Thomas Heath's A History of Greek Mathematics, Volume 2: From Aristarchus PDF

By Thomas Heath

ISBN-10: 0486240746

ISBN-13: 9780486240749

Volume 2 of an authoritative two-volume set that covers the necessities of arithmetic and lines each landmark innovation and every very important determine, including Euclid, Apollonius, and others.

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Additional info for A History of Greek Mathematics, Volume 2: From Aristarchus to Diophantus (Dover Books on Mathematics)

Example text

FIG. 16. E x a m p l e . U s e t h e D i v e r g e n c e t h e o r e m t o p r o v e t h a t d i v curl b = 0. Consider a n a r b i t r a r y closed surface S which is d i v i d e d i n t o t w o portions, 2 by S a c u r v e C (Fig. 1 6 ) . S, t A 48 C O U R S E O F M A T H E M A T I C S B u t , b y Stokes's t h e o r e m , w h e n a = curl b, /|β·(18= Jfcurlb- sl ST d S = ||a-dS=yy*cwWb-dS = - st st $b-ds, c (J)b-ds. c T h e n e g a t i v e sign for t h e integral o v e r 8 2 arises because of t h e relative directions of t h e n o r m a l vectors.

18 (ii). outwards from 2 Τ is opposite t o n x shown in Fig. 18 (ii). h. 58) is when V is harmonic in Τ. 61) Therefore the integral being taken over the complete bounding surface(s) of Τ. W h e n Ρ is inside S as in Fig. h. 61) must inl9 clude integration over S . h. 62) is zero when Ρ is outside but is independent of the shape of S when Ρ is inside S. E x a m p l e . Find

Note that the vector element of arc (dx 1} dx2) can be replaced b y an element of 'area' η as directed normal t o the curve, (Fig. 35) n o w becomes : 2 2 E x a m p l e . U s e Green's l e m m a t o e v a l u a t e φ (x y formed b y y dx -f- y dy), < where (J is t h e l o o p = x, y — χ (Fig. 1 2 ) . Therefore FIG. 12. 39), which are the usual forms of the Divergence and Stokes's theorems, are applied in a variety of ways in vector analysis. A l t h o u g h we placed no special interpretation o n the functions P i (or P,Q, R), we n o w take t h e m t o b e the c o m p o n e n t s of a vector field a .

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A History of Greek Mathematics, Volume 2: From Aristarchus to Diophantus (Dover Books on Mathematics) by Thomas Heath


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