※ [本文轉錄自 Math 看板 #1EY0qaSc ]
作者 pandajohn (貓熊醬) 看板 Math
標題 [工數] 對三度空間向量做全微分之證明
時間 Sun Oct 2 14:20:27 2011
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1
若Φ(x,y,z)= -------- 則: PS.(r與R為位置向量
)
|r一R|
2
▽。(▽Φ)=▽ Φ= 一4πσ(r一R) PS.σ:delta函數
請問,該如何證明?
Your problem is the special case of the following
general problem.
※ 引述《JohnMash (Paul)》之銘言：
※ 引述《deepwoody (快回火星吧)》之銘言：
: 2 2
: －D▽ φ(ｒ) + K Dφ(ｒ) = Qδ(ｒ)
- ▽^2 φ + K^2 φ = 0 whenever r≠0
and
▽^2 φ = (1/r)(d^2/dr^2) (rφ) + ...
(d^2/dr^2) (rφ) + K^2 (rφ) = 0
rφ= u = a cos(Kr) + b sin(Kr) when r≠0
▽= e_r (d/dr) + e_θ(1/r)(d/dθ) + e_φ(1/r sinθ)(d/dφ)
▽φ= e_r (-u/r^2 + u'/r)
and
－∮_Ω ▽ φ(ｒ)‧dA + K^2 ∫_V φ(ｒ) dV = Q/D when the origin in V
= 0 when the origin not in V
Take V a infinitesimal ball with the center origin
^^^^^^^^^^^^^
－∮_Ω ▽ φ(ｒ)‧dA = -4πa
K^2 ∫_V φ(ｒ) dV =0
hence, a=-Q/(4πD), b CAN BE ANY ARBITRARY CONSTANT
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
φ = - Q [cos(Kr) + b sin(Kr)] / (4πDr)
But, sometimes, we need a spherical wave solution,
φ = - Q e^{iKr} / (4πDr)
ps. I repost it just because it is a good question to me.
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※ 編輯: JohnMash 來自: 112.104.129.8 (11/24 23:49)