Difference between revisions of "0708-1300/Class notes for Tuesday, September 11"
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==Class Notes== | ==Class Notes== | ||
− | Let <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:''' |
− | <math>\lim_{h\rightarrow0}\frac{|f(x+h)-f(x)-L(h)|}{|h|}</math> | + | |
+ | We say that <math>f</math> is differentiable (''diffable'') 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> 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>. | ||
+ | |||
+ | '''Theorem''' | ||
+ | |||
+ | If <math>f:V\rightarrow W</math> and <math>g:U\rightarrow V</math> are ''diffable'' maps then the following asertions holds: | ||
+ | |||
+ | 1)<math>df_{x}</math> is unique. | ||
+ | 2)<math>d(f+g)_{x}=df_{x}+dg_{x}</math> | ||
+ | 3)If <math>f</math> is linear then <math>d(f)_{x}=f</math> | ||
+ | 4)<math>d(f\circ g)_{x}=df_{g(x)}\circ dg_{x}</math> |
Revision as of 16:37, 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)