant and b) an initial condition, for example [math] f(x,t=0)=\delta(x)[/math] c) a boundary condition [math]f(at,t)=f(-at,t)[/math] with 'a' nonzero it can be seen that * separation of variables does not work because of the time-dependence of the boundary * pseudo spectral method fails due to the non-orthogonality of the eigenvectors of the space-operator and their time derivatives. Is there another way to solve that kind of differe…

I'm using Newton's method to find the zero for this problem [math] \left( x^3 + 2x^2 - 4)^{\frac{1}{3}} [/math] Meh, I can't use latex. The Cube root of X^3 + 2x^2 - 4 Anyways, I've ran it through a few times with my initial guess being 1. The first itme I ran it through i got 2 as my answer, and then the second time my answer jumped over the actual root (which is 1.18 or something) to be .28942.... i checked my work and it's correct. Why would it do that?

I am having problems with understanding how the formula can have a geometrical interpretation by having an imaginary axis perpendicular to the real axis. I mean how can an exponent (of e) multiplied by the imaginary unit be related to an angle?

Hello all I'm new to these forums and hoping for a little help. I'm a software engineer and CNC machinist with a passion for electronics and physics. Currently I studying Maxwell's original treatise on Electromagnetism. I've been working on creating computer models based on his original work, but I'm having problems getting my head around the partial differentiations. Since I've been teaching myself the calculus used in his formulas, first I guess I would like to verify I'm on the right track. First I understand the Hamilton operator as follows: [math]H = -\nabla V[/math] Given the vector/quaternion position P, using the previous formula would solve…

Could anyone explain how this 3 * integral 1 to infinty ( rho^2 ( 1 - exp [ 1/(x * rho^6) ] ) ) d rho becomes this 3 * Sum from n=1 to infinity of ( x^-n / n!(6n-3) ) I'm a litle foggy when it comes to integral evaluation through series expansion I do understand that exp [ 1/x ] gives the series expansion 1 + x/2! + x^2/3!... how did they get the (6n-3) term

Hey guys, I am asking a question about my homework, but my point is to try and get general explanation about the method (since I am quite lost here, and the book isn't very helpful). Which is why I'm posting it here and not in the HW help section. I am using my hw as an example only; if I understand the point I will (hopefully) be able to solve this myself. Okay, then. I started a new advanced physics course (2, actually, expect questions about the math of the other one soon) and there's a lot of math that gets me quite confused. I am familiar with the general principles, but I think that somewhere I'm getting myself confused over the terms and permutations. Help.…

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

First, I have never really liked multi-index notation. I wish to find a neat expression for [math]\partial_{\mathbf{A}}(f(x))^{l} = \frac{\partial^{|\mathbf{A}|}}{\partial x^{A_{1}}\cdots \partial x^{A_{n}}} (f(x))^{l}[/math]. Here [math]\mathbf{A}[/math] is a multi-index, but [math]l[/math] is an exponent. One should be able to you the Leibniz rule recursively to get a nice expression. (But note that [math]\partial_{\mathbf{A}}[/math] is not a derivation). Has anyone seen the answer to this anywhere? I am sure it has been calculated before now. Can anyone save me the time and effort in doing it myself? I expect it is in a book on pseudo-differentia…

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesda…

I have used this way of approximating the integral of a function. It is pretty straight forward as it incorporates the sum of a product of the function to be integrated and a weight function. The weight function is a Legendre polynomial which is a solution to Legendre's differential equation. My problem is I do no understand how these solutions are connected in approximating the integral of a function? Anyone can use Gaussian quadrature once the solutions to the L. polynomials have been calculated, but understanding how Gauss saw a connection from a differential equation to a (very nice way) of approximating an integral is not apparent. I took an introduction…

Hello, my name is Calleigh and i am new to the forum! I am in Calculus II and have a few questions on some problems. I am using the textbook Calculus 8th edition by Larson, Hostetler and Edwards. Could someone please help me? The problem is on pg 922 in chapter 13.5 in the text, number 32. It reads: A triangle is measured and two adjacent sides are found to be 3 inches and 4 inches long, with an included angle of pi/4. The possible errors in measurement are 1/16 inch for the sides and .02 radian for the angle. Approximate the maximum possible error in the computation of the area. I haven't had any problems like this in class, so i don't know what to do. My pro…