0708-1300/Class notes for Tuesday, September 11: Difference between revisions

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'''Def:''' We say that <math>f</math> is differentiable (''diffable'') if there is a linear map <math>L</math> so that
'''Def:''' We say that <math>f</math> is differentiable (''diffable'') if there is a linear map <math>L</math> so that
<math>\lim_{h\rightarrow0}\frac{\right\|f(x+h)-f(x)-L(h)\left\|}{\right\|h\left\|}</math>
<math>\lim_{h\rightarrow0}</math>

Revision as of 15:26, 11 September 2007

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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

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

Def: We say that is differentiable (diffable) if there is a linear map so that

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