# Read e-book online ALGOL 60 Implementation: The Translation and Use of ALGOL 60 PDF

By Brian Randell

ALGOL 60 implementation: the interpretation and use of ALGOL 60 courses on a computer
Volume five of A.P.I.C. experiences in facts processing
Issue five of ALGOL 60 Implementation, Brian Randell

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Focusing on topos theory's integration of geometric and logical rules into the principles of arithmetic and theoretical computing device technological know-how, this quantity explores inner classification concept, topologies and sheaves, geometric morphisms, different subjects.

In dieser Arbeit wird die automatische Berechnung von Steigungstupeln zweiter und h? herer Ordnung behandelt. Die entwickelten Techniken werden in einem Algorithmus zur verifizierten globalen Optimierung verwendet. Anhand von Testbeispielen auf einem Rechner wird der neue Algorithmus mit einem Algorithmus aus der Literatur verglichen.

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St δf (x; x0 ) = x1 + (x0 )1 · x22 1 (x0 )21 + 1 · x2 + (x0 )2 0 = δf (x; x0 ) . ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ❊s s❡✐ f : D ⊆ Rn → R st❡t✐❣ ✉♥❞ x0 , x1 ∈ D ❜❡❧✐❡❜✐❣✱ ❛❜❡r ❢❡st✱ ♠✐t x0 = x1 ✳ ❊✐♥❡ ❋✉♥❦t✐♦♥ δ2 f : D → Rn×n ♠✐t ❉❡✜♥✐t✐♦♥ ✶✳✸✳✸✵ f (x) = f (x0 ) + δf (x1 ; x0 ) (x − x0 ) + (x − x1 )T δ2 f (x; x1 , x0 ) (x − x0 ) , x ∈ D, ✭✶✳✶✹✮ ❤❡✐ÿt ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x1 ✉♥❞ x0 ✳ ❉❛❜❡✐ ✐st δf (x1 ; x0 ) ∈ R1×n ❞❡r ❲❡rt ❡✐♥❡r ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ δf (x; x0 ) ❡rst❡r ❖r❞✲ ♥✉♥❣ ✈♦♥ f ❛♥ ❞❡r ❙t❡❧❧❡ x = x1 ✳ ❊✐♥❡ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ δ2 f ([x] ; x1 , x0 ) ✈♦♥ δf (x; x1 , x0 ) ❛✉❢ ❞❡♠ ■♥t❡r✈❛❧❧ [x]✱ ❞✳ ❤✳ ❡✐♥ δ2 f ([x] ; x1 , x0 ) ∈ IRn×n ♠✐t δ2 f ([x] ; x1 , x0 ) ⊇ {δ2 f (x; x1 , x0 ) | x ∈ [x] } , ♥❡♥♥❡♥ ✇✐r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x1 ✉♥❞ x0 ✳ ❋ür ❋✉♥❦t✐♦♥❡♥ f = (fi ) : D ⊆ Rn → Rm ✇✐r❞ ❡✐♥❡ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ❞❡✜♥✐❡rt✱ ✐♥❞❡♠ ❞✐❡ ●❧❡✐❝❤✉♥❣ ✭✶✳✶✹✮ ❦♦♠♣♦♥❡♥t❡♥✇❡✐s❡ ❛♥❣❡✇❡♥❞❡t ✇✐r❞✳ ✷✽ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ■st δ2 f ([x] ; x1 , x0 ) ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x]✱ s♦ ❣✐❧t ✇✐❡ ✐♥ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✷ ♠✐t ❍✐❧❢❡ ❞❡r ❊✐♥s❝❤❧✐❡ÿ✉♥❣s❡✐❣❡♥s❝❤❛❢t ✶✳✷✳✸ s♦✇✐❡ ❞❡r ❙✉❜❞✐str✐❜✉t✐✈✐tät ❞❡r ■♥t❡r✈❛❧❧r❡❝❤♥✉♥❣ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✸✶ f (x) ∈ f (x0 ) + δf (x1 ; x0 ) + ([x] − x1 )T δ2 f ([x] ; x1 , x0 ) ([x] − x0 ) ⊆ f (x0 ) + δf (x1 ; x0 ) ([x] − x0 ) + ([x] − x1 )T δ2 f ([x] ; x1 , x0 ) ([x] − x0 ) ❢ür ❛❧❧❡ x ∈ [x]✳ ✶✳✸✳✹ ❆♥✇❡♥❞✉♥❣❡♥ ◆❡❜❡♥ ❞❡r ❇❡st✐♠♠✉♥❣ ❜③✇✳ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ❡✐♥❡r ❋✉♥❦t✐♦♥ ✉♥❞ ♥❡❜❡♥ ❆♥✇❡♥❞✉♥❣❡♥ ✐♥ ❞❡r ❣❧♦❜❛❧❡♥ ❖♣t✐♠✐❡r✉♥❣ ✭s✐❡❤❡ ❑❛♣✐t❡❧ ✺✮ ❦ö♥♥❡♥ ❙t❡✐❣✉♥❣s✲ ❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❛✉❝❤ ③✉♠ ❊①✐st❡♥③♥❛❝❤✇❡✐s ✈♦♥ ◆✉❧❧st❡❧❧❡♥ ❡✐♥❡r ❋✉♥❦t✐♦♥ ✈❡r✇❡♥❞❡t ✇❡r❞❡♥✳ ❲✐r ❣❡❜❡♥ ✐♠ ❋♦❧❣❡♥❞❡♥ ❡✐♥✐❣❡ ❊①✐st❡♥③sät③❡ ❛♥✱ ✐♥ ❞❡♥❡♥ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ❜③✇✳ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ✈❡r✇❡♥❞❡t ✇❡r❞❡♥✳ ❇❛s✐❡r❡♥❞ ❛✉❢ ❞❡♠ ❋✐①♣✉♥❦ts❛t③ ✈♦♥ ❇r♦✉✇❡r ❣✐❧t ❞❡r ❢♦❧❣❡♥❞❡ ❙❛t③ ✈♦♥ ▼♦♦r❡✳ ❙❛t③ ✶✳✸✳✸✷ ✭❙❛t③ ✈♦♥ ▼♦♦r❡✱ ❬✷✻❪✮ ❊s s❡✐ f : D ⊆ Rn → Rn st❡t✐❣ ❛✉❢ ❞❡r ♦✛❡♥❡♥✱ ❦♦♥✈❡①❡♥ ▼❡♥❣❡ D ✉♥❞ [x] ⊆ D✳ ❋ür ❢❡st❡s x0 ∈ [x] s❡✐ δf ([x] ; x0 ) ∈ IRn×n ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ●✐❜t ❡s ❡✐♥❡ ♥✐❝❤ts✐♥❣✉❧är❡ ▼❛tr✐① A ∈ Rn×n ✱ s♦ ❞❛ss ❞❡r ❑r❛✇❝③②❦✲❖♣❡r❛t♦r K([x] , x0 , A) := x0 − A f (x0 ) + I − A δf ([x] ; x0 ) ([x] − x0 ) ❞✐❡ ■♥❦❧✉s✐♦♥ K([x] , x0 , A) ⊆ [x] , ✭✶✳✶✺✮ ❡r❢ü❧❧t✱ ❞❛♥♥ ❡♥t❤ä❧t [x] ❡✐♥❡ ◆✉❧❧st❡❧❧❡ ✈♦♥ f ✳ ❊✐♥ ✇❡✐t❡r❡r ❙❛t③✱ ❞❡r ❞✐❡ ❊①✐st❡♥③ ❡✐♥❡r ◆✉❧❧st❡❧❧❡ ✈♦♥ f : D ⊆ Rn → Rn ❣❛r❛♥t✐❡rt✱ ✐st ❞❡r ❙❛t③ ✈♦♥ ▼✐r❛♥❞❛✳ ❙❛t③ ✶✳✸✳✸✸ ✭▼✐r❛♥❞❛✱ ❬✷✺❪✮ ❊s s❡✐ f : D ⊆ Rn → Rn st❡t✐❣✱ x0 ∈ D ✉♥❞ [x] = [x0 − s, x0 + t] ⊆ D ♠✐t s, t ∈ Rn ✱ si ≥ 0✱ ti ≥ 0✳ ❉✐❡ s✐❝❤ ❣❡❣❡♥ü❜❡r❧✐❡❣❡♥❞❡♥✱ ♣❛r❛❧❧❡❧❡♥ ❙❡✐t❡♥ ✈♦♥ [x] s❡✐❡♥ ♠✐t ✉♥❞ [x]i,+ := {x ∈ [x] , xi = (x0 )i + ti } ✭✶✳✶✻✮ [x]i,− := {x ∈ [x] , xi = (x0 )i − si } ✭✶✳✶✼✮ ✶✳✸✳ ❙❚❊■●❯◆●❊◆ ✷✾ ❜❡③❡✐❝❤♥❡t✳ ❋❛❧❧s ❢ür ❛❧❧❡ i = 1, .

X0 )n = fi x1 , . . , xn − fi x1 , . . , xn−1 , (x0 )n + fi x1 , . . , xn−1 , (x0 )n −fi x1 , . . , xn−2 , (x0 )n−1 , (x0 )n + fi x1 , . . , xn−2 , (x0 )n−1 , (x0 )n − + · · · + fi x1 , (x0 )2 , . . , (x0 )n − fi (x0 )1 , . . , (x0 )n xj − (x0 )j . ❉✉r❝❤ P❡r♠✉t❛t✐♦♥ ❡r❣❡❜❡♥ s✐❝❤ ❡♥ts♣r❡❝❤❡♥❞ ❢ür ❡✐♥❡ ❋✉♥❦t✐♦♥ f : Rn → Rn ✐♥ ❥❡❞❡r ❑♦♠♣♦♥❡♥t❡ fi ❣❡♥❛✉ n! ▼ö❣❧✐❝❤❦❡✐t❡♥ ③✉r ❇✐❧❞✉♥❣ ❡✐♥❡r ❍❛♥s❡♥✲❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥✱ ✐♥s❣❡s❛♠t ❛❧s♦ n · n! ▼ö❣❧✐❝❤❦❡✐t❡♥✳ ✷✻ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ❇❡✐s♣✐❡❧ ✶✳✸✳✷✼ ✭❋♦rts❡t③✉♥❣✮✿ ❲✐r ❜❡tr❛❝❤t❡♥ ❞✐❡ ❋✉♥❦t✐♦♥ f ❛✉s ♦❜✐❣❡♠ ❇❡✐s♣✐❡❧ ✉♥❞ ❜❡r❡❝❤♥❡♥ ❡✐♥❡ ❍❛♥s❡♥✲ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x0 = (0, 0) ♠✐t ❍✐❧❢❡ ❛♥❞❡r❡r ❡✐♥❣❡❢ü❣t❡r ❙✉♠✲ ♠❛♥❞❡♥ ✭s✐❡❤❡ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✽✮✱ ✉♥❞ ③✇❛r δf (x, x0 )ij = fi (x1 , .

X0 ) | x ∈ [x] } , ♥❡♥♥❡♥ ✇✐r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ n−t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ xn−1 , . . , x0 ✳ ■st δ2 f ([x] ; x1 , x0 ) = δ2 f , δ2 f ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x]✱ s♦ ❣✐❧t ♠✐t ❞❡r ❊✐♥s❝❤❧✐❡ÿ✉♥❣s❡✐❣❡♥s❝❤❛❢t ✶✳✷✳✸ s♦✇✐❡ ❞❡r ❙✉❜❞✐str✐❜✉t✐✈✐tät ❞❡r ■♥t❡r✈❛❧❧r❡❝❤♥✉♥❣ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✷ f (x) = f (x0 ) + δf (x1 , x0 ) + δ2 f (x; x1 , x0 ) · (x − x1 ) · (x − x0 ) ∈ f (x0 ) + δf (x1 , x0 ) + δ2 f ([x] ; x1 , x0 ) · ([x] − x1 ) · ([x] − x0 ) ⊆ f (x0 ) + δf (x1 , x0 ) · ([x] − x0 ) + δ2 f ([x] ; x1 , x0 ) · ([x] − x1 ) · ([x] − x0 ) ❢ür ❛❧❧❡ x ∈ [x]✳ ❋❡r♥❡r s❝❤❧✐❡ÿ❡♥ ❞✐❡ ❜❡✐❞❡♥ P❛r❛❜❡❧♥ g (x) := f (x0 ) + δf (x1 , x0 ) · (x − x0 ) + δ2 f · (x − x1 ) · (x − x0 ) ✉♥❞ h (x) := f (x0 ) + δf (x1 , x0 ) · (x − x0 ) + δ2 f · (x − x1 ) · (x − x0 ) ❢ür ❥❡❞❡s x ∈ [x] ❞❡♥ ❋✉♥❦t✐♦♥s✇❡rt f (x) ❡✐♥ ✭s✐❡❤❡ ❆❜❜✐❧❞✉♥❣ ✶✳✷✮✳ ✷✵ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ f(x) h f g _x x0 _ x x x1 ❆❜❜✐❧❞✉♥❣ ✶✳✷✿ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ✈♦♥ f ♠✐t ❍✐❧❢❡ ❡✐♥❡r ■♥t❡r✈❛❧❧st❡✐✲ ❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x1 ✉♥❞ x0 δ2 f (x; x1 , x0 ) ❤ä♥❣t ✈♦♥ ❞❡r ❲❛❤❧ ❞❡r ❜❡✐❞❡♥ P✉♥❦t❡ x1 ✉♥❞ x0 ❛❜✳ ■♠ ❋♦❧❣❡♥❞❡♥ ❜❡tr❛❝❤t❡♥ ✇✐r ❞❡♥ ❋❛❧❧ x1 = x0 ✳ ❉❡✜♥✐t✐♦♥ ✶✳✸✳✷✸ ❊s s❡✐ f : D ⊆ R → R st❡t✐❣✱ [x] ⊆ D ✉♥❞ x0 ∈ [x] ❢❡st✳ ❛✮ ❊s ❡①✐st✐❡r❡ f (x0 )✳ ❊✐♥❡ ❋✉♥❦t✐♦♥ δ2 f : D → R ♠✐t f (x) = f (x0 ) + f (x0 ) · (x − x0 ) + δ2 f (x; x0 , x0 ) · (x − x0 )2 , x ∈ D, ❤❡✐ÿt ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x0 ✳ ❜✮ ❉✐❡ ✑●r❡♥③✲■♥t❡r✈❛❧❧st❡✐❣✉♥❣✑ δflim ([x0 ]) ❡①✐st✐❡r❡✳ ❊✐♥ δ2 f ([x] ; x0 , x0 ) ∈ IR ♠✐t f (x) ∈ f (x0 ) + δflim ([x0 ]) · (x − x0 ) + δ2 f ([x] ; x0 , x0 ) · (x − x0 )2 , x ∈ [x] , ✭✶✳✾✮ ❤❡✐ÿt ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ❩✉r ❆❜❦ür③✉♥❣ s❡✐ δ2 f (x; x0 ) := δ2 f (x; x0 , x0 ) s♦✇✐❡ δ2 f ([x] ; x0 ) := δ2 f ([x] ; x0 , x0 ) ❣❡s❡t③t✳ ❊s s❡✐ f : D ⊆ R → R st❡t✐❣✱ [x] ⊆ D ✉♥❞ f ③✇❡✐♠❛❧ st❡t✐❣ ❞✐✛❡r❡♥③✐❡r❜❛r ✐♥ ❡✐♥❡r ❯♠❣❡❜✉♥❣ ✈♦♥ x0 ∈ [x]✳ ❉❛♥♥ ✐st ❞✐❡ ❋✉♥❦t✐♦♥ δ2 f : D ⊆ R → R ♠✐t  f (x) − f (x0 ) − f (x0 ) (x − x0 )   ❢ür x = x0  (x − x0 )2 δ2 f (x; x0 ) = ✭✶✳✶✵✮    1 ❢ür x = x0 2 f (x0 ) ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✹ ✶✳✸✳ ❙❚❊■●❯◆●❊◆ ✷✶ ♥❛❝❤ ❞❡♠ ❙❛t③ ✈♦♥ ❚❛②❧♦r st❡t✐❣ ✐♥ x0 ✉♥❞ ❢♦❧❣❧✐❝❤ ❛✉❝❤ st❡t✐❣ ❛✉❢ ❣❛♥③ D✳ ❉❛♠✐t ❡①✐st✐❡rt ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ δ2 f ([x] ; x0 ) ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ❊s ❣❡❧t❡ ✭✶✳✾✮ ♠✐t δflim ([x0 ]) = δf x0 , δf x0 ✉♥❞ δ2 f ([x] ; x0 ) = δ2 f , δ2 f ✳ ❉❛♥♥ ✐st ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✺ f (x) ∈ f (x0 ) + δflim ([x0 ]) · ([x] − x0 ) + δ2 f ([x] ; x0 ) · ([x] − x0 )2 ✭✶✳✶✶✮ ❡✐♥❡ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ✈♦♥ f ❛✉❢ [x]✳ ❋❡r♥❡r ❢♦❧❣t ❛✉s ✭✶✳✾✮✱ ❞❛ss ❞✐❡ P❛r❛❜❡❧stü❝❦❡ gl (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x ≤ x ≤ x0 , hl (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x ≤ x ≤ x0 , gr (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x0 ≤ x ≤ x hr (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x0 ≤ x ≤ x ✉♥❞ ❢ür ❥❡❞❡s x ∈ [x] ❞❡♥ ❋✉♥❦t✐♦♥s✇❡rt f (x) ❡✐♥s❝❤❧✐❡ÿ❡♥ ✭s✐❡❤❡ ❆❜❜✐❧❞✉♥❣ ✶✳✸✮✳ f(x) hr hl f gr gl _x x0 _ x x ❆❜❜✐❧❞✉♥❣ ✶✳✸✿ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ✈♦♥ f ♠✐t ❍✐❧❢❡ ❡✐♥❡r ■♥t❡r✈❛❧❧st❡✐✲ ❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✷✷ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ✶✳✸✳✸ ❙t❡✐❣✉♥❣❡♥ ✈♦♥ ❋✉♥❦t✐♦♥❡♥ ♠❡❤r❡r❡r ❱❛r✐❛❜❧❡♥ ❊s s❡✐ f : D ⊆ Rn → Rm st❡t✐❣ ✉♥❞ x0 ∈ D ❜❡❧✐❡❜✐❣✱ ❛❜❡r ❢❡st✳ ❊✐♥❡ ❋✉♥❦t✐♦♥ δf : D → Rm×n ♠✐t ❉❡✜♥✐t✐♦♥ ✶✳✸✳✷✻ f (x) = f (x0 ) + δf (x; x0 ) · (x − x0 ) , ❤❡✐ÿt ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ ✭✶✳✶✷✮ x ∈ D, x0 ✳ ❊✐♥❡ ■♥t❡r✈❛❧❧♠❛tr✐① δf ([x] ; x0 ) ∈ IRm×n ♠✐t δf ([x] ; x0 ) ⊇ {δf (x; x0 ) | x ∈ [x] } ♥❡♥♥❡♥ ✇✐r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ❉✐❡ ❆✉ss❛❣❡♥ ❛✉s ❇❡♠❡r❦✉♥❣ ✶✳✸✳✶✷ ❣❡❧t❡♥ s♦♠✐t ❡♥ts♣r❡❝❤❡♥❞ ❢ür ❞✐❡ ❙t❡✐❣✉♥❣❡♥ ✈♦♥ ❋✉♥❦t✐♦♥❡♥ ♠❡❤r❡r❡r ❱❛r✐❛❜❧❡♥✳ ❇❡r❡❝❤♥✉♥❣ ✈♦♥ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❡rst❡r ❖r❞✲ ♥✉♥❣ ❋ür ❞✐❡ ❇❡r❡❝❤♥✉♥❣ ✈♦♥ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❡rst❡r ❖r❞♥✉♥❣ ❣✐❜t ❡s ✈❡rs❝❤✐❡❞❡♥❡ ▼ö❣❧✐❝❤❦❡✐t❡♥✿ ❛✮ ❊s s❡✐ D ⊆ Rn ♦✛❡♥✱ f = (fi ) : D → Rm st❡t✐❣ ❞✐✛❡r❡♥③✐❡r❜❛r ✉♥❞ [x] ⊆ D✳ ❆✉❢ ●r✉♥❞ ❞❡s ▼✐tt❡❧✇❡rts❛t③❡s ✶✳✶✳✶ ❣✐❧t ❢ür ❛❧❧❡ x ∈ [x]   f (x) − f (x0 ) =  ∂f1 ∂x1 (x0 ∂fm ∂x1 (x0 + t1 (x − x0 )) ✳✳ ✳ ··· + tm (x − x0 )) · · · ∂f1 ∂xn (x0  + t1 (x − x0 ))  ✳✳  (x − x0 ) ✳ ∂fm ∂xn (x0 + tm (x − x0 )) ♠✐t ti ∈ [0, 1]✳ ❲❡❣❡♥ ❞❡r ❙t❡t✐❣❦❡✐t ✈♦♥ f ❡①✐st✐❡rt ❡✐♥❡ ■♥t❡r✈❛❧❧♠❛tr✐① δf ([x] ; x0 ) ∈ IRm×n ♠✐t δf ([x] ; x0 ) ⊇ f (x) | x ∈ [x] .