Re: [複變] Laplace's equation
※ 引述《kuut (庫特)》之銘言：
: 1.Let f(re^(iθ))=u(r,θ)+iv(r,θ) be analytic on domain D show that
: the Laplace's equation in polar form i.e.
: => r^2*u_rr(r,θ)+ru_r(r,θ)+u_θθ(r,θ)=0
: 2.Show that u(r,θ) = r^n cos(nθ) is a harmonic function
: 拜託了QQ
: 一題500P
第二題 按照第一提公式算一下就是了...
u=^n cos (nθ)
=> u_r = n r^(n-1) cos(nθ)
u_rr = n (n-1) r^(n-2) cos(nθ)
u_θ = -n r^n sin(nθ)
u_θθ = -n^2 r^n cos (nθ)
=> r^2*u_rr(r,θ)+ru_r(r,θ)+u_θθ(r,θ)
n(n-1) r^n cos(nθ) + n r^n cos(nθ) - n^2 r^n cos(nθ) =0
第一題其實就導證Laplace operator in polar coordinate form...
書裡面翻翻肯定有證明的... 其實就是計算步驟..
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