北航张量分析课件03

张量分析Tensor第3课vzyx尹幸愉三馆[e-mail]能源与动力工程学院2aaeeeiiiia复习jjiibaggbajijibaggba适用右手系eijkijkabeee123123123aaabbbijk的偶排列123,231,312正循环排列ijk的奇排列132,321,213逆循环排列ijk的重复排列011ijkiijkkjeee3b复习3混合积与坐标系转向bca×ba,,abcabc123ijkijk123123111222333aaaabcbbbcccabcabcabc右手系混合积可用来表示行列式以及判断坐标系转向复习4坐标系转向e1e2e3e2e1e3•右手系•左手系,,gggGV12300VG1g2g3g21gg,,VeeeE12311•右手系•左手系复习复习5直角坐标系（左右手）e1e2e3e2e1e3iabeEjkVabijk,,abcEijkVabcijkeeeEVjkijki6指标一致性法则①表达式中各项（指标项）的自由标个数(为表达式展开的个数)、指标符号必须一致，各项指标的符号可同时替换。②各项中哑标必须成对出现（不能有两个以上相同的哑标符号），符号可任换，各项的展开项数由哑标数决定。③在指标符号不变的情况下指标式的代数运算规则与实数代数式运算规则相同(这是由代数运算的交换律、结合律、分配律所保证的。).仿代数特性复习7与的关系ijkijijklmkijmimj复习,,ai123jjjaaa=aa解析等8abceeeeeeeeeeeeeeeacbabcjjmmjmjmjmjmkmjmjkmjmijkiijkmjmijmijjiiiiabcabcabcabcabcabcabcacbabckkkkijmimj9第一章向量与坐标1.1向量与向量空间1.2点积与欧氏空间1.3叉积与轴向量1.4混合积与坐标系转向1.5并积与向量诱导空间1.6坐标系与坐标变换1.5并积与向量诱导空间10并积与向量诱导空间321321bbbaaa,,,,ab,,,,,,,,111213212223313233ijabababababababababab向量的解析定义（对应与自然基）有32个分量，按算法语言多重循环顺序排列（第一向量指标为外循环指标，第二向量指标为内循环指标）规定诱导向量满足向量基本运算法则，并可证明并积有如下运算法则bababaabbacbcacbacbcabacC诱导向量ab1.5并积与向量诱导空间11并积运算法则证明举例ab332313322212312111bababababababababa,,,,,,,,abbaba332313322212312111ababababababababab,,,,,,,,bababa333231232221131211bababababababababa,,,,,,,,1.5并积与向量诱导空间12bababababa),,,,,,,,(332313322212312111babababababababababa),,,,,,,,(332313322212312111babababababababababa),,,,,,,,(332313322212312111bababababababababa),,,,,,,,(332313322212312111bababababababababa1.5并积与向量诱导空间13jjiibaggbajijibaggCggijijC1212313121211111babababagggggggg33332332133132232222CCCCCggggggggggCjigg诱导向量并基诱导向量分量的集合为诱导向量空间C任意基由运算性质可以证明线性无关33332323131332322222bababababagggggggggg1221311321121111CCCCgggggggg问题5？ijC1.5并积与向量诱导空间14同理可定义,,,,,,,,,,,,,,,,,,,333233133323121311211111321332313322212312111cbacbacbacbacbacbacbacbacccbababababababababa＝cbacbakjikji333333233233133133311311211211111111cbacbacbacbacbacbacbagggggggggggggggggggggcba＋1.6坐标系与坐标变换15坐标系与坐标变换r坐标系的构成OP1x2x1x1i2i2xr21iixxx,ir坐标坐标系ii自然基坐标线矢径212211ysinyxycosyx:T变换可逆的条件是什么iOxryy2121222111yyxxyyxx:T,,ixjiiyxx:T正变换1.6坐标系与坐标变换16AC逆变换存在条件(可证)OOx1x2WWPPy1y2ABCBT:T-1:变换平面物理平面jiiyxx:Tjii1-xyy:T0yxyxyxyxyxJ22122111jidet坐标线y2y1y2y1Jacobi行列式1iiyxx2iiyxx212122211222111-xxyxxyxxyxxy:T,,1tg1.6坐标系与坐标变换17坐标系类型坐标系直角坐标系曲线坐标系直线坐标系斜角坐标系左手系右手系g3g1e1e2e3g2也称笛卡儿坐标系,基向量为常量也称一般坐标系,基向量为变量基向量为非标准正交基基向量为标准正交基1.6坐标系与坐标变换18直线变换333322311332332222112213312211111oxyyyxoxyyyxoxyyyxijjiioxyx与yi无关的常量2g1gO1i2iOrrorP1x2x2y2yorrr＋kkjjkkoxyxigikikjijkikoxyxijjiigiijijijiigig标准基点积为分量ikkijjikkoxyxiiigiiaaeeeiiiia1.6坐标系与坐标变换19旧系新系旋转OiieeOiieeOOiii平移i反常变换（改变转向，不连续）正常变换（连续）直线变换的分解变形OiigOg正交变换：平移，旋转，反射1.6坐标系与坐标变换20Jacobi行列式333322311332332222112213312211111oxyyyxoxyyyxoxyyyx332313332212312111yxyxyxyxyxyxyxyxyxyxJjidetdet112131122232132333ijjiioxyx)det(jiJacobi矩阵等于变换矩阵ijijxJy？1.6坐标系与坐标变换21微分运算指标式的仿代数特征仿代数特征——指标式的代数运算规则与实数代数式运算规则相同.微分运算仍满足指标式的仿代数特征ikikikijijjjxyyykkkjiyyxO1.6坐标系与坐标变换22的混合积表达式ijk,,,,,,gggggggggijk123ijkGVijk,,gggGijkVijkVG1g2g3g21gg体积的代数值1.6坐标系与坐标变换23基向量混合积关系式,,gggGijkVijkggijij,,,,,,det()det()gggggggggG123iijjkkijkijkGijkGijGjiG32VVVVVijkgijjii,,det,,gggiiiG123ji123IVJV（）基向量不共面,GIVVJ新旧1.6坐标系与坐标变换24正交变换旧系新系旋转O1e2e2e1eO1e2e2e1e反射OO1e2e22ee平移11eeiijijijiigig令iiijeiegijjiijjieeeejijieejeeiijeeeeeeeejjiiijiiji正变换逆变换问题6？1.6坐标系与坐标变换25正变换矩阵333232131332322212123132121111eeeeeeeeeeeeeeeeee111121312212223233132333eeeeee112233jijiee1.6坐标系与坐标变换26逆变换矩阵eeeeeeeeeeee111121231321212223233131232333eeeeee111213112122232231323333eeeeeeT112233eejijieeeTiijjjijjijiee1.6坐标系与坐标变换27ijkjkiijjkikeeeeeeikijjkijkjijkjikTikkjijeeeejkijikijikjkjijieeeejijiTikkjijETjeeiijTECABBB3ijikkjikkjk1CAA正交矩阵1.6坐标系与坐标变换28gijjiidetGjiIIVVJV（）jijiee正交基向量混合积关系式detEjiEEVVJV（）JJ同转向异转向29练习二2.1利用定义证明cbcacbajijieeiijjee2.2已知用指标法证明ijkjkiijjkik30练习二jijieea=aiei=lajlejiijjeejeeiij2.3已知证明ijijaajjiiaa312.4已知基向量为,,ggg1,,,,1,,求Jacobi行列式，并判别坐标系转向2.5将矩阵表达式写成指标式?EAx?TBTB?BA单位阵列