→ akrsw:(離題)不太明白為何用mutually而不用respectively？ 04/13 11:39
→ chungweitw:因為我用錯字了. 感謝. 04/13 11:52
→ chungweitw:然後, 應該是Laplace equation. 04/13 11:53
※ 引述《danny6430 (賴打)》之銘言：
: 這是我在我工數課本上面算conformal mapping裡面的題目
: 大概的意思就是要你求出空間中的電位
: 在兩個軸的電位是110V
: 還有一個X*Y=1這條雙曲線上的電位為60V
: 要你求出空間中的電位
: 答案是110-50X*Y
: 邊界條件帶進去正確...
: 可是完全沒有頭緒要怎麼算!
1.
w: z = x+iy -> z^2
w(z) = u + iv = z^2
=> u = x^2 - y^2
v = 2xy
Phi(x,y) = Psi(u,v) = Psi(w(z)),
whrer Phi and Psi are electrostatic potentials mutually
in the xy-plane and uv-plane.
2.
By the symmetry of the infinite plane,
the Laplacian equation in the uv-plane can be written as
( d^2/dv^2 ) Psi = 0,
the boundary condition of which is Psi(v=0) = 110.
This yields the solution Psi = av + 110, where a is a constant.
3.
xy = 1 => v = 2
Psi(v=2) = 60
=> 2a + 110 = 60
=> a = -25
=> Phi(x,y) = -25v + 100 = -50xy + 100.
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