If you're having trouble with the parabolas maybe try rotating the barrel 90 degrees? So you'll have more familiar y=ax^2+bx+c parabolas? So you have one parabolas that have a vertex at (0,-30) and two points at (45,-25) and (-45,-25) and another with the same points reflected across the x axis.

Do you know how to solve for a parabola given three points? It's just like solving for a line given two points.

Nope, not great in that area. One of my "leaks" in math
I'll look up for it.

First find the formula for the parabola with the apex in the origin (what does that mean for the formula?). Then shift it such that the barrel's axis of symmetry is on the x-axis. Finally use the formula for the volume of solids of rotation.

What's going on with part a)? Isn't it 1/4 of a circle with radius 2, hence giving it an area of π, but negative because it's below the x-axis?

you that should be correct, maybe it doesn't accept the symbol π and you should write pi or Pi or something?

y = ax^2. This is what you mean?
I can't figure this out. I seem to get that a = 1/405 in this case. Which would be true, since 1/405 * 45^2 = 5.
By "shifting axis of symmetry" do you mean y = ax^2 and x = ay^2?

at x=45 you want y to be 5. So a=1/405, yes. If you flip the barrel 90 degrees then put the origin at the center of the barrel, the parabola y=1/405x^2 is parallel with the bottom parabola. It needs to be shifted down by 30 cm. so, y=1/405x^2 - 30 is the parabola for the bottom of the barrel.

Lower case pi maybe. Upper case means product. It's pi in wolframalpha.

When calculating the volume of the barrel, are we only interested in the volume that the parabola produces? Since we could calculate the cylinders volume and then add the volume of parabola to it.
But well, i still haven't finished the task. I made a picture here. Since we are asked for a volume, is it good to use dm?
So first we should start by finding the formula for the parabola? The peak of the parabola is at (0,3). I read the book and it had examples where peak was at (0,0) and it suggested to use y = ax^2. Is it different in this case?
Since the parabola is this shape, it should have negative a.

If one told you what r(x) is (radius as function of x from -4.5 to 4.5) would you be able to find the volume of a barrel with height H=9m and min max radii 2.5, 3 m (or is it dm=0.1m that looks too small for a barrel but then again 2.5, 3m is too large as well)?

So find r(x) which is the y value of a parabola that is y=ax^2+c (no linear x term because we took the axis that way to be symmetric) if we know that it satisfies y(0)=3 and y(4.5)=2.5

(a general parabola is y=ax^2+bx+c but if it is symmetric with respect to y axis ie y(x)=y(-x) that forces b=0)


In any case if you do the calculation in dm expressing all lengths in these units the volume will be in dm^3 in the end that is 1/1000 of m^3.

You can check that your answer is between pi*2.5^2*9 and pi*3^2*9 dm^3. If you use dm like that, then remember to multiply the final volume by 1000 to get cm^3.

The answer behind the book is given in liters.
So which one of these should i use? I think the bottom one is correct. Since we calculate total volume of the barrel, we don't want to add same values twice, that already belong to the cylinder value?
The equation for volume of cylinder in this case would be pi * 2.5dm^2 * 9dm, which would equal 176.715. We need to add the volume of parabola part to this?

You want the top one so you can integrate pi*y^2 dx where y is the whole radius. You don't want to just do the top part and add the cylinder separately.

If you want liters, then just leave it in dm^3 since that's what a liter is.

Anyone here good with empirical statistics? I'm trying to run copula functions in stata and having a hard time understanding what the hell is going on.

Anyone knows some differential geometry? I need to find the squared line element ds^2 under a coordinate transformation such that y[i]=x[i]/r^2 (in 3 dimensions) where r is the radial distance.

The answer is

You were probably supposed to express x[i] via y[i] - that's easy:

- then write dx[i] as

and tediously calculate the partial differentials using the fraction and chain rules, and then assemble terms in dx[i]^2

But there's an easier way using the specific property of this transformation (called a spherical inversion, btw), namely that, for any vector y, the origin (0), y and x(y) lie on the same line (where x(y) is the image of y under the inverse transformation), thus, for any vectors y and dy, the five points 0, y, x(y), y+dy and x(y+dy) lie in the same plane. (Let's denote these points as O, A, B, C, D respectively, so A is mapped into B and C - into D.)

Having observed that, also note that the triangles AOB and DOC are similar by side-angle-side, and so BD = AC * OB / OC = |dy| * |x(y)| / |y+dy| = |dy| / (|y| * |y+dy|) (as |x(y) = 1/|y|; |.| denotes lengths of vectors here). Now find the limit of |x(y+dy)-x(y)|/|dy| as |dy|->0: |x(y+dy)-x(y)|/|dy| = 1/ (|y| * |y+dy|) -> 1/|y|^2.

[Curiously, in this plane, the transformation acts like a circle inversion; in particular, the above triangle similarity proves that it maps any straight line not passing through the origin into a circle passing through the origin.]

Here's yet another proof using inner products of vectors.

Let y and y+dy denote vectors like above and |.| denote lengths, x(y)=y/|y|^2, x(y+dy)=(y+dy)/|y+dy|^2. Then



Edit: I understand that my specific proofs won't teach you the general method of rewriting metrics in new coordinates, but you haven't told about the exact step where your solution attempts ran into problems. Is it the general relation between the metrics in the old and new coordinates (see Wiki: Metric Tensor) then, or is it calculation of the partial derivatives (separately for i=j and i=/=j) in the expression of dx[i] in terms of dy[j], or assembling terms in ds^2 = \sum_{i} dx[i]^2 (grouping terms with dy[k]dy[j] with the same k and j), which requires just some patience and 'calculational literacy' but no bright ideas? It's hard to tell as I don't know specifics of your course of diff. geom., how deep you are into the course and even notations used there.

First of all thx a bunch! However I'm having some difficulties understanding how you express x[i] via y[i]. Why is x[i]=y[i]/r^2 and not x[i]=y[i]*r^2? As far as notation goes I think we can express the radial distance squared as kroneckerdelta[jl]*x[j]*x[l].

oooh I see!! That makes so much sense! Thx again! I'll probably come back later on complaining about the computations!

Trying to connect the dots between 4 topics here: Signal, fourier series, spectrum and bandwidth. All pretty important things, so i try to figure these out before getting to next topics.
One of the questions we might have in our telecom. exam is to define Fourier series. Not on a mathematical level really. We made an experiment with excel where we summed up a lot of waves. So that's the formula we used, no integrals or anything more complex. We will have fourier transforms etc. next year.
From: http://en.wikipedia.org/wiki/Frequency_spectrum
"The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency."
I wonder how these are connected?

A Fourier series is for periodic signals. When you compute the Fourier series coefficients for a periodic signal, you are computing the contributions for the discrete sine wave frequencies contained in that signal. The Fourier transform of that periodic signal will consist of individual lines, called a line spectrum, though the word spectrum usually refers to power which is the magnitude squared of the Fourier transform (the Fourier transform generally consists of complex numbers which represent magnitude and phase). The lines are actually mathematically represented by impulses. They go off to infinity at a single frequency, but they have a finite area so you can integrate over them to get energy and power. In real life there are no real impulses, only high narrow things that approximate them. Now if your signal was not periodic, you can still compute the Fourier transform instead of the Fourier series, and you get a continuous spectrum, not discrete lines. A Fourier series is a sum, while a Fourier transform is an integral. So periodic signals always have a line spectrum, while aperiodic signals have a continuous spectrum.

Finite duration signals have a spectrum that requires frequencies all the way out to infinity, though the contributions generally become small as the frequencies become very large. It follows then that any signal that only occupies a finite range of frequencies must have infinite extent in time. A sine wave or finite sum of sine waves only has frequency content over a finite range of frequencies, so they must exist for all times.

It's important to be able to think in both domains simultaneously and know what signals look like in each domain, and what effect things have in each domain. That's typically done in a signals and systems class taken before a digital telecom class.

I need more help with differential equations.
the line element of a 2-sphere is given by ds^2=d(theta)^2+sin^2 (theta) d(phi)^2.
I need to formulate the variational principle that extermizes the square length of the tangent vector (rather than just the length of the tangent vector).

And, I need to find the geodesics.

If you don't know how to do it in general, read the Wiki here and here (and here to learn how the Euler-Lagrange equations for the variational principle are reduced to the ones with the Christoffel symbols). Then try to solve the equations on your own.

If you fail, don't worry - this exercise has been done zillions of times and many cheat sheets, like this, can be readily found on the Internet.

thx!