推 monene5566:原來是這樣 謝謝:) 118.166.64.183 07/08 00:33
※ 引述《monene5566 (機掰凱文)》之銘言：
: 2 2
: Let f:| R --> | R be differentiable . Given u=(3/5,4/5), v=(4/5,-3/5)
: 2 ' 2 ' 2
: prove that | gradient f | = |f_u| + |f_v|
: ' '
: where f_u and f_v are the directional derivative of f in the direction
: of u and v
: 這題不知道該怎樣證 連第一步要怎樣進行都無頭緒
: 請版友解題了 謝謝:)
let ▽f=(f1)i+(f2)j => |▽f|^2 = (f1)^2 +(f2)^2
f'u=▽f．u = ((f1)i+(f2)j)．(3/5,4/5) = 3f1/5 + 4f2/5
f'v=▽f．v = ((f1)i+(f2)j)．(4/5,-3/5) = 4f1/5 - 3f2/5
|f'u|^2 + |f'v|^2 = (3f1/5 + 4f2/5)^2 + (4f1/5 - 3f2/5)^2
= (f1)^2 +(f2)^2 = |▽f|^2 得證
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◆ From: 118.161.144.169
※ 編輯: LeoRen 來自: 118.161.144.169 (07/08 00:29)