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1:
Δ = nabla
Gauss' Theorem
2009-02-28 8:41 pm
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)
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)
回答 (1)
2009-03-01 3:11 am
✔ 最佳答案
(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
收錄日期: 2021-04-26 13:15:08
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