=2πa(?1a222)ln(a?ρ)|20?πaln2.
16.设?是有界闭区域?的光滑边界曲面,函数u在?上有二阶连续偏导数,记
22?u??uu2u?x2???y2???z2.
试证明：???u?ndS?????udxdydz (n是的外法线方向向量).
??证 应用两种曲面积分的关系和高斯公式,得
???udS???(?u?u??n??xcos???ycos???u?zcos?)dS
=???u?u?u??xdydz??ydzdx??zdxdy
222 =???(?u?u??u2??x2??y?z2)dxdydz.
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