[理學(xué)]多元積分學(xué)(編輯修改稿)
2025-03-17 23:10 本頁(yè)面
【文章內(nèi)容簡(jiǎn)介】 x(39。xf)x(f x2 ?????01)()11()(39。 2 ???? xexxfxxfxCee)x(f xx2 ??,1)x(fl i m0x???又,1C, ??故.xee)x(f xx2 ??所以，二 . 證明題 :,1||||,)(證明確定由區(qū)域?yàn)檫B續(xù)函數(shù)設(shè) ?? xyDuf例 1 dxxxfdxxxfd x d yfIxDyx)(a r c c o s4()()(211022)1???????????分析 : 要證明的等式右端是定積分 , 且被積函數(shù)中有 項(xiàng) , 故需將 看成一整體 .)( xf yx22?xy ?)(22,22)1,1(D12D11證明 : 采用極坐標(biāo) .1r,DDD,DD121111??? 分界線為在第一象限的部分為設(shè)將式中 r的換成 x,即得證 . ?? ??1D22 d x d y)yx(f4I由對(duì)稱性知 ]d x d y)yx(fd x d y)yx(f[41211 D22D22 ???? ?????????????? 4r1ar c c os211040]d)r(rfdrdr)r(rfd[4?? ????? 2110 dr)r(rf)r1a r c c o s4(dr)r(rf例 2 ??????? d x d yexfyxyfxf1)()(22,]1,0[)( 證明上可積在設(shè)證 )0(1!212 之間于介于 xxxexe x ???????)y(f)(f1)y(f)(f ???? xe x????????? ???1x1x)y(f)(f2222))y(f)(f1(yyx dxdyxdxdye?? ???? ???????? 1x1x 2222)y(f)x(fyyd x d yd x d y.Ryx:D)ba(R21d xd y)y()x()y(b)x(a)t(222D2??????????????其中為連續(xù)正值函數(shù)，證明設(shè)證明： 由積分區(qū)域 D關(guān)于 y=x對(duì)稱，所以 ，?? ???????????D Dd x d y)y()x()y(d x d y)y()x()x( 從而 ?? ??? ???Dd xd y)y()x( )y(b)x(a例 3 d xd y)y()x( )y()x()ba(D?? ??? ????? )ba(R21 2 ???.)()(1)(:,0)()(2],[abdxxfdxxfxfCxfbababa??????證明且設(shè)例 4 解 ： dxxfdxxfIbaba?? ?? )(1)(dyyfdxxfbaba?? ?? )(1)(d x d y)y(f1)x(fbyabxa?????????? ??Ddx dyxfyfyfxfI ))()()()((21???Dd x dy2212)( ab ??例 4’ .1dyedxe:],1,0[C)x(f 10)y(f10)x(f ??? ?? ?證明設(shè)dyedxe yfxf ?? ??10)(10)(:證????????1y01x0)()( dxd ye yfxf???????? ??1y01x0)()()()( )(21d x d yee xfyfyfxf1221 ?? ??Ddxd y例 5 ).0t(,dxxt)x(ft2dxxt)x(f:],1,0[C)x(f 1022221022 ?????????????? ??證明設(shè):證dxxt)x(fxtdx 102221022 ?? ????210222221022 dxxt1xt)x(fdxxt)x(f????????????????????? ??dxxt)x(ft1ar c t ant1 10222? ???dxxt)x(ft210222? ????例 6 ,)x(f 單調(diào)減少且恒大于零在上連續(xù)設(shè)函數(shù)?????1010210102dx)x(fdx)x(fdx)x(xfdx)x(xf證明：分析： ???? ?1021010102 dx)x(xfdx)x(fdx)x(xfdx)x(f只要證明???? ? 10 2101010 2 dy)y(yfdx)x(fdy)y(yfdx)x(f即證0d x d y)]y(f)x(f][y)y(f)x(f[I 10 10 ??? ? ?即證d x d y)]x(f)y(f][x)x(f)y(f[I 1010?? ? ?同理d x d y)]y(f)x(f][xy)[y(f)x(fI2 10 10 ??? ? ?于是0)]y(f)x(f)[xy(0f)x(f???? 可保證的單調(diào)性及由則本題得證 . 例 7 :,1yxD 22 試證明不等式為設(shè) ??.52d x d y)yx(s i n1 6 561D322 ????? ??證明 drrs i nr2d x d y)yx(s i nI103D322
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