Algebraic Topology (Colloquium Publications, Volume 27) by Solomon Lefschetz

By Solomon Lefschetz

Because the booklet of Lefschetz's Topology (Amer. Math. Soc. Colloquium courses, vol. 12, 1930; spoke of under as (L)) 3 significant advances have stimulated algebraic topology: the improvement of an summary advanced self reliant of the geometric simplex, the Pontrjagin duality theorem for abelian topological teams, and the strategy of Cech for treating the homology concept of topological areas via structures of "nerves" every one of that's an summary complicated. the result of (L), very materially further to either by way of incorporation of next released paintings and by way of new theorems of the author's, are the following thoroughly recast and unified when it comes to those new ideas. A excessive measure of generality is postulated from the outset.

The summary perspective with its concomitant formalism allows succinct, unique presentation of definitions and proofs. Examples are sparingly given, typically of an easy variety, which, as they don't partake of the scope of the corresponding textual content, may be intelligible to an user-friendly pupil. yet this is often essentially a e-book for the mature reader, during which he can locate the theorems of algebraic topology welded right into a logically coherent complete

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Example text

G =G +i·G Der Speichermodul ist proportional zur gespeicherten Energie, der Verlustmodul proportional zur dissipierten Energie. 5 τˆ0 τˆ0 · cos δ und G = · sin δ γˆ0 γˆ0 Weitere komplexe Größen und etwas Mathematik Da das mechanische Verhalten eines isotropen, inkompressiblen viskoelastischen Körpers im linearen Deformationsbereich durch den komplexen Scher- bzw. den komplexen Elastizitätsmodul (siehe Gl. 8) eindeutig festgelegt ist, können alle mechanischen Eigenschaften aus dem komplexen Modul abgeleitet werden.

Wirkt eine Spannung τik in Richtung k auf eine Fläche mit der Normalen i, so wird das Volumenelement deformiert. Die resultierende Deformation ist im allgemeinen Fall nicht nur auf die Richtung der wirkenden Spannung beschränkt, sondern wird Beiträge in allen Raumrichtungen besitzen. Der formale Zusammenhang zwischen der resultierenden Deformation und der angelegten Spannung ergibt sich aus dem linearen Ansatz (siehe Gl. 5) l=1 m=1 = cik11 · ε11 + cik12 · γ12 + cik13 · γ13 + cik21 · γ21 + cik22 · ε22 + cik23 · γ23 + cik31 · γ31 + cik32 · γ32 + cik33 · ε33 Abb.

Z = |z | · eiδ = |z | · cos δ + i · |z | · sin δ = z +i·z ↓ z = |z | · cos δ z = |z | · sin δ tan δ = z z Natürlich können die Summe, das Produkt und der Quotient zweier komplexer Zahlen in Polarkoordinaten ebenso berechnet werden wie in der Darstellung mit Real- und Imaginärteil. Wirklich vorteilhaft ist dies aber nur beim Produkt und beim Quotienten zweier komplexer Zahlen. Das Produkt zweier komplexer Zahlen u und v berechnet sich in Polarkoordinaten sehr einfach aus dem Produkt der Beträge und der Summe der Phasenwinkel.

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