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

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(Class Notes)
(Class Notes)
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==Class Notes==
 
==Class Notes==
Let <math>V</math> and <math>W</math> be two normed finite dimensional vector spaces and let <math>f:V\rightarrow W</math> be a function defined on a neighborhood of the point <math>x</math>
+
Let <math>U</math>, <math>V</math> and <math>W</math> be two normed finite dimensional vector spaces and let <math>f:V\rightarrow W</math> be a function defined on a neighborhood of the point <math>x</math>
  
'''Def:''' We say that <math>f</math> is differentiable (''diffable'') if there is a linear map <math>L</math> so that
+
'''Definition:'''  
<math>\lim_{h\rightarrow0}\frac{|f(x+h)-f(x)-L(h)|}{|h|}</math>
+
 
 +
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{|f(x+h)-f(x)-L(h)|}{|h|}.</math> In this case we will say that <math>L</math> is a differential of <math>f</math> and will denote it by <math>df_{x}</math>.
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 +
'''Theorem'''
 +
 
 +
If <math>f:V\rightarrow W</math> and <math>g:U\rightarrow V</math> are ''diffable'' maps then the following asertions holds:
 +
 
 +
1)<math>df_{x}</math> is unique.
 +
2)<math>d(f+g)_{x}=df_{x}+dg_{x}</math>
 +
3)If <math>f</math> is linear then <math>d(f)_{x}=f</math>
 +
4)<math>d(f\circ g)_{x}=df_{g(x)}\circ dg_{x}</math>

Revision as of 16:37, 11 September 2007

Announcements go here

In Small Scales, Everything's Linear

06-240-QuiltBeforeMap.png \longrightarrow 06-240-QuiltAfterMap.png
z \mapsto z^2

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 U, V and W be two normed finite dimensional vector spaces and let f:V\rightarrow W be a function defined on a neighborhood of the point x

Definition:

We say that f is differentiable (diffable) if there is a linear map L so that \lim_{h\rightarrow0}\frac{|f(x+h)-f(x)-L(h)|}{|h|}. In this case we will say that L is a differential of f and will denote it by df_{x}.

Theorem

If f:V\rightarrow W and g:U\rightarrow V are diffable maps then the following asertions holds:

1)df_{x} is unique. 2)d(f+g)_{x}=df_{x}+dg_{x} 3)If f is linear then d(f)_{x}=f 4)d(f\circ g)_{x}=df_{g(x)}\circ dg_{x}