![0708-1300-ClassPhoto.jpg](/images/thumb/d/d4/0708-1300-ClassPhoto.jpg/215px-0708-1300-ClassPhoto.jpg) Add your name / see who's in!
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Week of...
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Links
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Fall Semester
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1
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Sep 10
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About, Tue, Thu
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2
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Sep 17
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Tue, HW1, Thu
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3
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Sep 24
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Tue, Photo, Thu
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4
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Oct 1
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Questionnaire, Tue, HW2, Thu
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5
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Oct 8
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Thanksgiving, Tue, Thu
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6
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Oct 15
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Tue, HW3, Thu
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7
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Oct 22
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Tue, Thu
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8
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Oct 29
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Tue, HW4, Thu, Hilbert sphere
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9
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Nov 5
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Tue,Thu, TE1
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10
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Nov 12
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Tue, Thu
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11
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Nov 19
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Tue, Thu, HW5
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12
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Nov 26
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Tue, Thu
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13
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Dec 3
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Tue, Thu, HW6
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Spring Semester
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14
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Jan 7
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Tue, Thu, HW7
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15
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Jan 14
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Tue, Thu
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16
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Jan 21
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Tue, Thu, HW8
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17
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Jan 28
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Tue, Thu
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18
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Feb 4
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Tue
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19
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Feb 11
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TE2, Tue, HW9, Thu, Feb 17: last chance to drop class
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R
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Feb 18
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Reading week
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20
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Feb 25
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Tue, Thu, HW10
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21
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Mar 3
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Tue, Thu
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22
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Mar 10
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Tue, Thu, HW11
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23
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Mar 17
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Tue, Thu
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24
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Mar 24
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Tue, HW12, Thu
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25
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Mar 31
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Referendum,Tue, Thu
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26
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Apr 7
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Tue, Thu
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R
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Apr 14
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Office hours
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R
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Apr 21
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Office hours
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F
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Apr 28
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Office hours, Final (Fri, May 2)
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Register of Good Deeds
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Errata to Bredon's Book
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Announcements go here
In Small Scales, Everything's Linear
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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
,
and
be two normed finite dimensional vector spaces and let
be a function defined on a neighborhood of the point
Definition:
We say that
is differentiable (diffable) if there is a linear map
so that
In this case we will say that
is a differential of
and will denote it by
.
Theorem
If
and
are diffable maps then the following asertions holds:
is unique.
![{\displaystyle d(f+g)_{x}=df_{x}+dg_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/909ed24a009f70190b36ebf219c773b425a967e0)
- If
is linear then ![{\displaystyle df_{x}=f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f350db34895e0f06a5a83535713ded62233dbfbd)
![{\displaystyle d(f\circ g)_{x}=df_{g(x)}\circ dg_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcb68399d9499e6f1b8b3ef0b526c85644c8d9f)
- For every scalar number
it holds ![{\displaystyle d(\alpha f)_{x}=\alpha df_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00b394ea03e6ef1f1108e4bc714e93b3aee63359)
Implicit Function Theorem
Example
Although
does not defines
as a function of
, in a neighborhood of
we can define
so that
. Furthermore,
is differentiable with differential
. 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
will be the class of all functions defined on
with continuous partial derivatives up to order
Theorem(Implicit function theorem)
Let
be a
function defined on a neighborhood
of the point
and such that
and suppose that
is non-singular then, the following results holds:
There is an open neighborhood of
,
, and a
function
such that for every
.