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_}] :=
Module[
  {dx, dy, grid, ix, iy},
  SeedRandom[1];
  dx=(xmax-xmin)/nx;
  dy=(ymax-ymin)/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^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

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) if there is a linear map so that

In this case we will say that is a differential of and will denote it by .

Theorem

If and are diffable maps then the following asertions holds:

1) is unique.

2)

3)If is linear then

4)

5)For every scalar number it holds

Implicit Function Theorem

Example Although does not defines 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 f:X\times Y\rightarrow Z then given x\in X we will define f_{[x]}:Y\rightarrow Z by f_{[x]}(y)=f(x;y)

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 non-singular then, the following results holds:

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