thhzpvx的部落格

標題:

Gauss' Theorem

發問:

R=(x,y,z) r = (x^2 + y^2 + z^2)^(1/2) i): Show that (Δ ? F) = 0 if F has the form kR/r^3 where r not equal to 0 (ii): Show that if F has the form kR/r^3 ∫∫ F ? ndS = 4pi where the boundary is the sphere of radius 1 center at (0,0,0) 更新: Δ = nabla

最佳解答:

• 日本agnes b 及 buberry一問
• c朗迷請入
• a-maths_0@1@
• A.maths .......Vector@1@
• K至Z的食物名 (快~~~!)@1@

此文章來自奇摩知識+如有不便請留言告知

(i) R=(x,y,z) ?. F =k[?/?x(x/r^3)+?/?x(x/r^3)+?/?x(x/r^3)] =k{[r^3-x3r^2(x/r)]/r^6+[r^3-y3r^2(y/r)]/r^6+[r^3-z3r^2(z/r)]/r^6} =k{[r^3-3x^2r]/r^6+[r^3-3y^2r]/r^6+[r^3-3z^2r]/r^6} =k{[3r^3-3r^3]/r^6} =0 (ii) Since n=(x/r,y/r,z/r), F ? n=(x^2+y^2+z^2)/r^4 ∫∫ F ? ndS =∫∫(x^2+y^2+z^2)/r^4 dS =r^(-2) ∫∫ dS =4 pi [The boundary is the sphere of radius r center at (0,0,0)] ∫∫ F ? ndS

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