# Advanced courses of mathematical analysis II: proceedings of by M. V. Velasco, A. Rodriguez-Palacios

By M. V. Velasco, A. Rodriguez-Palacios

This quantity includes a suite of articles by way of major researchers in mathematical research. It presents the reader with an intensive evaluation of recent instructions and advances in themes for present and destiny learn within the box.

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Additional resources for Advanced courses of mathematical analysis II: proceedings of the 2nd international school, Granada, Spain, 20-24 September 2004

Example text

Punkt, Strecke und Dreiecke im Raum. Übersicht A-. (Fortsetzung). Punkte und Strecken im Raum Entfernung P1 P2 (P1 (x 1 , y 1 , z 1 )) ; (P2 (x 2 , y 2 , z 2 )) P1 P2 = (x 1 − x 2 )2 + (y 1 − y 2 )2 + (z 1 − z 2 )2 Teilung von P1 P2 im Verhältnis λ xP = λ 0 λ<0 x 1 + λx 2 y 1 + λy 2 z 1 + λz 2 ; yP = ; zP = ; 1+λ 1+λ 1+λ innerer Teilpunkt äußerer Teilpunkt Mittelpunkt M x1 + x2 xM = ; 2 yM y1 + y2 ; = 2 yS = Für Punktmassen m 1 , m 2 , m 3 m1 x1 + m2 x2 + m3 x3 xS = ; m1 + m2 + m3 m1 y1 + m2 y2 + m3 y3 ; yS = m1 + m2 + m3 m1 z1 + m2 z2 + m3 z3 zS = m1 + m2 + m3 Fläche zM z1 + z2 ; = 2 Dreiecke im Raum Schwerpunkt x1 + x2 + x3 ; xS = 3 z1 + z2 + z3 zS = 3 Dreiecke im Raum y1 + y2 + y3 ; 3 A= A1 = A21 + A22 + A23 , 1 2 1 A3 = 2 y1 z1 1 y2 z2 1 y3 z3 1 x1 y1 1 x2 y2 1 x3 y3 1 mit ; A2 = 1 2 z1 x1 1 z2 x2 1 z3 x3 1   A Mathematik Übersicht A-.

8 Flächen und Körper Übersicht A-. Inhalt von Flächen.  Flächen und Körper  Übersicht A-. (Fortsetzung). Art der Fläche Flächeninhalt A Ellipse A= π Dċd = aċbċπ 4 Umfang U . Guldin’sche Regel 0,75π(D + d) − 0,5 π Dd Rotation der ebenen Kurve C um die x-Achse ergibt einen (räumlichen) Rotationskörper. Dessen Mantelﬂäche habe den Flächeninhalt A. Es sei L die Länge von C, und S R2 sei der Schwerpunkt von C mit dem Abstand rs von der Drehachse. Dann ist A = 2πrs ċL. Weg des Schwerpunktes bei Rotation Übersicht A-.

Schnittwinkel β zweier Geraden allgemeine Gleichung Schnitt zweier beliebiger Ebenen m 1 = tan φ 1 ; m 2 = tan φ 2 A1 x + B 1 y + C 1 z + D 1 = 0 A2 x + B 2 y + C 2 z + D 2 = 0 senkrechte Geraden: m 1 m 2 = −1 A1 A2 + B 1 B 2 = 0 parallele Geraden: m1 = m2 A1 A2 = B 1 B 2 Winkelhalbierende zweier Geraden A21 + B 21 Zwei-Punkte-Form x − x1 y − y1 z − z1 = = x2 − x1 y2 − y1 z2 − z1 m2 − m1 1 + m1 m2 A1 B 2 − A2 B 1 tan β = A1 A2 + B 1 B 2 tan β = A1 x + B 1 y + C 1  A2 x + B 2 y + C 2 A22 + B 22 =0 Winkel zwischen Gerade und Achsen 1 B1 C1 1 C 1 A1 E1 = 0 cos α = ; cos β = E2 = 0 N B2 C2 N C 2 A2 N: Normalenvektor cos γ = 1 A1 B 1 N A2 B 2 N2 = B1 C1 B2 C2 2 + C 1 A1 C 2 A2 2 + A1 B 1 A2 B 2 2 cos2 α + cos2 β + cos2 γ = 1 Gerade durch Punkt P1 (x 1 , y 1 , z 1 ) x − x1 y − y1 z − z1 = = cos α cos β cos γ in Parameterform: Hesse’sche Normalform x (cos β 1 cos β 2 ) + y (sin β 1 − (p1 p2 ) = 0 x = x 1 + t cos α ; z = z 1 + t cos γ sin β 2 ) y = y 1 + t cos β Parameterdarstellung x = a1 t + a2 ; y = b1 t + b2 ; z = c1 t + c2 Schnittwinkel zweier Geraden cos β = cos α 1 cos α 2 + cos β 1 cos β 2 + cos γ 1 cos γ 2  A Mathematik Übersicht A-.