0708-1300/Class notes for Tuesday, September 11: Difference between revisions
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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'') if there is a linear map <math>L</math> so that |
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<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|}.</math> |
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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>. |
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'''Theorem''' |
'''Theorem''' |
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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 asertions holds: |
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1)<math>df_{x}</math> is unique. |
1)<math>df_{x}</math> is unique. |
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2)<math>d(f+g)_{x}=df_{x}+dg_{x}</math> |
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4)<math>d(f\circ g)_{x}=df_{g(x)}\circ dg_{x}</math> |
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3)If <math>f</math> is linear then <math>d(f)_{x}=f</math> |
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5)For every scalar number <math>\alpha</math> it holds <math>d(\alpha f)_{x}=\alpha df_{x}</math> |
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Revision as of 15:38, 11 September 2007
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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
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:
1) is unique.
2)
3)If is linear then
4)
5)For every scalar number it holds
