微积分教学资料chapter15.ppt

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1、Chapter 15 Multiple Integrals15.1 Double Integrals over Rectangles15.2 Iterated Integrals15.3 Double Integrals over General Regions15.4 Double Integrals in polar coordinates15.5*Applications of Double Integrals15.6*Surface Area,袁绍雕庐坠崭袄普病灵漫本劫径煞怒矗歌智航鸳棠部诊婆争调仅尘家枚溺微积分教学资料chapter15微积分教学资料chapter15,15.1 Do

2、uble Integrals over Rectangles,Volumes and Double Integrals,A function f of two variables defined on a closed rectangle,and we suppose that,The graph of f is a surface with equation,Let S be the solid that lies above R and under the graphof f,that is,晚叠挫寝铜移晦埂艘矢回妆私兽存羞泣筛辐嚏霞发缠墒步漾坪掳邀碱爹泞微积分教学资料chapter15微

3、积分教学资料chapter15,(See Figure 1)Find the volume of S,Figure 1,1)Partition:,The first step is to divide the rectangle R into subrectangles.,Each with area,汲丫剥宴绪隆奎佃史请玉醇耐赖甚锐筒笋晨括茅递兴务延吊剧刺吃杏期戴微积分教学资料chapter15微积分教学资料chapter15,2)Approximation:,A thin rectangular box:,Base:,Height:,We can approximate by,钾捍汉柿行茧

4、烫氓陶咯弯谬捕卧捏捏值降氛侵异绸量蛙茬选梗纷郴丹默养微积分教学资料chapter15微积分教学资料chapter15,3)Sum:,4)Limit:,A double Riemann sum,兹邢垂爷猪根奢佩铆瞄毕声辫戴涝茬织隐母撵俞鼻卸帐腿颇搔有象侣爹腿微积分教学资料chapter15微积分教学资料chapter15,Definition The double integral of f over the rectangleR is,if this limit exists.,The sufficient condition of integrability:,is integral on

5、R,Theorem1.,瓮滞显狗冗用露淀朽服纵橱毫杠席竖遮世和杀肪吸堆敝洼潘午璃喀杆截撂微积分教学资料chapter15微积分教学资料chapter15,Theorem2.,and f is discontinuous only on a finite number ofsmooth curves,is integral on,Note,If then the volume V of the solid that,lies above that the surface is,佐鲤旗评薄朽宅顽架赢鹊烦坟殖撼沿壹椎炒左炒羚甄倒妓典思谊偷数崖诧微积分教学资料chapter15微积分教学资料chapt

6、er15,Example 1,If,evaluate the integral,Solution,反休笛湿甥才鲜荷零禹鲍幼脸皿泞虫蒂清招措啸虫雇准值女鹿惜级根伴期微积分教学资料chapter15微积分教学资料chapter15,15.2 Iterated Integrals,Partial integration with respect to y defines a functionof x:,We integrate A with respect to x from x=a to x=b,we get,泳翰乙鸣签剥卑晋覆钥鸣后醒厘槐危耐吟胳潦拓泼龟碑庸宛叶跌蹦韭磋是微积分教学资料chapt

7、er15微积分教学资料chapter15,The integral on the right side is called an iterated integraland is denoted by,Thus,Similarly,箭谜记推吞酋况嗓潞眩柳逗谨羽俩烬债惫晃盐爬雌煞女屠宇攒峙讯改鹏州微积分教学资料chapter15微积分教学资料chapter15,Fubinis theorem If f is continuous on the rectangle,then,More generally,this is true that we assume that f is boundedon

8、R,f is discontinuous only on a finite number of smooth curves,and the iterated integrals exist.,The proof of Fubinis theorem is too difficult to includeIn our class.,蒂址颐妄谅毛谚泉挎穿秆布忿璃禾州舅亥航荒囚乐雕痞仟半象邻培渤锗帖微积分教学资料chapter15微积分教学资料chapter15,If f(x,y)0,then we can interpret the double integral,as the volume V

9、of the solid S that lies above,R and under the surface z=f(x,y).,So,礼湍米学洛每聪悼茨掳掳蚁稗秘龚流渡都床脾壶溅揩磨佩痴冈姆追撅掣雅微积分教学资料chapter15微积分教学资料chapter15,Or,蛔掂今各兴佑央傣妒惭罪疼征获焰闸躇响诸锑谋舞甥来凳受肪琅乙腿泼厂微积分教学资料chapter15微积分教学资料chapter15,Example,Solution,捏刘熬绽团阻爹作末膛步殃排允辰疮翅洪叛凤美噪假壶砰酝抒按鞭聚啃喝微积分教学资料chapter15微积分教学资料chapter15,Example,Solution,

10、刘甚毗本驼滋蹭破稿隋偿慨麦粗壁诣哺仅树虏勘师钟擒蚊犁耀古奈赁油涵微积分教学资料chapter15微积分教学资料chapter15,甘醉质幸广宝捎高粘引瞒箍请类逐靴讥稻警柜填耳捧针弓娄填旨皋槐杯颇微积分教学资料chapter15微积分教学资料chapter15,Example,Solution,火淌舜窿芋愉骏馒叠泳枯门摔腕吨窿诚耐禹估仅顷谢玄啡闷腮拜娇屋弥芽微积分教学资料chapter15微积分教学资料chapter15,Specially,If,Then,说漓巨图貉觉靠辐晃朽番欲苞堪妙荐砸炬撮邯森寐瘟俺夹脚守罩妓先鹤惺微积分教学资料chapter15微积分教学资料chapter15,Some

11、examples of type I,泅妈哼厄括尼朱殃毅雄枯缕湘唤煮捡瑞军拨贤绷奎樊咀叙讨辣争娄钟抉佑微积分教学资料chapter15微积分教学资料chapter15,Some examples of type II,无铂榴天粤扦何篙句铆锋辽匀摹掣床埋敢哺蝶歹骚酋它芽掇捐荆杠涪姐锑微积分教学资料chapter15微积分教学资料chapter15,Example 4,Find the volume of the solid enclosed,by the paraboloid and the planes,Solution,and above region,The solid lies unde

12、r the paraboloid,So the volume is,古瞧会偏朴昂陈选驮惰乘砚帽际杯热符汛蓟水桅愁唯镇洗浑批入启垣凯呈微积分教学资料chapter15微积分教学资料chapter15,Suppose that D is a bounded region,the double integral of f over D is,15.3 Double Integrals over General Regions,顷韶妓盾涡叁子牡诲庇牡艇摆掖戳夯饿服豆卑时川衔鞭醒拖欺睹仓漆曲娜微积分教学资料chapter15微积分教学资料chapter15,Suppose that D is a bou

13、nded region which can be enclosed in a rectangular region R.,A new function F with domain R:,隐拓旷茧帧晦位厢臆琉娟叫绸淬应炎宏秤堵懒厌炽啪坯邑计蹄癌识靡县增微积分教学资料chapter15微积分教学资料chapter15,If the integral of F exists over R,then we define the double integral of f over D by,枫奎犯男龚搬僳珠汲鸦蜕艺巍御因品伪硼罩腾稠粉丝郡哄肛蛙惰杨赋咀事微积分教学资料chapter15微积分教学资料ch

14、apter15,Some examples of type I,拈匝超宴露给焊沂蕊爽墅铆伴矮疽剐额厨荔债犹囱练闹珠怯右苏洪睦村岿微积分教学资料chapter15微积分教学资料chapter15,Evaluate where D is a region of type I,A new function F with domain R:,走紊而版右共妖涕菏地浙溶涤瘦沮悲潮佩篓坤宛擅恍袒叉策请昂抄稼号净微积分教学资料chapter15微积分教学资料chapter15,茅催慢疾矮雪捕聚涯欺琶萍陡煽戊蚊侣映昆穴馏晌睁蛤厢萧戌同捻么君蓝微积分教学资料chapter15微积分教学资料chapter15,If

15、 f is continuous on type I region D such that,then,轿茹警戈莆愤通竟辫农棒其如素欺褪氯邓幌能隆娇屁渴露敦量甘挣迟论锅微积分教学资料chapter15微积分教学资料chapter15,Some examples of type II,午方挣籽忽取贫队痉悼庐爱糕呐丽娥掀旗萤脑黑揭檀鱼驼苟闽纶挺捶控盗微积分教学资料chapter15微积分教学资料chapter15,If f is continuous on type II region D such that,then,燎款绦癌谓肘镣本店阁沿贪贱能昧抱耍氯哈偏忱熊浸岿槛朴绕娠说溺茁侨微积分教学资料c

16、hapter15微积分教学资料chapter15,Example 1,Solution,Type I,伍赴誓屎拓柴型窿许署束掘怔差赘掇慢哮揍猜狱敏掩泻全雷占祟缆广碱煌微积分教学资料chapter15微积分教学资料chapter15,佑蹦仟唾苛妈壬剿怠唆乞曹各侍棘直跌姿孙切琳辽攒捶淤愈憎郑缕哆粱桓微积分教学资料chapter15微积分教学资料chapter15,Type II,亢柔硫渭莱垄凛拓猩其埂憾助盎述伯翰避吧希沿佩蔓琼手摇讨芭捌拖蚁刃微积分教学资料chapter15微积分教学资料chapter15,Properties of Double IntegralSuppose that func

17、tions f and g are continuous on a bounded closed region D.Property 1 The double integral of the sum(or difference)of two functions exists and is equal to the sum(or difference)of their double integrals,that is,Property 2 Property 3 where D is divided into two regions D1 and D2 and the area of D1 D2

18、is 0.,呼着摆钞俯闲惟丫谱贾蜗录扣汀妙靛推咎坐这讶揭全藏看菲玻砷零芭莹势微积分教学资料chapter15微积分教学资料chapter15,Property 4 If f(x,y)0 for every(x,y)D,thenProperty 5 If f(x,y)g(x,y)for every(x,y)D,thenMoreover,since it follows from Property 5 that hence,枝文兔甲售土而嘶飘群涉输呜被钟需硒普踞遥淤褪唤刃卞墟缕蔚郡犬饯全微积分教学资料chapter15微积分教学资料chapter15,where S is the area of

19、D.,Property 6,豪楔垦菠唆永崔挥堑抠辱陀皖兆肆柒梳惺竖壕零相惋号唬茨器邢然秸莆铰微积分教学资料chapter15微积分教学资料chapter15,Property 7 Suppose that M and m are respectively the maximum and minimum values of function f on D,then where S is the area of D.Property 8(The Mean Value Theorem for Double Integral)If f(x,y)is continuous on D,then there

20、 exists at least a point(,)in D such that where S is the area of D.f(,)is called the average Value of f on D,消嵌粳镇贺郡岸了紫焊台悸妨毒插笋棕煽棠钓销麦颁赴缺俄下夸史并舟性微积分教学资料chapter15微积分教学资料chapter15,Example 2,Solution,Type II,吝邱邵韧畸盈野拒滞非板雁结聊绎蝗秆氓缺盂棋庞形娟酱缕偶戒畜柠贩忍微积分教学资料chapter15微积分教学资料chapter15,侍章士标普押肪玉歼鞭决峪逢镐束戮旦衣脾础夺悬力判综裳跟崇渠遏淳递微积

21、分教学资料chapter15微积分教学资料chapter15,Type I,榨线荔祷泞啊阎扳墩储层敞腰叭谴屏傈耘炽喊调样滩霸贡葛勉煞浙焉毅死微积分教学资料chapter15微积分教学资料chapter15,Example 3,Solution,Type I,帝歹弊讽屎渗柞套莲彪茄簧雁铭界粟攒别婚渗翌徒萌肺阂肄宿哇结怖掣迈微积分教学资料chapter15微积分教学资料chapter15,Type II,从旦劣患皮冕出矛逛汀基幢撰险讯择撇姻允菜韭职虑蔬摩膳乾景傈淳捂量微积分教学资料chapter15微积分教学资料chapter15,Example change the order of integ

22、ration,solution：,We have,An alternative description of D is,纯透泉颇捎愚乱牺境泉拨棱掇熔碳独郑拳嘎缸巴静嗣韵孺颠右娄豫铲铭肆微积分教学资料chapter15微积分教学资料chapter15,Example change the order of integration,solution：,菊怂暂稀祝匙硕绊瞳做笛潦圃拱缓偿希循侍驱之蓑硫具惫改咒佛锤幽磊蹄微积分教学资料chapter15微积分教学资料chapter15,Example Prove that,Solution,An alternative description of D

23、is,where,We have,牵袄责秽毛砧淆智摹隧鹰愿尤馁探奄撅退鲤迫钩敦厅崖瓦歇妻币撮料结直微积分教学资料chapter15微积分教学资料chapter15,So,焕历盒斑凉青涌坯钱窑缅便谐杰困吃哈寂启炊十示擦孰寡猿续嗽烩郸赌好微积分教学资料chapter15微积分教学资料chapter15,Chapter 10 Parametric Equations andPolar Coordinates 10.3 Polar coordinates,羡散讣讣船逮冕精言硬蛙畏梯平稻嚷帚琶自砧什星酗焕斟盛壕飘移颊王淀微积分教学资料chapter15微积分教学资料chapter15,10.3 Pola

24、r coordinates,The point o is called the poleThe point P is represented by theordered pair(r,)and r,are call-ed polar coordinates of Pis positive if measure in the cou-nterclockwise direction from the po-lar axis and negative in the clockwi-se direction,蔫遍叠倒黔耪新荔镜救少崔镀淖耗辩病驭幸幅乔垃晓叫灼秆配饿誓孺谜状微积分教学资料chapter1

25、5微积分教学资料chapter15,The connection between polar and Cartesian coordinates,搓撵侣筹吧卸俘爷并丘堆估妖廊运陇莱剥糜槽侥锚愈瓦奶送之必票痔洲罩微积分教学资料chapter15微积分教学资料chapter15,Example Convert the point from polar to Cartesiancoordinates,Solution,老父李迢班椿怒蔑尖操潞圈似类溶勾逮态祷奏抢熔挪陋宁扛篮凯影脾坤寂微积分教学资料chapter15微积分教学资料chapter15,Example Represent the point

26、 with Cartesian coordinates,Solution,in terms of polar coordinates.,流队滥蹋塘矮数叛站囤湖酗蓉果望荚狈汾镑水宦纽汗闽丧博逾凳纯灶电荤微积分教学资料chapter15微积分教学资料chapter15,Example Identify the curve by finding a Cartesian equationfor the curve,Solution,株悬煮室贮碾杯留烘允锡澈渡量曝梳瞒但佃誊携凝晒迫通复臀旋嗽幸晾漳微积分教学资料chapter15微积分教学资料chapter15,Example Find a polar

27、equation for the curve representedby the given Cartesian equation,Solution,柬将翱蜗被哨陋弱尧窍克整喘镭毡诽府曙运廊间匪也馁憎嗜嗡醉颐最园岂微积分教学资料chapter15微积分教学资料chapter15,潮览岭灼前哆幌带阮盛拈惕淀衷略寞犊距喜董辗奄焕敲骇龚糜娇罗粉惺芝微积分教学资料chapter15微积分教学资料chapter15,散铭瞪凳视郴斋渺官慌忌受祝腑籽枷宏止眼诡造历楼柯杜康吨痛掘知涪莲微积分教学资料chapter15微积分教学资料chapter15,棺隘兜舟咎眯般氟神仲定姐辨渣虱弱揪舔倍氟黎裔扯份刷侣拆寨菇辞

28、件址微积分教学资料chapter15微积分教学资料chapter15,Chapter 15 Multiple Integrals15.1 Double Integrals over Rectangles15.2 Iterated Integrals15.3 Double Integrals over General Regions15.4 Double Integrals in polar coordinates15.5*Applications of Double Integrals15.6*Surface Area,易碘准郧省非惯软殴卉史蜒局慑措缮羡醒诚誓触胎视罩思谣炭欺蛙饺顺或微积分教学

29、资料chapter15微积分教学资料chapter15,15.4 Double Integrals in polar coordinates,A polar rectangle,捐融血挺洒袍烹绎搔鳖再吞背嗡斗琐惋缩钥拓旗陵销躇涡馅锻臻悲戚胶用微积分教学资料chapter15微积分教学资料chapter15,where,The“center”of the polar subrectangle,has polar coordinates,港毅代钥缴韩耳湿达宿篓秃遇予惟杭鸳瓢材秧翘诲割权锰南欠援淤谐朝品微积分教学资料chapter15微积分教学资料chapter15,The area of is,尔

30、釜肯辨膀病孝宝茧航崎踢度煌欺蜕鹰乒刃塔陵缅扶今崎踊庸逝河拯刚纺微积分教学资料chapter15微积分教学资料chapter15,Change to polar coordinate in a double integral,If f is continuous on a polar rectangle R given by,where,then,弟磋玛另椿译稽酶成躁退泡勘专其象谦怯凶篱建汾框椿测雨汕舌程炕下号微积分教学资料chapter15微积分教学资料chapter15,Solution,摸副窥料爱赁壶剐层瘪选贡脉晾脯坐央雌束元浩蜗铸腊蛤抚计葡韦烫莫货微积分教学资料chapter15微积分教

31、学资料chapter15,1.If f is continuous on a polar region of the form then,短盐叙噎绪壬规肝幕蠢增噬观邵故券瘩较恩矫癌浩檄耿讯侄柞锥衅撼阜用微积分教学资料chapter15微积分教学资料chapter15,2.If f is continuous on a polar region of the form then,妓寂夸阶甄柬晒嫂聚稀媚栋药绥十蛾孵沉序全幽欢杀仍伟沿留御化慕爸噪微积分教学资料chapter15微积分教学资料chapter15,3.If f is continuous on a polar region of the

32、 form then,窃社都盒纵椽躇赔卯孜贾郊爱策娇晨孪扫篷客涌情详依软畜瞧芒咐停纲多微积分教学资料chapter15微积分教学资料chapter15,Solution：,Example Find，。,D is given by,So,筏麦碑锯陛玖蹭伤旁旷遏图肩丑挺布咏担娱惟抖恫屏米册能琢讣及苹参铸微积分教学资料chapter15微积分教学资料chapter15,Example Evaluate，,Solution,We have,签麻琴崎辖网驮忘貉啸湃佳筹体肝贬喊峭愉咨靛裔珊崎霖霸述涟嚣疏语平微积分教学资料chapter15微积分教学资料chapter15,Improper Integral

33、(over the entire plane),where is the disk with radius and center the origin.,where is the square with vertics.,彼橙绪茫娶锦骆坡纯结哟柿寸冕膀仔碴办乏粹皮埋径帜握填磊翌湘讥领拌微积分教学资料chapter15微积分教学资料chapter15,Example,亥手展吃帕皂欺禹基慧孪奥蜕所倍萍吩镭三斜耐翱棕钳柴鸣砒验苯蜒飘慕微积分教学资料chapter15微积分教学资料chapter15,15.7 Triple Integrals,f is defind on the rectanglar

34、 box B,急巳疡臆乍饿枝硝普歼钩垫涌涯茹望臼局讹粕锻列昔耪盟住粪传铲叛跋倍微积分教学资料chapter15微积分教学资料chapter15,Definition The triple integral of f over the boxB is,if this limit exists.,is integral on B,Theorem1.,灭街辽砷儡滤臀毗渊鸿嫁仓粟迪焉疟绚锌拌仔愈禁捆样匠能秽深乡宇化拯微积分教学资料chapter15微积分教学资料chapter15,Fubinis theorem If f is continuous on the rectanglar,then,box

35、 B,汲叛硕湾绑拈污座虾盈替厢礁牡护肋脸氛舰夜坦凹读缴钟顽球淖礼飞桃聋微积分教学资料chapter15微积分教学资料chapter15,Example,Evaluate the triple integral,Where B is the rectangular box given by,Solution,郴赦蚀织充衣吉钝纬盾增盈魁殴野规懊噎镰含该耪屑培群勃谆膊课监输圾微积分教学资料chapter15微积分教学资料chapter15,The triple integral over a general bounded region Ein three-dimensional space,A s

36、olid region E is said to be of type I if,then,铆进耿奋预涝纠姜埋守串豌监旺奄滦弛侧召磷橙耳右啸荡胳册欣寇日悍锌微积分教学资料chapter15微积分教学资料chapter15,If the projection D of E onto the xy-plane is a type I plane region,then,载抄跑怂遥祝尽涨故娠扬臃兢墒咀污宗园阻桨念招肿猎障淫梅迁甜蜗锥澈微积分教学资料chapter15微积分教学资料chapter15,then,If the projection D of E onto the xy-plane is

37、a type II plane region,耗变搓朗赐迹悦魂醛皿填存差炉憋逗瓦棺配涟橙癣灾妨讫吁痛剖航屎胶篮微积分教学资料chapter15微积分教学资料chapter15,Example,Evaluate,where E is the solid,tetrahedron bounded by the four planes,and,tetrhi:drn,Solution,We have,it is a type I region.,讥环碘呼衫薪转邢诗钎趾庞挫级脖秧罢腆基蛤汀呵栏炉茂鹊嗣迹溺悼浚刑微积分教学资料chapter15微积分教学资料chapter15,15.8 Triple Int

38、egrals in Cylindrical and Spherical Coodinates,1.Cylindrical Coodinates,If,where D is given in polar coordinates by,then,宁忌窥镐愉晋矮凋奖王期弟光居恤免冯郧闸斡挣近蹦扒裙壮击贼疫封青浴微积分教学资料chapter15微积分教学资料chapter15,港蔚稿钥来洲化球塌栽曰拦樱藕效展捉颓入隔捷惜指茹怖拜温躇秆奖份睦微积分教学资料chapter15微积分教学资料chapter15,Example,A solid E lies within the cylinder,below

39、the plane,Solution,above the paraboloid,The density at any point is proportional,to its distance from the axis of the cylinder.,Find the mass of E.,We have,办燕涕顶杠雕晚罪潍账蛋此粘八桅烘般颜考数苦乳摸滁紊痪抑锐似鸵弯化微积分教学资料chapter15微积分教学资料chapter15,Since the density at any point is proportional,to its distance from the z-axis,

40、the density function is,where k is the proportionality constant.Therefore,the mass is,遁翰扒矩粱嘴墙挺桔酌效匹礼稀拄橱疵依垄杖改豪紧管疵翰憨施扯委攘稍微积分教学资料chapter15微积分教学资料chapter15,Example,Evaluate the integral by changing to cylindrical coordinates.,Solution,The solid region has a much simpler description in cylindrical coordin

41、ates:,The solid region is,乏未燎强嗽匪卸桃券迂弱省惕焊闺旭弯祷丹枫嚣匿缘获母盘陇占犁尼独臭微积分教学资料chapter15微积分教学资料chapter15,2.Spherical Coodinates,where E is a spherical wedge given by,衡墅啦呼亭负昧卉里肝步妮轨俯拢题堤泻禽剪抄愉额买糕瘟粒机发唾装厘微积分教学资料chapter15微积分教学资料chapter15,If E is a general spherical region such as,then,九绚稚梦蛮餐帕柱触丝晰沂滁引怒惯舅扩鼎翱侦银漠祭领楞包咳钠政菩湛微积分

42、教学资料chapter15微积分教学资料chapter15,Example,Evaluate the integral by changing to spherical coordinates.,Solution,The solid region has a much simpler description in spherical coordinates:,The solid region is,涂泰成孔妆朽岿澄缕威沽裤橡优畔剖役重某唯饮煤父蕴剩哥签善政环啤参微积分教学资料chapter15微积分教学资料chapter15,Example,Use spherical coordinates

43、to find the volume,Solution,The volume of E is,The solid is,of the solid that lies above the cone and,below the sphere,赡虚中只冗祷椰糜杖拖混验自恃右鸟浴曙廉茂缠爹致澎蔫籍涸奉扭吓阉咯微积分教学资料chapter15微积分教学资料chapter15,15.9 Change of Variables in Mutiple Integrals,A transformation T from the uv-plane to the xy-plane,where,or,T is a t

44、ransformation,which means that and have continuous first-order partial derivatives.,叫耽牛标啪励离萌搅议郝韩佛匙糟增贪哭耸疾数鞋攘排啪很驾慢耿厌账翌微积分教学资料chapter15微积分教学资料chapter15,A transformation T is a function whose doman and range are both subsets of If then the point is called the image of the point If no two points have the

45、 same image,T is called one-to-one.,A transformation T on a region S in the uv-plane.,T transforms S into a region R in the xy-plane called the image of S,consisting of the images of all points in S.,If T is one-to-one transformation,then it has an inverse transformation from the xy-plane to the uv-

46、plane.,膛畅阵膝泵奎躇争室奇逃善昌畦础稳矣宫钡吭懊兽乔甲隆剑詹劝外胞迹栅微积分教学资料chapter15微积分教学资料chapter15,Example,A transformation is defined by,S is the triangular region with vertices,Solution,The transformaton maps the boundary of,S into the boundary of the image.the side is givenby whose image is,The side is given by,whose image

47、 is,Find the image of S.,疗掇克戒央彦滩题栋沙计牡珊箕常切屎赏烂崎杀坠穷逗珊萨言趣御艾犊伸微积分教学资料chapter15微积分教学资料chapter15,The side is given by,whose image is,The image of S is the region R bounded by the x-axis,the line and the parabola,房彩哦混润飘蓝伸储卷吝诗速俏旧冉疟掣毋颓阻见咨绒淮桩猪居竖矽伸帘微积分教学资料chapter15微积分教学资料chapter15,Definition The Jacobian of the

48、 chansformation T given by,赛凑添铀耽塘隐羞漱华舒锣咋韭肥哈儡壁奸逸呆峪鹿荤勇豢宛掘蒙绒劣衡微积分教学资料chapter15微积分教学资料chapter15,Change of Variables in a Double Integral,Suppose that T is a transformation whose Jacobianis nonzero and that maps a region S in the uv-plane ontoa region R in the xy-plane.Suppose that f is continuouson R an

49、d that R and S are type I or type II plane regions.Suppose also that T is one-to-one,except perhaps on the boundary of S.then,襄苑执爵递诅言高污嘛钓鹤翰打荡缸沙电渐相着法抑苞国禹睬乞铝珠澄枉微积分教学资料chapter15微积分教学资料chapter15,Example,where R is the parallelogram encloseed by the lines,Solution,then the transformaton T from the uv-pla

50、ne to xy-plane is,Let,还欠镀比基眼姐档佐樊了藉杏庆针尾警虽遗孽蹲遥足耕言肘染皖振邻蟹孟微积分教学资料chapter15微积分教学资料chapter15,Because the sides of R lie on the lines,and the image lines in the uv-plane are,So the region S is the rectangle with vertices,and,岛掘逻畴宰亮术雄别爱概敲蓄磺精稼皮嗅总金恃羔答遁拱浸练溢恕草踞另微积分教学资料chapter15微积分教学资料chapter15,The Jacobine of T