07081300/Class notes for Tuesday, September 11

Contents 
In Small Scales, Everything's Linear
Code in Mathematica:
QuiltPlot[{f_,g_}, {x_, xmin_, xmax_, nx_}, {y_, ymin_, ymax_, ny_}] := Module[ {dx, dy, grid, ix, iy}, SeedRandom[1]; dx=(xmaxxmin)/nx; dy=(ymaxymin)/ny; grid = Table[ {x > xmin+ix*dx, y > ymin+iy*dy}, {ix, 0, nx}, {iy, 0, ny} ]; grid = Map[({f, g} /. #)&, grid, {2}]; Show[ Graphics[Table[ { RGBColor[Random[], Random[], Random[]], Polygon[{ grid[[ix, iy]], grid[[ix+1, iy]], grid[[ix+1, iy+1]], grid[[ix, iy+1]] }] }, {ix, nx}, {iy, ny} ]], Frame > True ] ] QuiltPlot[{x, y}, {x, 10, 10, 8}, {y, 5, 10, 8}] QuiltPlot[{x^2y^2, 2*x*y}, {x, 10, 10, 8}, {y, 5, 10, 8}]
See also 06240/Linear Algebra  Why We Care.
Class Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
Differentiability
Let , and be two normed finite dimensional vector spaces and let be a function defined on a neighborhood of the point .
Definition:
We say that is differentiable (diffable) at if there is a linear map so that
In this case we will say that is a differential of at and will denote it by .
Theorem
If and are diffable maps then the following assertions hold:
 is unique.
 If is linear then
 For every scalar number it holds
Implicit Function Theorem
Example Although does not define as a function of , in a neighborhood of we can define so that . Furthermore, is differentiable with differential . This is a motivation for the following theorem.
Notation
If then given we will define by
Definition
will be the class of all functions defined on with continuous partial derivatives up to order
Theorem(Implicit function theorem)
Let be a function defined on a neighborhood of the point and such that and suppose that is nonsingular then, the following results holds:
There is an open neighborhood of , , and a diffable function such that and for every .