Unformatted text preview: R n , write F = ( F 1 ,F 2 ,...,F n ) where F i : R n â†’ R . For f F to make sense we must have f : R n â†’ R also. Then âˆ‡ Â· ( f F ) = âˆ‡ Â· ( fF 1 ,fF 2 ,...,fF n ) = n X i =1 âˆ‚ ( fF i ) âˆ‚x i = n X i =1 Â² âˆ‚f âˆ‚x i F i + f âˆ‚F i âˆ‚x i Â¶ = n X i =1 Â² âˆ‚f âˆ‚x i F i Â¶ + n X i =1 Â² f âˆ‚F i âˆ‚x i Â¶ = ( âˆ‡ f ) Â· F + f ( âˆ‡ Â· F ) = F Â· âˆ‡ f + f âˆ‡ Â· F . 6. We calculate f,f ,f 00 ,f 000 as follows: f ( x ) = âˆš x, f (1) = 1; f ( x ) = 1 2 x1 / 2 , f (1) = 1 2 ; f 00 ( x ) =1 4 x3 / 2 , f 00 (1) =1 4 ; f 000 ( x ) = 3 8 x5 / 2 , f 000 (1) = 3 8 . Thus p 3 ( x ) = f (1)+ f (1)( x1)+ f 00 (1) 2! ( x1) 2 + f 000 (1) 3! ( x1) 2 = 1+ 1 2 ( x1)1 8 ( x1) 2 + 1 16 ( x1) 3 . 1...
View
Full Document
 Spring '11
 wang
 Multivariable Calculus, Vector Calculus, Cos, 0j, Department of Mathematics, University of Hong Kong, DT DT DT

Click to edit the document details