![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
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}
,
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) at
if there is a linear map
so that
In this case we will say that
is a differential of
at
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
then given
we will define
by
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 diffable function
such that for every
.