Vector Calculus 4

(a)
C: (x,y,z)=(√5 cost, √5 sint, 2), t=0~2π
∫C F∙dr =∫[0~2π] [10sint cost(-√5 sint) -4√5 cost ] dt =0
S: R =(x,y,z)=(r cost, rsint, √(9-5r^2) ), r=0~√5, t=0~2π
ndS=(∂R/∂r)x(∂R/∂t) drdt=( 5r^2 cost /√(9-5r^2), 5r^2 sint /√(9-5r^2), r)drdt
∫∫S curf(F)∙ndS
=∫[0~2π]∫[0~√5] [10r^2 cost - 5r^2 sint/√(9-5r^2) + 2r^2 cost) dr dt
=0
故Stokes's thm成立

(b)
(i) ∂g/∂n = grad(g)∙n
div( f grad(g)) = grad(f)∙grad(g)+ f ∆g
By the div. thm,∙
∫∫S f(∂g/∂n)dS=∫∫∫V grad(f)∙grad(g) dV +∫∫∫V f∆g dV -------(A)
故得證
(ii)設 f=g, 代入(A)式,得
∫∫S f(∂f/∂n) dS=∫∫∫V ( |grad(f)|^2+ f∆f ]dV
0 = ∫∫∫V ( |grad(f)|^2 +λ f^2 ) dV
故 |grad(f)|=0 on V
then f= constant on V
而f=0 on S, 故f=constant=0 on V
註:∆g= div[grad(g)]=Laplacian of g

2011-07-31 23:36:02 補充：
Hah! Hah! Hah! 我也有點角色混淆不清吔!

哈哈..Lotus不是男的名字嗎?

2011-07-31 20:03:14 補充：
咁Rebecca你是男定女?

怎知是兄不是姊? .