# 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

The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

### 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

$\lim_{h\rightarrow0}\frac{|f(x+h)-f(x)-L(h)|}{|h|}=0.$

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 assertions hold:

1. $df_{x}$ is unique.
2. $d(f+g)_{x}=df_{x}+dg_{x}$
3. If $f$ is linear then $df_{x}=f$
4. $d(f\circ g)_{x}=df_{g(x)}\cdot dg_{x}$
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 define $y$ 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 $x_0$, $V\subset U$, and a diffable function $g:V\rightarrow\mathbb{R}^m$ such that $g(x_0)=y_0$ and for every $x\in V$ $f(x;g(x))=0.$.