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Date: 23-7-2018
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Date: 13-7-2018
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Date: 12-7-2018
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On the surface of a sphere, attempt separation of variables in spherical coordinates by writing
(1)
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then the Helmholtz differential equation becomes
(2)
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Dividing both sides by ,
(3)
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which can now be separated by writing
(4)
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The solution to this equation must be periodic, so must be an integer. The solution may then be defined either as a complex function
(5)
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for , ..., , or as a sum of real sine and cosine functions
(6)
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for , ..., . Plugging (4) into (3) gives
(7)
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(8)
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which is the Legendre differential equation for with
(9)
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giving
(10)
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(11)
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Solutions are therefore Legendre polynomials with a complex index. The general complex solution is then
(12)
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and the general real solution is
(13)
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Note that these solutions depend on only a single variable . However, on the surface of a sphere, it is usual to express solutions in terms of the spherical harmonics derived for the three-dimensional spherical case, which depend on the two variables and .
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تفوقت في الاختبار على الجميع.. فاكهة "خارقة" في عالم التغذية
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أمين عام أوبك: النفط الخام والغاز الطبيعي "هبة من الله"
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قسم شؤون المعارف ينظم دورة عن آليات عمل الفهارس الفنية للموسوعات والكتب لملاكاته
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