0708-1300/Class notes for Tuesday, September 11

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In Small Scales, Everything's Linear

	$\longrightarrow$
$z$	$\mapsto$	$z^{2}$

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

Differentiability

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) at $x$ if there is a linear map $L$ so that

$\lim _{h\rightarrow 0}{\frac {|f(x+h)-f(x)-L(h)|}{|h|}}=0.$

In this case we will say that $L$ is a differential of $f$ at $x$ 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 $df_{x}=f$
4. $d(f\circ g)_{x}=df_{g(x)}\circ dg_{x}$
5. For every scalar number $\alpha$ it holds $d(\alpha f)_{x}=\alpha df_{x}$

Implicit Function Theorem

Example Although $x^{2}+y^{2}=1$ does not defines $y$ as a function of $x$, in a neighborhood of $(0;-1)$ we can define $g(x)=-{\sqrt {1-x^{2}}}$ so that $x^{2}+g(x)^{2}=1$. Furthermore, $g$ is differentiable with differential $dg_{x}={\frac {x}{\sqrt {1-x^{2}}}}$. 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

$C^{p}(V)$ will be the class of all functions defined on $V$ with continuous partial derivatives up to order $p.$

Theorem(Implicit function theorem)

Let $f:\mathbb {R} ^{n}\times \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{m}$ be a $C^{1}(\mathbb {R} ^{n}\times \mathbb {R} ^{m})$ function defined on a neighborhood $U$ of the point $(x_{0};y_{0})$ and such that $f(x_{0};y_{0})=0$ and suppose that $d(f_{[x]})_{y}$ is non-singular then, the following results holds:

There is an open neighborhood of $x$, $V\subset U$, and a diffable function $g:V\rightarrow \mathbb {R} ^{m}$ such that for every $x\in V$ $f(x;g(x))=0.$.