Difference between revisions of "0708-1300/Class notes for Tuesday, September 11"
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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 | ||
− | <math> | + | |
+ | <math>\lim_{h\rightarrow0}\frac{|f(x+h)-f(x)-L(h)|}{|h|}.</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> and will denote it by <math>df_{x}</math>. | ||
Revision as of 16:40, 11 September 2007
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In Small Scales, Everything's Linear
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