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Analytical geometry of three dimensions by William H. McCrea

By William H. McCrea

Written via a exceptional mathematician and educator, this short yet rigorous textual content is aimed at complicated undergraduates and graduate scholars. It covers the coordinate process, planes and features, spheres, homogeneous coordinates, normal equations of the second one measure, quadric in Cartesian coordinates, and intersection of quadrics. 1947 version.

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S ∂s c T. Guardia Geod´ esicas 47 Para t = 0, H(s, 0) = X(β(s)) = α(s). Por lo tanto la funci´on l : (−ε, ε) dl alcanza un m´ınimo en t = 0 por tanto dt (0) = 0. d dl d (t) = dt dt c d < ∂ 2 H ∂H , ∂t∂s ∂s < ∂H ∂H , ∂s ∂s < Es decir dl (0) = dt Como H(s, t) t=0 > > t=0 1 2 ds = 0. 47) t=0 = α(s) entonces ∂H ∂H 1 , >2 ∂s ∂s < c ∂H ∂H 1 , > 2 ds. ∂s ∂s t=0 dH dα (s, 0) = (s) = 1. 47 obtenemos dl (0) = dt d < c ∂ 2 H ∂H , > ∂t∂s ∂s t=0 ds. 49) Ahora bien dl (0) = dt d < c ∂ 2 H ∂H ∂H ∂ 2 H ∂H ∂ 2 H , >+< , > − < , > ∂t∂s ∂s ∂t ∂s2 ∂t ∂s2 t=0 ds.

Demostraci´ on: Sea M una superficie y sean p = q ∈ M . / M una curva regular tal que α(a) = p y α(b) = q. Sea α : [a, b] Asumamos que α est´a parametrizada por longitud de arco. Supongamos que adem´as la longitud de arco de α minimiza la distancia entre p y q es decir l(α) = inf{longitudes de arcos entre p y q}. Veamos que α es una geod´esica. Supongamos que κg (s0 ) = 0. Como κg es una funci´on cont´ınua existen c, d ∈ [a, b] tales que s0 ∈ [c, d] y κg (s) = 0 para todo s ∈ [c, d]. Es claro que α minimiza la distancia entre p = α(c) y q = α(d).

Si ∇F (p) = 0 para todo p ∈ R3 entonces M es una superficie. c T. Guardia Superficies Demostraci´ on: 27 Sea p ∈ R3 entonces ∇F (p) = ( ∂F ∂F ∂F (p), (p), (p)) = 0 ∂x ∂y ∂z Queremos encontrar una superficie regular que contenga a un entrorno de p. Como ∇F (p) = 0 entonces al menos una derivada parcial no es cero. Sea = 0. Por el teorema de la funci´on impl´ıcita existen un entorno abierto /V U de (p1 , p2 ), un entorno abierto V de p3 y una u ´nica funci´on g : U tal que g(p1 , p2 ) = p3 y F (x, y, g(x, y)) = 0 para todo (x, y) ∈ U .

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