By T. J. Willmore

Part 1 starts by means of applying vector ways to discover the classical idea of curves and surfaces. An creation to the differential geometry of surfaces within the huge offers scholars with principles and strategies enthusiastic about worldwide learn. half 2 introduces the idea that of a tensor, first in algebra, then in calculus. It covers the elemental thought of absolutely the calculus and the basics of Riemannian geometry. labored examples and routines seem during the text.

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T ( (o) is continuous, a contradiction is reached, and therefore pn satisfies the continuity principle. 48 II. Domains of Holomorphy H artogs Convexity . A domain G c en is called Hartogs convex if the following holds: If (P, H ) is a general Hartogs figure with H C G, then P C G. Definition. An immediate consequence of the definition is the following: The biholomorphic image of a Hartogs convex domain is again Hartogs convex. 5 Theorem. Let G c en be a domain that satisfies the continuity prin ciple.

Let G c e n be a domain, Zo E G a point, and f : B --+ e a holomorphic fu nction. If f ( z) = :Z::: v >o av (z - zo) v is the (uniquely deter m ined) power series expansion near z0 -E G, then a, = � . Dv f(zo ) , fo r each v. v E N� 4 . 9 Cor ollary (C auchy ' s in equ alities). Let G c e n be a doma in, f : G --+ e holomorphic, z0 E G a point, and P = p n (z 0 , r ) C C G a polydisk with distinguished boundary T. Th en Let f (z) = :Z::: v >O a v (z - z0 ) v be the power series expansion of f at z 0 .

Then Df (z ) = (8f )z , which is complex linear. In this case we have the Cauchy Riemann equations PROOF: and therefore If n = m, A + i C. then JIR, f = (� B D ) w1th. B = -C and A = D, and Jr = 32 I . Holomorphic Functions By elementary transformations, ( A +C i C ( A + iC det C d - et l det ( A + ;cw . • It follows that holomorphic map s are orientation preserving ! Chain Rules. f f . Then g o f : B ---+ e is differentiable, and the following holds: 7 . 3 P r op o sition (comp lex chain r u le).

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