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[da_nang]

​Does anything worth writing up happen when you start with a 3D vector field Field[x, y, z] and calculate the curl of curlField[x, y, z] which is given by
curlcurlField[x, y, z] = ∇⨯(∇⨯Field[x, y, z])?​

Other than ∇⨯(∇⨯F) = ∇(∇·F) - ∇²F ? Nothing that comes to mind right now. Nothing special pertaining to that particular calculation, anyway.

[DreamThorn]

My math skills... are quite rusty.  Especially integration, of course, but in this case it's simply summation of series.  I haven't worked with series much in my life, and now I'm a bit stuck.

Maybe one of you can guide me a bit.

b_n = sum(k_m × a_n+m, for all m)

I have all the values of b_i and of k_i, but need to find all the values of a_i.

Any ideas?

EDIT: Clarified some things.

[Sergarr]

Without n, probably no.

[DreamThorn]

I mean: I have to get a_n for all n.

I will edit my post for clarity.

[da_nang]

My math skills... are quite rusty.  Especially integration, of course, but in this case it's simply summation of series.  I haven't worked with series much in my life, and now I'm a bit stuck.

Maybe one of you can guide me a bit.

b_n = sum(k_m × a_n+m, for all m)

I have all the values of b_i and of k_i, but need to find all the values of a_i.

Any ideas?

EDIT: Clarified some things.

It looks like a variant of a linear inverse problem. Problematic of course if any of the ranges of the indices are infinite, but solvable for finite ranges. (Assuming n and m are indices.)

b = Ka, where b and a are column vectors. Let k be a row vector and dim(K) = NxN.

Let 1 <= m <= M and 1 <= n <= N.
If N > M, then K will be a band matrix with the first row featuring k followed by zeroes and for each succeeding row, k is moved to the right.
If N<=M then K will be a upper right triangular matrix, with the first row featuring k (truncated at the Nth element) and moves similarly to the right for each succeeding row.

Either way, the naive solution is a = K^-1b. However, since inverse problems are typically ill-conditioned, you'd probably need to use a truncated singular value decomposition or other method for handling ill-conditioned matrices such as CGLS.

[DreamThorn]

Thanks da_nang!

Using a matrix is a very good idea.

I'll look into that CGLS and truncated singular value decomposition you mentioned.

[Skyrunner]

The area of a circular sector is r^2*theta*pi (theta is the central angle), right? Would a *b*theta have any meaning? Imagine something similar to a sector, but with uneven sides.

[da_nang]

The area of a circular sector is r^2*theta*pi (theta is the central angle), right? Would a *b*theta have any meaning? Imagine something similar to a sector, but with uneven sides.

Eh, I believe it's ½*θ*r², when θ is in radians ((θ/360º)*π*r², when in degrees). This is due to the symmetry of the circle so that the area of a circular sector is just a fraction of the area of the circle. That fraction is determined by the central angle.

Incidentally, ½*θ*r² = ½*b*r, where b is the arc length, again due to symmetry.

As for the second question, you'd need to specify what a and b are.

[Pnx]

So quick question, does anyone know of any cool uses of polar coordinates in nature, or the world in general? It's for an optional paper I'm thinking of writing.

[da_nang]

Navigation mostly. But I almost always see modified or generalized variants.

[Arx]

Objects in orbit around anything.

[palsch]

Any spherical or circular systems that need to be described mathematically.

Other than the two stated, the first to pop to mind is chemistry or quantum mechanical descriptions of atoms. Atomic orbitals (or probability distributions) are described in polar terms.

When working with electromagnetism you tend to use polar coordinates based on the shape of the system. A lot of the time that means cylindrical polar coordinates. Examples would range from a simple electrical wire (with fields radiating out) to the various fields within a fusion Tokamak.

[Pnx]

Navigation mostly. But I almost always see modified or generalized variants.

This could be what I go with. I was honestly considering talking about WWII technology. I was looking at a LP of Silent Hunter 3, and thinking about the whole range finding, angle, and speed system for finding torpedo firing solutions. Bomb sights, the battle of the beams, and various aspects of naval warfare all seem like they could be interesting problems to talk about, in fact in general I think the whole problem of "How do I get this moving object to collide with this other moving object" is an interesting problem mathematically.

Any spherical or circular systems that need to be described mathematically.

Other than the two stated, the first to pop to mind is chemistry or quantum mechanical descriptions of atoms. Atomic orbitals (or probability distributions) are described in polar terms.

When working with electromagnetism you tend to use polar coordinates based on the shape of the system. A lot of the time that means cylindrical polar coordinates. Examples would range from a simple electrical wire (with fields radiating out) to the various fields within a fusion Tokamak.

This is also extremely interesting, I may have to read up on this.

[da_nang]

Navigation mostly. But I almost always see modified or generalized variants.

This could be what I go with. I was honestly considering talking about WWII technology. I was looking at a LP of Silent Hunter 3, and thinking about the whole range finding, angle, and speed system for finding torpedo firing solutions. Bomb sights, the battle of the beams, and various aspects of naval warfare all seem like they could be interesting problems to talk about, in fact in general I think the whole problem of "How do I get this moving object to collide with this other moving object" is an interesting problem mathematically.

Well according to Silent Hunter, all you need is position, Angle-on-Bow (AoB) and speed, with the sub in a (near)-stationary position. Basically, course and velocity relative to the sub. But there's stuff hidden under the hood, such as gyro angles, torpedo speed, running depth and the most cursed problem of all: faulty torpedoes. That Mk-14 torpedo fiasco is legendary.

And based on my experience with that series, the range and AoB needs to be damn accurate to get a good firing solution. Especially range as it's used to find the speed and thus compounding errors.

Range-finding in that game can either be based on similar triangles or trigonometry.
Trigonometric case:
d = h/tan(a)

Similar triangles:
d = h*b/c

Where d is the distance, h is a reference height of a ship met from the waterline, a is the angle measured, b is the lower catheter of the smaller triangle and c is the measured catheter.

Either way, the smaller the measured quantity is, the greater the error. This is then used to calculate the speed as well.

[GrizzlyAdamz]

"To celebrate a colleague's graduation, the 'm' coworkers in an office agreed to contribute equally to a catered lunch that costs a total of 'y' dollars. If 'p' of the coworkers fail to contribute, which of the following represents the additional amount, in dollars, that each of the remaining coworkers must contribute to pay for the lunch?"

The correct answer is
(py) / (m(m-p))

I'd like to know how to get to that.