DEL APPLIED TO PRODUCTS OF POINT FUNCTIONS To prove that (1) grad (fg) = f (grad g) +g (grad f) (2) div (f G) = (grad f) · G+f (div G) (3) curl (f G) = (grad f) x G +f (curl G) (4) grad (F · G) = (F - V) G + (G V) F +Fx Gurl Gt i.e. V (fg) = f Vg +gVf. i.e. V (f G) = Vf . G +f V · G i.e. V x (f G) = Vf × G +f V x G %3D %3D
DEL APPLIED TO PRODUCTS OF POINT FUNCTIONS To prove that (1) grad (fg) = f (grad g) +g (grad f) (2) div (f G) = (grad f) · G+f (div G) (3) curl (f G) = (grad f) x G +f (curl G) (4) grad (F · G) = (F - V) G + (G V) F +Fx Gurl Gt i.e. V (fg) = f Vg +gVf. i.e. V (f G) = Vf . G +f V · G i.e. V x (f G) = Vf × G +f V x G %3D %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:DEL APPLIED TO PRODUCTS OF POINT FUNCTIONS
To prove that
(1) grad (fg) = f (grad g) +g (grad A)
(2) div (f G) = (grad f) · G+f (div G)
(3) curl (f G) = (grad f) x G +f (curl G)
(4) grad (F G) = (F - V) G + (G. V) F + F x curl G+ Gx curl F
i.e. V (fg) = f Vg +gVf.
i.e. V (f G) = Vf · G +fV · G
i.e. V x (f G) = Vf× G + f V x G
%3D
%3D
V (F . G) = (F . V) G + (G V) F + F x (V × G) +G × (V × F)
%3D
i.e. V. (F x G) =G· (V × F) – F (V x G)
(5) div (F x G) = G (curl F) – F (curl G)
(6) curl (F x G) = F (div G) – G (div F) + (G · V) F – (F - V) G
V x (F x G) = F (V · G) – G (V · F) + (G - V) F – (F. V) G
%3D
%3D
|
%3D
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