لو سمحتم أبيكم تساعدون في إثبات معادلات لابلاس في الإحداثيات الكروية والأسطوانية مع شرح خطوات الاثبات بالتفصيل، تعبت وأنا أحاول في حلها وماقدرت أوصل للحل .
عرض للطباعة
لو سمحتم أبيكم تساعدون في إثبات معادلات لابلاس في الإحداثيات الكروية والأسطوانية مع شرح خطوات الاثبات بالتفصيل، تعبت وأنا أحاول في حلها وماقدرت أوصل للحل .
هاي المعادلة
In spherical coordinates, the scale factors are http://mathworld.wolfram.com/images/...es/Inline1.gif, http://mathworld.wolfram.com/images/...es/Inline2.gif, http://mathworld.wolfram.com/images/...es/Inline3.gif, and the separation functions are http://mathworld.wolfram.com/images/...es/Inline4.gif, http://mathworld.wolfram.com/images/...es/Inline5.gif, http://mathworld.wolfram.com/images/...es/Inline6.gif, giving a Stäckel determinant of http://mathworld.wolfram.com/images/...es/Inline7.gif.
The Laplacian is
To solve Laplace’s equation in spherical coordinates, attempt separation of variables by writing
Then the Helmholtz differential equation becomes
Now divide by http://mathworld.wolfram.com/images/...es/Inline8.gif,
The solution to the second part of (5) must be sinusoidal, so the differential equation is
which has solutions which may be defined either as a complex function with http://mathworld.wolfram.com/images/...es/Inline9.gif, ..., http://mathworld.wolfram.com/images/...s/Inline10.gif
or as a sum of real sine and cosine functions with http://mathworld.wolfram.com/images/...s/Inline11.gif, ..., http://mathworld.wolfram.com/images/...s/Inline12.gif
The radial part must be equal to a constant
But this is the Euler differential equation, so we try a series solution of the form
Then
This must hold true for all powers of http://mathworld.wolfram.com/images/...s/Inline13.gif. For the http://mathworld.wolfram.com/images/...s/Inline14.gif term (with http://mathworld.wolfram.com/images/...s/Inline15.gif),
which is true only if http://mathworld.wolfram.com/images/...s/Inline16.gif and all other terms vanish. So http://mathworld.wolfram.com/images/...s/Inline17.gif for http://mathworld.wolfram.com/images/...s/Inline18.gif, http://mathworld.wolfram.com/images/...s/Inline19.gif. Therefore, the solution of the http://mathworld.wolfram.com/images/...s/Inline20.gif component is given by
Plugging (17) back into (◇),
which is the associated Legendre differential equation for http://mathworld.wolfram.com/images/...s/Inline21.gif and http://mathworld.wolfram.com/images/...s/Inline22.gif, ..., http://mathworld.wolfram.com/images/...s/Inline23.gif. The general complex solution is therefore
where
are the (complex) spherical harmonics. The general real solution is
Some of the normalization constants of http://mathworld.wolfram.com/images/...s/Inline24.gif can be absorbed by http://mathworld.wolfram.com/images/...s/Inline25.gif and http://mathworld.wolfram.com/images/...s/Inline26.gif, so this equation may appear in the form
where
are the even and odd (real) spherical harmonics. If azimuthal symmetry is present, then http://mathworld.wolfram.com/images/...s/Inline27.gif is constant and the solution of the http://mathworld.wolfram.com/images/...s/Inline28.gifcomponent is a Legendre polynomial http://mathworld.wolfram.com/images/...s/Inline29.gif. The general solution is then
انا صراحة ماعرف شو لابلاس لاكن شفتلك من الانترنيت