/sci/ - Science & Math

[Anonymous No.6898913]

Explain Jacobians.
I read about a Jacobian matrix in some research paper I'm trying to understand.
The term "Jacobian matrices of the mappings that connect different spaces" came up in the abstract.

First glance it has something to do with differential equations an matrix transformations. Though I'm sure you guys have a better explanation.
Have at it chumps, Thanks in advance.

[Anonymous No.6898967]

>>6898913
http://en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi

http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

I will explain, iff you post some cool pics

[Anonymous No.6898985]

Will do

[Anonymous No.6898988]

Southpole/?

[Anonymous No.6898998]

Yellowstone/?

[Anonymous No.6899022]

Margot Gerritson makes interesting art using algebra, quasi-newtonian PE reduction models, and interesting data-sets.
Here is a visual model of tidal flow simulations.

[Anonymous No.6899024]

>>6899022
Here is her work on shallow water flow

[Anonymous No.6899030]

>>6899024
Here is her model of the library of congress.

Her model treats nodes as repelling point charges, as well as treat relations between nodes as simple springs.

Her models "wiggle" into a reduced potential energy state.

[Anonymous No.6899048]

>>6899030
Financial portfolio optimization

[Anonymous No.6899050]

>>6898985
>>6898988
>>6898998
Okay, decent....

Jacobians are used for coordinate transforms.

Example:
Say I have a function in Cartesian coordinates, f(x,y). But, I want to transform it into polar coordinates. So I need to know x(r,theta) and y(r, theta) to get f( x(r,theta), y(r,theta)) = f (r, theta). Simple enough, right?

But what if for some godforsaken reason I have dx and dy? how the fuck would I get dx(dr,dtheta) and dy(dr,dtheta)? You may say "when the hell would you get that?". Well, you get it when you are transforming integrals. Integrals have dx and dy and shit, remember?

So when I have integral( f(x,y) dx dy) and need to change coordinate systems, I don't just change to f(r, theta), I also change to dr and dtheta. The Jacobian is just a matrix made up of the transformations between the various dx,dy,dr,dtheta, that way you don't continually have to derive them.

.....

[Anonymous No.6899057]

>>6899048
Here is an un-mapped Comanche helicopter mesh.

[Anonymous No.6899060]

>>6899057
Here is the stable 2d model of that mesh.

[Anonymous No.6899078]

>>6899050
....continued

So for x,y ---> r, theta
Our Jacobian is 2x2 (order is kinda irrelevant)

It has four entries by defintion
dx/dr = something1
dy/dr = something2
dx/dtheta = something3
dy/dtheta = something4

We derive the "something's" from the x(r,theta) and y(r,theta) functions

if
x = r cos theta
y = r sin theta
then
dx/dr = cos theta
dy/dr = sin theta
dx/dtheta = -r sin theta
dy/dtheta = r cos theta
Those would be the values in our Jacobian matrix

So the integral transform is then
Integral( f(x,y) dx dy ) = Integral ( f(r, theta) r dr dtheta ). Notice how that "r" just appeared!

This is because dx dy = r dr dtheta!. We get this from the jacobian and the rules of differentiation.

We know that dx = r* dx/dr + theta * dx/dtheta
And dy = r* dy/dr + theta * dy/dtheta
And we have all the d../d.. form our jacobian!

[Anonymous No.6899092]

>>6899078
I hope that explanation made sense?

To recap: The Jacobian is the matrix that contains the transforms between the partial derivatives of one set of coordinates and another set of coordinates.

The entries are always of the form d_original_coordinate/d_new_coordinate (these are partial derivatives). Particular expressions for these entries depend on your coordinate systems. They are found using basic calculus and the functions original_coordinate(new_coordinate).

Once we have the jacobian matrix (all the partial derivatives transforms), we can get the differential/total derivative (the kind used in integrals), by it's definition and the entries in the jacobian
http://en.wikipedia.org/wiki/Differential_%28infinitesimal%29

[Anonymous No.6899097]

>>6899078
>dx = dr* dx/dr + dtheta * dx/dtheta
>dy = dr* dy/dr + dtheta * dy/dtheta

Fixed: Where the solitary dx,dy,dr,dtheta are the "differentials", and the d../d.. are the partial derivatives.

[Anonymous No.6899107]

>>6898913
Fuck Jacobians
>The Jacobian rebels have risen!

[Anonymous No.6900860]

Almost all *my* work in the field of Computational Physics at the moment involves algorithms which heavily employ the determination of Jacobian Derivative matrices for systems of coupled non-linear equations.

Have a picture from my previous project to whet your appetite. I'm currently working on something in 3D, which I can't elaborate on at all before completion and publication.