Wirtingerkalkül < komplex < Analysis < Hochschule < Mathe < Vorhilfe
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| Status: |
(Frage) überfällig  | | Datum: | 00:54 Mo 22.04.2013 | | Autor: | quasimo |
| Aufgabe | | Zeige die Produkt und kettenregel für das Wirtingerkalkül |
Hallo
f: [mm] \IC [/mm] -> [mm] \IC [/mm] reell linear
f(z)=f(x+iy)= u(x,y) + i v(x,y)
ZuZeigen:
[mm] \frac{\partial fg}{\partial z} [/mm] = f [mm] \frac{\partial g}{\partial z} [/mm] + [mm] \frac{\partial f}{\partial z} [/mm] g
[mm] \frac{\partial fg}{\partial \overline{z}} [/mm] = f [mm] \frac{\partial g}{\partial \overline{z}} [/mm] + [mm] \frac{\partial f}{\partial \overline{z}} [/mm] g
Bew.:
[mm] \frac{\partial fg}{\partial z} [/mm] = [mm] (fg)_z [/mm] = 1/2 [mm] ((fg)_x [/mm] - [mm] i(fg)_y [/mm] )= 1/2 [mm] (fg_x [/mm] +f_xg - i [mm] (fg_y +f_y [/mm] g))= 1/2 [mm] (fg_x [/mm] - i [mm] fg_y) [/mm] + 1/2 [mm] (f_x [/mm] g - i [mm] f_y [/mm] g) = f [mm] g_z [/mm] + g [mm] f_z
[/mm]
zweites Analog.
Kettenregel
[mm] \frac{\partial f \circ g}{\partial z} =(\frac{\partial f}{\partial w} \circ [/mm] g ) [mm] \frac{\partial g}{\partial z} +(\frac{\partial f}{\partial \overline{w}} \circ [/mm] g ) [mm] \frac{\partial \overline{g}}{\partial z} [/mm]
WIe mach ich das am besten?
http://matheplanet.de/
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| Status: |
(Mitteilung) Reaktion unnötig  | | Datum: | 01:20 Mi 24.04.2013 | | Autor: | matux |
$MATUXTEXT(ueberfaellige_frage)
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