0708-1300/Class notes for Tuesday, September 11

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In Small Scales, Everything's Linear

06-240-QuiltBeforeMap.png 06-240-QuiltAfterMap.png

Code in Mathematica:

QuiltPlot[{f_,g_}, {x_, xmin_, xmax_, nx_}, {y_, ymin_, ymax_, ny_}] :=
  {dx, dy, grid, ix, iy},
  grid = Table[
    {x -> xmin+ix*dx, y -> ymin+iy*dy},
    {ix, 0, nx}, {iy, 0, ny}
  grid = Map[({f, g} /. #)&, grid, {2}];
        RGBColor[Random[], Random[], Random[]],
          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^2-y^2, 2*x*y}, {x, -10, 10, 8}, {y, 5, 10, 8}]

See also 06-240/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.


Let , and be two normed finite dimensional vector spaces and let be a function defined on a neighborhood of the point .


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 .


If and are diffable maps then the following assertions hold:

  1. is unique.
  2. If is linear then
  3. 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.


If then given we will define by


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 non-singular then, the following results holds:

There is an open neighborhood of , , and a diffable function such that and for every .