文档介绍:该【微分几何彭家贵课后题答案 完整版2025 】是由【小屁孩】上传分享,文档一共【24】页,该文档可以免费在线阅读,需要了解更多关于【微分几何彭家贵课后题答案 完整版2025 】的内容,可以使用淘豆网的站内搜索功能,选择自己适合的文档,以下文字是截取该文章内的部分文字,如需要获得完整电子版,请下载此文档到您的设备,方便您编辑和打印。:..去留无意,闲看庭前花开花落;宠辱不惊,漫随天外云卷云舒。——《幽窗小记》可编辑修改习题一(P).设a(t)是向量值函数,证明:()a?常数当且仅当a(t),a?(t)?;()a(t)的方向不变当且仅当a(t)?a?(t)?。a??a???a(t),a(t)??()证明:常数常数常数??a?(t),a(t)???a(t),a?(t)????a(t),a?(t)????a(t),a?(t)??。()注意到:a(t)?,所以a(t)a(t)的方向不变?单位向量e(t)??常向量。a(t)a(t)若单位向量e(t)??常向量,则e?(t)??e(t)?e?(t)?。a(t)反之,设e(t)为单位向量,若e(t)?e?(t)?,则e(t)//e?(t)。由e(t)为单位向量??e(t),e(t)????e(t),e?(t)???e(t)?e?(t)。e(t)//e?(t)?从而,由??e?(t)??e(t)?常向量。e(t)?e?(t)?a(t)所以,a(t)的方向不变?单位向量e(t)??常向量a(t)a(t)?a?(t)d??e(t)?e?(t)?????()a(t)????a(t)a(t)dta(t)??d??a(t)?a?(t)??()?a(t)?a(t)??a(t)dta(t)a(t)?a(t)?a?(t)?。即a(t)的方向不变当且仅当a(t)?a?(t)?。补充:精选doc:..先天下之忧而忧,后天下之乐而乐。——范仲淹可编辑修改定理r(t)平行于固定平面?的充要条件是?r(t),r?(t),r??(t)??。证明:"?":若r(t)平行于固定平面?,设n是平面?的法向量,为一常向量。于是,?r(t),n????r?(t),n??,?r??(t),n???r(t),r?(t),r??(t)共面??r(t),r?(t),r??(t)??。"?":若?r(t),r?(t),r??(t)??,则r(t),r?(t),r??(t)共面。若r(t)?r?(t)?则r(t)方向固定,从而平行于固定平面?。若r(t)?r?(t)?,则r??(t)??r(t)??r?(t)。令n(t)?r(t)?r?(t),则n?(t)?r?(t)?r?(t)?r(t)?r??(t)?r(t)?r??(t)?r(t)???(t)r(t)??(t)r?(t)???(t)?r(t)?r?(t)???(t)n(t)?n(t)?n?(t)?,又n(t)??n(t)有固定的方向,又n(t)?r(t)?r(t)平行于固定平面。..。性质.()证明:设v?(x,x,x),v?(y,y,y),v?(z,z,z),v?v?(w,w,w),则ijk?yyyyyy?v?v?yyy?,,??w,w,w???zzzzzz??zzz?w?yz?yz,w?yz?yz,w?yz?yz,ijk?xxxxxx??左=v?(v?v)?xxx??,,?wwwwww??www??xw?xw,xw?xw,xw?xw???x[yz?yz]?x[yz?yz],x[yz?yz]?x[yz?yz],x[yz?yz]?x[yz?yz]???[xz?xz]y?[xy?xy]z,[xz?xz]y?[xy?xy]z,[xz?xz]y?[xy?xy]z???[xz?xz]y,[xz?xz]y,[xz?xz]y???[xy?xy]z,[xy?xy]z,[xy?xy]z???[xz?xz?xz]y,[xz?xz?xz]y,[xz?xz?xz]y???[xy?xy?xy]z,[xy?xy?xy]z,[xy?xy?xy]z??[xz?xz?xz]?y,y,y??[xy?xy?xy]?z,z,z???v,v?v??v,v?v?右()证明:设v?(x,x,x),v?(y,y,y),v?(z,z,z),v?(w,w,w),则精选doc:..学而不知道,与不学同;知而不能行,与不知同。——黄睎可编辑修改ijk?xxxxxx?v?v?xxx?,,??X,X,X???yyyyyy??yyy?X?xy?xy,X?xy?xy,X?xy??zzzzzz?v?v?zzz?,,??Y,Y,Y?????www?Y?zw?zw,Y?zw?zw,Y?zw?zw.?左=?v?v,v?v??XY?XY?XY?(xy?xy)(zw?zw)?(xy?xy)(zw?zw)?(xy?xy)(zw?zw)?[xzyw?ywxz?ywxz?xzyw?xzyw?ywxz]?[xwyz?yzxw?yzxw?xzyw?xwyz?xwyz?yzxw]?[(xyzw?xyzw?xyzw)?xzyw?ywxz?ywxz?xzyw?xzyw?ywxz]?[(xyzw?xyzw?xyzw)?xwyz?yzxw?yzxw?xwyz?xwyz?yzxw]?(xz?xz?xz)(yw?yw?yw)?(xw?xw?xw)(yz?yz?yz)=?v,v??v?v???v,v??v?v??右()证明:设v?(x,x,x),v?(y,y,y),v?(z,z,z),,则ijk?xxxxxx?v?v?xxx?,,??X,X,X???yyyyyy??yyy?X?xy?xy,X?xy?xy,X?xy?xy??v,v,v???v,v?v??zX?zX?zX?z(xy?xy)?z(xy?xy)?z(xy?xy)?(zxy?yzx?xyz)?(zyx?xzy?yxz)同理,ijk?zzzzzz?v?v?zzz?,,??Y,Y,Y???xxxxxx??xxx?Y?zx?zx,Y?zx?zx,Y?zx?zx??v,v,v???v,v?v??yY?yY?yY?y(zx?zx)?y(zx?zx)?y(zx?zx)?(zxy?yzx?xyz)?(zyx?xzy?yxz)??v,v,v?精选doc:..非淡泊无以明志,非宁静无以致远。——诸葛亮可编辑修改ijk?yyyyyy?v?v?yyy?,,??Z,Z,Z???zzzzzz??zzz?Z?yz?yz,Z?yz?yz,Z?yz?yz??v,v,v???v,v?v??xZ?xZ?xZ?x(yz?yz)?x(yz?yz)?x(yz?yz)?(zxy?yzx?xyz)?(zyx?xzy?yxz)??v,v,v?所以,?v,v,v???v,v,v???v,v,v?。?f?f?f???证明:()??(?f)???(,,)??x?y?z?x?y?z?f?f?f?x?y?z????f????f????f????f????f????f?????????,?????,????????y??z??z??y??z??x??x??z??x??y??y??x????f?f?f?f?f?f????,?,???(,,)?.??y?z?z?y?z?x?x?z?x?y?y?x?ijk?????R?Q?P?R?Q?P?证明:()??,??F????,????,??,?,????x?y?z??y?z?z?x?x?y?PQR???R?Q????P?R????Q?P??????????????x??y?z??y??z?x??z??x?y??R?Q?P?R?Q?P???????.?x?y?x?z?y?z?y?x?z?x?z?y?O;e,e,e???,,?.设是正交标架,是的一个置换,证明:??()O;e,e,e是正交标架;?()?()?()????()O;e,e,e与O;e,e,e定向相同当且仅当?是一个偶置换。?()?()?()()证明:当i?j时,?(i)?(j)??e,e??;??(i)?(j)当i?j时,?(i)??(j)??e,e??,?(i)?(j)精选doc:..志不强者智不达,言不信者行不果。——墨翟可编辑修改?O;e,e,e?所以,是正交标架。?()?()?()()证明:A)当??()??()?,?()?,?()????????????e,e,e??e,e,e???e,e,e?,det??;?()?()?()????????????B)当??()??()?,?()?,?()????????????????e,e,e?e,e,e?e,e,e,det??;?()?()?()????????????C)当??()??()?,?()?,?()????????????e,e,e??e,e,e???e,e,e?,det??;?()?()?()????????????????D)当??()?()(),此时,O;e,e,e?O;e,e,e;?()?()?()E)当??()?()()??()?,?()?,?()?,??????????e,e,e??e,e,e???e,e,e?,det?;?()?()?()????????????F)当??()?()()??()?,?()?,?()?,??????????????e,e,e?e,e,e?e,e,e,det?.?()?()?()????????????????所以,O;e,e,e与O;e,e,e定向相同当且仅当?是一个偶置换。?()?()?()习题二(P).求下列曲线的弧长与曲率:()y?axxx解:r(x)?(x,ax)?r?(x)?(,ax)?l(x)??r?(t)dt???atdt精选doc:..去留无意,闲看庭前花开花落;宠辱不惊,漫随天外云卷云舒。——《幽窗小记》可编辑修改令|a|t?tan?,?at?sec?,则??atdt=?sec?d??Ia|a|I?sec?d???(sec?tan??sec?)d???tan?dsec??sec?d??tan?sec???sec?d???sec?d??tan?sec??I??sec?d??I?[tan?sec??ln|sec??tan?|]?C???|a|t?at?ln|a|t??at?C所以,??atdt??=?sec?d??I?|a|t?at?ln|a|t??at?Ca|a||a|xx???l(x)??r?(t)dt???atdt?|a|x?ax?ln|a|x??ax|a|.设曲线r(t)?(x(t),y(t)),证明它的曲率为x?(t)y??(t)?x??(t)y?(t)?(t)?.??(x?)?(y?)证明:r(t)?(x(t),y(t))?r?(t)?(x?(t),y?(t))?r??(t)?(x??(t),y??(t))drdtdt?t(s)??r?(t)?(x?(t),y?(t))dsdsdsdt?n(s)?(?y?(t),x?(t))dsdrd?dt??dt?dt?t(s)??r?(t)?r??(t)?r?(t)????dsds?ds??ds?ds?t(s)??(s)n(s)?dt?dtdt?r??(t)?r?(t)??(s)(?y?(t),x?(t))???ds?dsds??dt?dtdt?x??(t)?x?(t)???(s)y?(t)????ds?dsds??dtdtdt???y??(t)???y?(t)??(s)x?(t)???ds?dsds精选doc:..不飞则已,一飞冲天;不鸣则已,一鸣惊人。——《韩非子》可编辑修改dtdtdtdt????x??(t)?x?(t)y??(t)?y?(t)?????ds?ds?ds?ds??(s)???dtdty?(t)x?(t)dsds?dt??dt?x?(t)y??(t)?x??(t)y?(t)?????ds??ds?x?(t)y??(t)?x??(t)y?(t)????dt??ds(y?)?(x?)(y?)?(x?)dsdtds由?|r?(t)|?(x?)?(y?)dtx?(t)y??(t)?x??(t)y?(t)??(s)?,即??(y?)?(x?)x?(t)y??(t)?x??(t)y?(t)?(t)?。????(y)?(x).设曲线C在极坐标下的表示为r?f(?),证明曲线C的曲率表达式为dfdf??f(?)??f(?)???d??d??(?)?.?????df??????f(?)??????d?????证明:x?rcos??f(?)cos?,y?rsin??f(?)sin??r(?)?(f(?)cos?,f(?)sin?)?r?(?)?(f?(?)cos??f(?)sin?,f?(?)sin??f(?)cos?)r??(?)?(f?(?)cos??f(?)sin?,f?(?)sin??f(?)cos?)?(f??(?)cos??f?(?)sin??f(?)cos?,f??(?)sin??f?(?)cos??f(?)sin?)所以,x??f?(?)cos??f(?)sin?;y??f?(?)sin??f(?)cos?;x???f??(?)cos??f?(?)sin??f(?)cos?;y???f??(?)sin??f?(?)cos??f(?)sin?。因此,精选doc:..先天下之忧而忧,后天下之乐而乐。——范仲淹可编辑修改x?y???x??y???f?(?)cos??f(?)sin???f??(?)sin??f?(?)cos??f(?)sin????f?(?)sin??f(?)cos???f??(?)cos??f?(?)sin??f(?)cos???f(?)??f?(?)??f(?)f??(?)(y?)?(x?)??f?(?)cos??f(?)sin????f?(?)sin??f(?)cos????f?(?)???f(?)?dfdf??f(?)??f(?)??x?(?)y??(?)?x??(?)y?(?)?d??d???(?)??.??(x?)?(y?)?????df??????f(?)?????d??????????.求下列曲线的曲率与挠率:a()r(t)?(at,alnt,)(a?)taaaaaa解:r?(t)?(a,,?),r??(t)?(,?,),r???(t)?(,,?);ttttttijkaa?aaa??r?(t)?r??(t)?a???,?,????tt?ttt?aa?ttaaaaa???r?(t)?r??(t)?????t?t?t?tttttaaa??r?(t)?a???t?tttaaaaaa?r?,r??,r??????(,?,?),(,,?)??。tttttt所以,aa????t?t?r?(t)?r??(t)ttt?(t)????;r?(t)aa????????at?t?t???t?t?精选doc:..天将降大任于斯人也,必先苦其心志,劳其筋骨,饿其体肤,空乏其身,行拂乱其所为。——《孟子》可编辑修改????????r,r,ra?a?t???(t)???t???。r?(t)?r??(t)tt????at?.证明:E的正则曲线r(t)的曲率与挠率分别为r?(t)?r??(t)?r?,r??,r?????(t)?,?(t)?。r?(t)r??r??drdrdtdt证明:??t(s)?r(s)?r?(t)dsdtdsdsdtdt???t(s)?r??(t)???r?(t)?ds?dsdtdtdtdt???t(s)?r???(t)???r??(t)?r?(t)?ds?dsdsds根据弗雷内特标架运动方程?t?????t?d??????n????n,得:??????ds???????b??????b?t(s)??(s)n(s)?n(s)?t(s)?b(s)?t(s)?n(s)??t(s)?t(s)??(s?(s?dt?dtdt??????r?(t)??r??(t)?r?(t)???????(s)?ds?ds?ds?????dt????r?(t)?r??(t)????(s)?ds?dt?????r?(t)?r??(t)????(s)?ds??r?(t)?r??(t)?r?(t)?r??(t)?dt???(s)??r?(t)?r??(t)??????ds?dsr?(t)?????dt?t(s)??(s)n(s)?t(s)??(s)n(s)??(s)n(s)由n(s)=??(s)t(s)??(s)b(s)?t(s)??(s)n(s)??(s)???(s)t(s)??(s)b(s)???(s)n(s)??(s)t(s)??(s)?(s)b(s)??t(s),b(s)???(s)?(s)精选doc:..乐民之乐者,民亦乐其乐;忧民之忧者,民亦忧其忧。——《孟子》可编辑修改?dt?dtdtdt?dt?因为?t(s),b(s)???r???(t)?r??(t)?r?(t),?r?(t)?r??(t)???????ds?dsdsds?(s)?ds?dt??=?r?,r??,r???????(s)?ds?dt?r?,r??,r????dt?r?,r??,r????????所以,?(s)?(s)=?r?,r??,r??????(s)=?。?????(s)?ds??(s)?ds?r??r??.证明:曲线???(?s)(?s)s?r(s)?,,(??s?)??????以s为弧长参数,并求出它的曲率,标架。???(?s)(?s)?证明:)r?(s)?,?,(??s?)??????(?s)(?s)所以,r?(s)????(??s?)?该曲线以s为弧长参数。?????(?s)(?s)?t(s)?r??(s)?,,(??s?)??????(?s)?(?s)???(s)???(?s)???n(s)?t(s)?(s)??(?s)(?s),(?s)(?s),???ijk(?s)(?s)?b(s)?t(s)?n(s)??(?s)(?s)(?s)(?s)?????(?s)(?s),(?s)(?s),(?s)?????s?s?由n(s)???,,?及??s?s?精选doc:..子曰:“知者不惑,仁者不忧,勇者不惧。”——《论语》可编辑修改??b(s)???(?s)(?s),(?s)(?s),(?s)?得???(s)??n(s),b(s)???s?s??????,,,?(?s)(?s),(?s)(?s),(?s)???????s?s????s?s??(?s)(?s)??(?s)(?s)?s?s?(?s)(?s)?(?s)(?s)?(?s)所以,)?(s)?,(??s?);?(s)??(?s),(??s?)。(?s)?r(s);t(s),n(s),b(s)?)标架是,其中???(?s)(?s)?t(s)?,?,(??s?),????????n(s)??(?s)(?s),(?s)(?s),?(??s?),????b(s)???(?s)(?s),(?s)(?s),(?s)?(??s?)。??.设(X)?XT?P是E中的一个合同变换,detT??。r(t)是E中的正则曲线。求曲线r?r与曲线r的弧长参数、曲率、挠率之间的关系。ttd(r)td(rT?P)tt解:()S(t)??r?(?)d???d???d???r?(?)Td??r?(?)d??S(t)d?d?可见,r?r与曲线r除相差一个常数外,有相同的弧长参数。r?(t)?r??(t)r?(t)T?r??(t)T()?(t)??r?(t)r?(t)Tsgn(detT)?r?(t)?r??(t)?Tr?(t)?r??(t)????(t)r?(t)r?(t)可见,r?r与曲线r有相同的曲率。精选doc:..英雄者,胸怀大志,腹有良策,有包藏宇宙之机,吞吐天地之志者也。——《三国演义》可编辑修改?r?,r??,r?????r?T,r??T,r???T??r?T,r??T?r???T?()?(t)???r?(t)?r??(t)r?(t)T?r??(t)Tr?(t)?r??(t)?r?T,sgn(detT)(r???r???)T??r?T,sgn(detT)(r???r???)T???sgn(detT)r?(t)?r??(t)r?(t)?r??(t)?r?T,(r???r???)T??r?,(r???r???)??sgn(detT)?sgn(detT)r?(t)?r??(t)r?(t)?r??(t)?r?,r??,r?????r?,r??,r?????sgn(detT)?????(t)r?(t)?r??(t)r?(t)?r??(t)可见,r?r与曲线r的曲率相差一个符号。a.()求曲率?(s)?(s是弧长参数)的平面曲线r(s)。a?s解:设所求平面曲线r(s)??x(s),y(s)?因为s是弧长参数,所以|r?(s)|???x?(s)???y?(s)???可设x?(s)?cos?,x?(s)?sin?,由曲率的定义,知d?aaas??(s)??d??ds????ds?arctandsa?sa?sa?sass?x?(s)?cos(arctan),x?(s)?sin(arctan)aasx(s)??cos(arctan)ds??dsas?tan(arctan)a??ds?a?ds?aln(s?a?s)sa?s?assy(s)??sin(arctan)ds???cos(arctan)dsaa???ds???dssssec(arctan)?tan(arctan)aas??ds?a?sa?s??所以,所求平面曲线r(s)?aln(s?a?s),a?s)。.证明:曲线r(t)?(t?sint,cost,t?sint)与曲线精选doc:..操千曲尔后晓声,观千剑尔后识器。——刘勰可编辑修改ttr(t)?(cos,sin,?t)是合同的。证明:)对曲线C:r?r(t)作参数变换t?u,则r?(cosu,sinu,?u)。可知C是圆柱螺线(a?,b??),它的曲率和挠率分别为??,???。因此,只要证明曲线C:r?r(t)的曲率??,挠率???,从而根据曲线论基本定理,它们可以通过刚体运动彼此重合。)下面计算曲线C的曲率?与挠率?。由r?(t)?(?cost,?sint,?cost)?|r?(t)|?,进而r??(t)?(?sint,?cost,sint)?r?(t)?r??(t)?(cost?,?sint,??cost)??(?cost,sint,?cost)|r?(t)?r??(t)|????。r???(t)?(?cost,sint,cost)??r?(t),r??(t),r???(t)???????。.证明:?(s)?是连续可微函数,则()存在平面E的曲线r(s),它以s为弧长参数,?(s)为曲率;()上述曲线在相差一个刚体运动的意义下是唯一的。证明:先证明(),为此考虑下面的一阶微分方程组?dr?e(s)?ds??de(.)???(s)e(s)ds??de???(s)e(s)??ds??给定初值r,e,e,其中e,e是E中的一个与自然标架定向相同的正交标架,以及??s?(a,b),则由微分方程组理论得,(.)有唯一一组解r(s);e(s),e(s)满足初始条件:????r(s);e(s),e(s)|?r;e,e。s?s??若r(s)为所求曲线,则e(s),e(s)标架。因此,我们首先证明??e(s),e(s)?s?(a,b)均是与自然定向相同的正交标架。将微分方程组(.)改写成精选doc:..百学须先立志。——朱熹可编辑修改de(.)i??ae(s),i?,dsijjj?其中??(s)???a?。??ij???(s)??是一个反对称矩阵,即a?a?i,j?,.令ijji(.)g(s)??e(s),e(s)?(?g)i,j?,.ijijij对(.)求导,并利用(.)有:dddd(.)g(s)??e(s),e(s)???e(s),e(s)???e(s),e(s)?dsijdsijdsijidsj??e(