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

Latest revision as of 17:23, 1 November 2007

#	Week of...	Links
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Fall Semester
1	Sep 10	About, Tue, Thu
2	Sep 17	Tue, HW1, Thu
3	Sep 24	Tue, Photo, Thu
4	Oct 1	Questionnaire, Tue, HW2, Thu
5	Oct 8	Thanksgiving, Tue, Thu
6	Oct 15	Tue, HW3, Thu
7	Oct 22	Tue, Thu
8	Oct 29	Tue, HW4, Thu, Hilbert sphere
9	Nov 5	Tue,Thu, TE1
10	Nov 12	Tue, ~~Thu~~
11	Nov 19	Tue, Thu, HW5
12	Nov 26	Tue, Thu
13	Dec 3	Tue, Thu, HW6
Spring Semester
14	Jan 7	Tue, Thu, HW7
15	Jan 14	Tue, Thu
16	Jan 21	Tue, Thu, HW8
17	Jan 28	Tue, Thu
18	Feb 4	Tue
19	Feb 11	TE2, Tue, HW9, Thu, Feb 17: last chance to drop class
R	Feb 18	Reading week
20	Feb 25	Tue, Thu, HW10
21	Mar 3	Tue, Thu
22	Mar 10	Tue, Thu, HW11
23	Mar 17	Tue, Thu
24	Mar 24	Tue, HW12, Thu
25	Mar 31	Referendum,Tue, Thu
26	Apr 7	Tue, Thu
R	Apr 14	Office hours
R	Apr 21	Office hours
F	Apr 28	Office hours, Final (Fri, May 2)
Register of Good Deeds
Errata to Bredon's Book

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}]

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

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

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 assertions hold:

$df_{x}$ is unique.
$d(f+g)_{x}=df_{x}+dg_{x}$
If $f$ is linear then $df_{x}=f$
$d(f\circ g)_{x}=df_{g(x)}\cdot dg_{x}$
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 $x\in V$ $f(x;g(x))=0.$ .

@@ Line 49: / Line 49: @@
 See also [[06-240/Linear Algebra - Why We Care]].
-==Class Notes==
+{{0708-1300/Class Notes}}
 ===Differentiability===
-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>
+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>.
 '''Definition:'''
-We say that <math>f</math> is differentiable (''diffable'') if there is a linear map <math>L</math> so that
+We say that <math>f</math> is differentiable (''diffable'') at <math>x</math> 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>
+<math>\lim_{h\rightarrow0}\frac{|f(x+h)-f(x)-L(h)|}{|h|}=0.</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>.
+In this case we will say that <math>L</math> is a differential of <math>f</math> at <math>x</math> and will denote it by <math>df_{x}</math>.
 '''Theorem'''
-If <math>f:V\rightarrow W</math> and <math>g:U\rightarrow V</math> are ''diffable'' maps then the following asertions holds:
+If <math>f:V\rightarrow W</math> and <math>g:U\rightarrow V</math> are ''diffable'' maps then the following assertions hold:
 # <math>df_{x}</math> is unique.
 # <math>d(f+g)_{x}=df_{x}+dg_{x}</math>
 # If <math>f</math> is linear then <math>df_{x}=f</math>
+# <math>d(f\circ g)_{x}=df_{g(x)}\cdot dg_{x}</math>
-# <math>d(f\circ g)_{x}=df_{g(x)}\circ dg_{x}</math>
 # For every scalar number <math>\alpha</math> it holds <math>d(\alpha f)_{x}=\alpha df_{x}</math>
@@ Line 78: / Line 74: @@
 '''Example'''
-Although <math>x^2+y^2=1</math> does not defines <math>y</math> as a function of <math>x</math>, in a neighborhood of <math>(0;-1)</math> we can define <math>g(x)=-\sqrt{1-x^2}</math> so that <math>x^2+g(x)^2=1</math>. Furthermore, <math>g</math> is differentiable with differential <math>dg_{x}=\frac{x}{\sqrt{1-x^2}}</math>. This is a motivation for the following theorem.
+Although <math>x^2+y^2=1</math> does not define <math>y</math> as a function of <math>x</math>, in a neighborhood of <math>(0;-1)</math> we can define <math>g(x)=-\sqrt{1-x^2}</math> so that <math>x^2+g(x)^2=1</math>. Furthermore, <math>g</math> is differentiable with differential <math>dg_{x}=\frac{x}{\sqrt{1-x^2}}</math>. 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)
+If <math>f:X\times Y\rightarrow Z</math> then given <math>x\in X</math> we will define <math>f_{[x]}:Y\rightarrow Z</math> by <math>f_{[x]}(y)=f(x;y).</math>
 '''Definition'''
@@ Line 92: / Line 88: @@
 Let <math>f:\mathbb{R}^n \times \mathbb{R}^m\rightarrow \mathbb{R}^m</math> be a <math>C^{1}(\mathbb{R}^n \times \mathbb{R}^m)</math> function defined on a neighborhood <math>U</math> of the point <math>(x_0;y_0)</math> and such that <math>f(x_0;y_0)=0</math> and suppose that <math>d(f_{[x]})_{y}</math> is non-singular then, the following results holds:
-There is an open neighborhood of <math>x</math>, <math>V\subset U</math>, and a <math>diffable</math> function <math>g:V\rightarrow\mathbb{R}^m</math> such that for every <math>x\in V</math>  <math>f(x;g(x))=0.</math>.
+There is an open neighborhood of <math>x_0</math>, <math>V\subset U</math>, and a ''diffable'' function <math>g:V\rightarrow\mathbb{R}^m</math> such that <math>g(x_0)=y_0</math> and for every <math>x\in V</math> <math>f(x;g(x))=0.</math>.

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

Latest revision as of 17:23, 1 November 2007

Contents

In Small Scales, Everything's Linear

Class Notes

Differentiability

Implicit Function Theorem

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