(完整版)张量分析中文翻译

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(完整版)《张量分析》报告

一爱因斯坦求和约定1.1指标变量的集合：n n y y y x x x ,...,,,...,,2121表示为：n j y n i x j i ...,3,2,1,,...,3,2,1,==写在字符右下角的指标，例如xi 中的i 称为下标。

写在字符右上角的指标，例如yj 中的j 称为上标；使用上标或下标的涵义是不同的。

用作下标或上标的拉丁字母或希腊字母，除非作了说明，一般取从1到n 的所有整数，其中n 称为指标的范围。

1.2求和约定若在一项中，同一个指标字母在上标和下标中重复出现，则表示要对这个指标遍历其范围1，2，3，…n 求和。

这是一个约定，称为求和约定。

例如：333323213123232221211313212111bx A x A x A b x A x A x A bx A x A x A =++=++=++筒写为：ijijbx A =ｊ——哑指标ｉ——自由指标，在每一项中只出现一次，一个公式中必须相同遍历指标的范围求和的重复指标称为“哑标”或“伪标”。

不求和的指标称为自由指标。

1.3 Kronecker-δ符号（克罗内克符号）和置换符号Kronecker-δ符号定义j i ji ij ji ≠=⎩⎨⎧==当当01δδ置换符号ijkijk e e =定义为:⎪⎩⎪⎨⎧-==的任意二个指标任意k j,i,当021)(213,132,3的奇置换3，2，1是k j,i,当112)(123,231,3的偶置换3，2，1是k j,i,当1ijk ijke ei,j,k 的这些排列分别叫做循环排列、逆循环排列和非循环排列。

置换符号主要可用来展开三阶行列式：231231331221233211231231133221332211333231232221131211a a a a a a a a a a a a a a a a a a a a a a a a a a a a ---++==因此有：ijmjimii i i jijAA aa a a a ==++=δδδδδ332211kijjkiijkkjiikjjikijkee e e e e e ==-=-=-=同时有：ijjijij iiiijijijkj ikilkljkijjjiiijijijkjikiie e aa aa a a a aa δδδδδδδδδδδδδδδδδδδ=⋅=++=========++=332211332211331001010100131211232221333231321333222111321321321-=====δδδδδδδδδδδδδδδδδδδδδδδδδδδe e k j i k j i k j i k k k j j j i i i ijk333222111321321321r q p r q p r q p k k k j j j i i i pqr ijke e δδδδδδδδδδδδδδδδδδ⋅=ipp i p i p i p i δδδδδδδδδ==++11332211krkqkpjrjqjpiriqippqrijke e δδδδδδδδδ=jqirjriqjrjqiriqkqrijke e kp δδδδδδδδ-===321321322311332112312213322113312312332211333231232221131211k j i ijkkjiijkaa a e a a a e aa a a a a a a a a a a a a a a a a aaaa a aaa a A ==---++==Kronecker-δ和置换符号符号的关系为：itjsjtiskstkije e δδδδ-=二张量代数2.1张量的加法(减法)两个同阶、同变异(结构) 的张量可以相加(或相减)。

机械工程学专业词汇英语翻译（T）

t beam t 型梁 t type tail t 型尾翼 tab 补翼 tachometer 转速表 tachymeter 视距仪 tackle block 滑车组 tactile sensor 接触式传感器触觉感受器 tail 尾部 tail area 尾部⾯积 tail band 拖尾带 tail shock wave 尾部激波 tail spin 尾旋 tail water 尾⽔ tailplane ⽔平尾翼 tailrace 尾⽔沟 take off distance 起飞距离 taking of power 功率提取 taking off speed 起飞速度 talweg 深泓线 tammann principle 塔曼原理 tamping 捣固 tangent length 切线长度 tangent line 切线 tangent method 切向法 tangent modulus 切线模数 tangent plane 切平⾯ tangential acceleration 切向加速度 tangential component 切向分量 tangential displacement 切向位移 tangential force 切向⼒ tangential load 切向荷载 tangential pressure 切向压⼒ tangential resistance 切向阻⼒ tangential speed 切向速度 tangential stress 切向应⼒ tangential vector 切向量 tangential velocity 切向速度 tangential wind stress 切向风应⼒ tank 槽 taper of the wing 机翼根梢⽐ tare ⽪重 target ⽬标 tautochrone 等时降落轨迹 taxiing 滑⾏ taylor effect 泰勒效应 taylor expansion 泰勒展开 taylor flow 泰勒流 taylor formula 泰勒公式 taylor instability 泰勒不稳定性 taylor microscale 泰勒微尺度 taylor series 泰勒级数 taylor vortex 泰勒涡流 taylor vortex system 泰勒平⾏涡系 tear testing 撕裂试验 tearing 裂开 tearing instability 撕裂不稳定性 tearing mode crack 撕开型裂纹 technical diagnostics 诊断⼯程学 tectonic 构造的 tectonic analysis 构造分析 tectonic earthquake 构造地震 tectonic stress 构造应⼒ telecontrol 远程控制 telegage 遥测计 telemanipulator 遥控机械⼿ telemeasuring 遥测 telemeter 遥测计 telemetering 遥测 teleoperator 遥控机械⼿ telescopic 望远镜的 telethermometer 远程温度计 temper brittleness 回⽕脆性 temperature boundary layer 温度边界层 temperature coefficient 温度系数 temperature conductivity 温度传导率 temperature control 温度第 temperature correction 温度补偿 temperature curve 温度曲线 temperature difference 温度差 temperature distribution 温度分布 temperature drop 温度降差 temperature effect 温度效应 temperature fall 温度降差 temperature fluctuation 温度涨落 temperature grade 温度梯度 temperature lapse rate 温度下减率 temperature of equilibrium 平衡温度 temperature oscillation 温度振荡 temperature profile 温度剖⾯图 temperature range 温度范围 temperature reduction 温度下降 temperature relaxation 温度弛豫 temperature rise 温升 temperature stress 热应⼒ temperature stresses 热应⼒ temperature wave 温度波 tempering 回⽕ temporary hardness 暂时硬度 temporary strain 暂时应变 tenacity 粘性 tensile 拉的 tensile deformation 拉伸应变 tensile diagram 拉⼒图 tensile force 张⼒ tensile fracture 拉伸破坏 tensile impact strength 拉伸冲豢度 tensile load 拉伸载荷 tensile modulus of elasticity 拉伸弹性模量 tensile rigidity 抗拉刚度 tensile stiffness 抗拉刚度 tensile straining 抗张应变 tensile strength 抗拉强度 tensile stress 拉伸应⼒ tensile stress field 抗拉应⼒场 tensile test 张⼒试验 tensile test at elevated temperature ⾼温张⼒试验 tensile test piece 抗张试样 tensile testing machine 拉⼒试验机 tensile wave 张⼒波 tensile yield strength 抗拉屈服强度 tensiometer 张⼒计 tension 张⼒ tension diagonal 受拉斜杆 tension dynamometer 拉⼒测⼒计 tension field 张⼒场 tension fracture 拉伸破坏 tension impact testing 张⼒冲辉验 tension load 拉伸载荷 tension relief 张⼒消除 tension side 紧边 tension specimen 抗张试样 tension spring 拉簧 tension strain 拉伸应变 tensional strain 拉伸应变 tensional wave 张⼒波 tensor 张量 tensor analysis 张量分析 tensor coupling 张量耦合 tensor density 张量密度 tensor field 张量场 tensor force 张量⼒ tensor of finite rotation 有限转动张量 tensor of inertia 惯性张量 tensor potential 张量势 tensor product 张量积 tera 兆兆 terminal 端的 terminal ballistics 终段弹道学 terminal power 终端功率 terminal pressure 最终压⼒ terminal resistance 终端阻抗 terminal speed 终速度 terminal velocity 终速 test cock 试验旋塞 test condition 试验条件 test data 试验数据 test piece 试样 test point 试验点 test pressure 试验压⼒ test pulse 试验脉冲 test result 试验结果 test run 试车 test stand 试验台 tester 试验仪器 testing apparatus 试验装置 testing equipment 试验设备 testing load 试验荷重 testing machine 试验机 tetragonal crystal system 四⽅晶系 tetrahedral 四⾯的 tetrahedral angle 四⾯⾓ tetrahedral structure 四⾯体结构 textural stress 织构应⼒ thalweg 深泓线 theodolite 经纬仪 theorem 定理 theorem of corresponding state 对应态定理 theorem of equivalence 等效定理 theorem of minium strain energy 最⼩应变能定理 theorem of moments 矩定理 theorem of momentum 动量定理 theoretical ceiling 理论升限 theoretical curve 理论曲线 theoretical density 理论密度 theoretical mechanics 理论⼒学 theoretical model 理论模型 theoretical strength 理论强度 theoretical value 理论值 theoretical water power 理论⽔⼒ theory of dimension 量纲理论 theory of elasticity 弹性理论 theory of errors 误差理论 theory of failure 断裂理论 theory of heat 热理论 theory of liquids 液体理论 theory of non linear vibration ⾮线性振动理论 theory of nuclear forces 核⼒理论 theory of oscillations 振荡理论 theory of plasticity 塑性理论 theory of relativity 相对论 theory of similarity 相似理论 theory of structures 结构⼒学 theory of vibrations 振荡理论 thermal absorption 热吸收 thermal ageing 热时效 thermal analysis 热分析 thermal balance 热平衡 thermal behavior 热⾏为 thermal bending 热弯曲 thermal boundary layer 热边界层 thermal capacity 热容量 thermal conduction 热传导 thermal conductivity 导热性 thermal convection 热对流 thermal creep 热蠕变 thermal creep velocity 热蠕变速度 thermal cutting 热切割 thermal cycle 热循环 thermal deformation 热形变 thermal diffuse scattering 热扩散散射 thermal diffusion 热扩散 thermal diffusion coefficient 热扩散系数 thermal diffusion effect 热扩散效应 thermal diffusion flow 热扩散流 thermal diffusion potential 热扩散势 thermal diffusivity 热扩散率 thermal distribution 热分布 thermal disturbance 热扰动 thermal efficiency 热效率 thermal energy 热能 thermal equilibrium 热平衡 thermal equivalent 热当量 thermal expansion 热膨胀 thermal fatigue 热疲劳 thermal flux 热流 thermal flux vector 热通量⽮量 thermal inertia 热惯性 thermal instability 热不稳定性 thermal insulation 热绝缘 thermal load 热负载 thermal loss 热损失 thermal molecular flow 热分⼦流 thermal motion 热运动 thermal phenomenon 热现象 thermal plasma 热等离⼦体 thermal polymerization 热聚合 thermal pressure 热压⼒ thermal radiation 热辐射 thermal relaxation 热弛豫 thermal resistance 热阻 thermal response 热响应 thermal shock 热冲击 thermal stability 热稳定性 thermal stress 热应⼒ thermal transmission 传热 thermal unit 热单位 thermal vibration 热振动 thermal wave 热波 thermal wind 热风 thermic wind 热风 thermistor 热敏电阻器 thermo inelasticity 热滞弹性 thermobalance 热天平 thermocline 温度跃层 thermoconvection 热对流 thermocouple 热电偶 thermodiffusion 热扩散 thermodynamic 热⼒学的 thermodynamic efficiency 热⼒学的效率 thermodynamic equation 热⼒学⽅程 thermodynamic equation of state 热⼒学状态⽅程 thermodynamic factor 热⼒学因⼦ thermodynamic flux 热⼒学通量 thermodynamic function 热⼒函数 thermodynamic perturbation theory 热⼒学微扰理论 thermodynamic power cycle 热⼒学动⼒循环 thermodynamic pressure 热⼒学压强 thermodynamic principle 热⼒学原理 thermodynamic probability 热⼒学概率 thermodynamic process 热⼒学过程 thermodynamic quantity 热⼒学量 thermodynamic similarity 热⼒学相似 thermodynamic system 热⼒学系统 thermodynamical equilibrium 热⼒学平衡 thermodynamical potential 热⼒势 thermodynamics 热⼒学 thermoelastic attenuation 热弹性阻尼 thermoelastic coefficient 热弹性系数 thermoelastic constant 热弹性常数 thermoelastic coupling 热弹性耦合 thermoelastic damping 热弹性阻尼 thermoelastic effect 热弹性效应 thermoelastic material 热弹性材料 thermoelastic stress 热弹性应⼒ thermoelastic wave 热弹性波 thermoelasticity 热弹性 thermograph 温度记录器 thermohydrodynamic 热铃动⼒学的 thermohydrodynamic equation 热铃动⼒学⽅程 thermokinetics 热动⼒学 thermomechanical curve 热⼒学曲线 thermomechanical effect 热⼒学效应 thermomechanics 热⼒学 thermometer 温度计 thermometry 温度测量 thermomolecular pressure 热分⼦压⼒ thermoplastic material 热塑性材料 thermoplastic resin 热塑性⼫ thermoplasticity 热塑性 thermoresonance 热共振 thermoscope 测温计 thermosetting 热固的 thermosetting resin 热硬化⼫ thermosphere 热层 thermostat 恒温箱 thermostatics 热静⼒学 thermoviscoelastic material 热粘弹性材料 thermoviscometer 热粘度计 thermoviscous fluid 热粘性铃 thick plate 厚板 thick walled pipe 厚壁管 thick walled tube 厚壁管 thickness of shock layer 冲汇厚度 thickness of the shell 壳厚度 thiele transformation 梯勒变换 thin airfoil theory 薄翼理论 thin film 薄膜 thin lamination 薄层 thin layer 薄层 thin membrane 薄膜 thin plate 薄板 thin section 薄⽚ thin shell 薄壳 thin slab 壳 thin walled beam 薄壁梁 thin walled pipe 薄壁管 thin walled structures 薄壁结构 third boundary condition 第三类边界条件 third cosmic velocity 第三宇宙速度 third law of dynamics 动⼒学第三定律 thixotropy 触变性 thread tension 丝张⼒ three body problem 三体问题 three dimensional elasticity 三维弹性 three dimensional flow 三维流 three dimensional motion 三维运动 three dimensional strain 三维应变 three hinged arch 三铰拱 three moment equation 三弯矩⽅程 three point bending test 三点弯曲试验 threshold 阚界限 threshold energy 阈能 threshold field 阈场 threshold frequency 阈频率 threshold in fatigue 疲劳阈 threshold value 阈值 threshold velocity 阈速度 threshold wave number 阈波数 threshold wavelength 临界波长 throttle 节璃 throttle valve 节璃 through thickness 全厚度 throw 抛射 throw of eccentric 偏⼼距 throwing range 投掷距离 thrust 推⼒ thrust journal ⽌推轴颈 thrust line 推⼒线 thrust nozzle 推⼒喷管 thrust strength 推⼒强度 thunderstroke 雷击 tidal component 分潮 tidal current 潮流 tidal instability 潮汐不稳定性 tidal motion 潮了动 tidal power 潮汐⼒ tidal power station 潮⼒发电站 tidal wave 潮汐波 tidal wind 潮汐风 tide energy 潮能 tide generating forces 潮汐⼒ tide generating potential 引潮势 tides 潮汐 tie 连接杆 tightness 紧密性 tilt 倾斜 tilt angle 倾⾓ tilting level 微倾⽔准仪 tilting mechanism 倾翻机构 tilting moment 倾翻⼒矩 timber ⽊材 time average 时间平均 time average holography 时间平均全息照相术 time behavior 时间⾏为 time dependent method 时间相关法 time domain 时区域 time history 时间推移 time integral of force 冲⼒ time interval 时间间隔 time lag 时间延迟 time measurement 测时 time of collision 碰撞时间 time of operation 动妆间 time relay 时间继电器 time temperature equivalence principle 时温等效原理 tip 尖端 tip velocity ratio 叶尖速度⽐ to and fro motion 往复运动 tokamak 托卡马克 tolerance 公差 tolerance plug gage 极限塞规 toms effect 汤姆斯效应 top 蛇螺 top chord 上弦 top speed 速度 top view 上视图 toroidal magnetic field 环向磁场 toroidal surface 圆环⾯ torque 转矩 torque converter 液⼒变扭器 torque load 扭转负载 torque meter 扭矩计 torque moment 扭矩 torque of couple ⼒偶矩 torrent 急流 torsiometer 扭⼒计 torsion 扭转 torsion angle 扭转⾓ torsion balance 扭秤 torsion bar 扭杆 torsion constant 扭转常数 torsion curve 扭转曲线 torsion damper 扭转振动减震器 torsion dynamometer 扭转式测⼒计 torsion endurance test 扭转疲劳试验 torsion fatigue test 扭转疲劳试验 torsion frequency 扭振频率 torsion meter 测扭计 torsion moment 扭矩 torsion of bar 杆的扭转 torsion of cylinder 圆筒扭转 torsion pendulum 扭摆 torsion radius 扭转半径 torsion resistant 抗扭的 torsion rod 扭杆 torsion tensor 扭张量 torsion testing machine 扭转试验机 torsional center 扭转中⼼ torsional deflection 扭转变形 torsional elasticity 扭转弹性 torsional fatigue 扭转疲劳 torsional fissure 扭转裂缝 torsional flexibility 扭转挠性 torsional flow 扭转怜 torsional force 扭转⼒ torsional frequency 扭转频率 torsional impact 扭转冲击 torsional load 扭转载荷 torsional mode 扭振模式 torsional modulus 抗扭模量 torsional moment 扭矩 torsional resonance frequency 扭转共振频率 torsional rigidity 抗扭刚度 torsional spring 扭簧 torsional strain 扭转应变 torsional strength 抗扭强度 torsional stress 扭转应⼒ torsional test 扭转试验 torsional vibration 扭转振动 torsional vibration damper 扭转振动减震器 torsional wave 扭转波 torsional work 扭转功 torsionless stress ⽆扭应⼒ tortuosity 弯曲度 tortuosity factor 曲折因⼦ total acceleration 总加速度。

张量分析(Tensor Analysis)

ds 2 (dx1 ) 2 (dx 2 ) 2 (dx3 ) 2
利用克罗内克符号，上式可写成：
ds ij dx dx
2 i
j
克罗内克符号的一些常用性质：
i j xi x j
x j ij x i
i
j i k
j k
D) 置换符号
置换符号eijk=eijk定义为:
r i dr i dx x
空间一点P的位置矢量可用直角坐标表示为：
r z ji j
式中 ij 为沿坐标轴 zj 方向的单位矢量。
r r z j z j j i i ij i x z x x
r 上式表明， i 是单位矢量 ij 的线性组合，因此也是矢量。 x
基矢量（续）
r r i 变化时位置矢量r的变化，因此 i i 表征当 x i 的方向是沿坐标曲线 x x x r 的切线方向。矢量 i 可以取作曲线坐标系的基矢量（协变基矢量）： x
r z j gi i i i j x x
注意：对于在曲线坐标系中的每一点，都有三个基矢量。基矢量一般不是单位矢量，彼此也不正交；基矢量可以有量纲，但一点的三个基矢量的量纲可以不同；
1 张量的概念
在三维空间，一个矢量（例如力矢量、速度矢量等）在某参考坐标系中，有三个分量；这三个分量的集合，规定了这个矢量；当坐标变换时，这些分量按一定的变换法则变换。
在力学中还有一些更复杂的量。例如受力物体内一点的应力状态，有9个应力分量，如以直角坐标表示，用矩阵形式列出，则有：
xx xy xz ij yx yy yz zx zy zz
克罗内克符号 i j 的定义是:

张量分析——初学者必看精选全文

§ A-1 指标符号三、Kronecker-符号和置换符号(Ricci符号)
Ricci符号定义
偶次置换
1 若i, j, k 1,2,3, 2,3,1, 3,1,2 eijk 1 若i, j, k 3,2,1, 2,1,3, 1,3,2
0 若有两个或三个指标相等
e123 e231 e312 1 e213 e132 e321 1 e111 e112 e113 0
§A-4 张量的代数运算三、矢量与张量的叉积
A 张量分析
右叉乘
T a (Tijeie j ) (akek ) Tij akeie jkrer e T jkr ij akeier B
§A-4 张量的代数运算
A 张量分析
四、两个张量的点积
两个张量点积的结果仍为张量。新张量的阶数是原两个张量的阶数之和减 2
坐标变换式 xi ii xi xi ii xi
ii cos(xi, xi ) ii cos(xi , xi )
§A-3 坐标变换与张量的定义 A 张量分析
[ii ], [ii ]
互逆、正交矩阵
ii ii
ij
1 0
0 1
基矢量变换式
ei iiei ei iiei
坐标变换系数
v 任意向量变换式 i vii i vii i
ip iq ir eijk epqr jp jq jr
kp kq kr
pk
eijk ekqr
iq jq
ir jr
iq jr ir jq
a11 a12 a13 A a21 a22 a23 a11a22a33 a12a23a31
a31 a32 a33 a13a21a32 a13a22a31 a12a21a33 a11a23a32 eijk a1ia2 j a3k eijk ai1a j2ak3

张量分析翻译英文原文

TensorTensors are geometric objects that describe linearrelations between vectors, scalars, and other tensors.Elementary examples of such relations include thedot product, the cross product, and linearmaps.Vectors and scalars themselves are also tensors.A tensor can be represented as a multi-dimensionalarray of numerical values. The order (also degree orrank )of a tensor is the dimensionality of the arrayneeded to represent it, or equivalently, the number ofindices needed to label a component of that array. For example, a linear map can be represented by a matrix, a 2-dimensional array, and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a1st-order tensor. Scalars are single numbers andare thus 0th-order tensors.Tensors are used to represent correspondences between sets of geometric vectors. For example, the Cauchy stress tensor T takes a direction v as input and produces the stress T (v ) on the surfacenormal to this vector for output thus expressinga relationship between these two vectors, shown in the figure (right).Because they express a relationship between vectors, tensors themselves must beindependent of a particular choice of coordinate system. Taking a coordinate basis or frame of reference and applying the tensor to it results in an organized multidimensional array representing the tensor in that basis, or frame of reference. The coordinate independence of a tensor then takes the form of a "covariant" transformation law that relates the array computed in one coordinate system to that computed in another one. This transformation law is considered to be built into the notion of a tensor in a geometric or physical setting, and the precise form of the transformation law determines the type (or valence ) of the tensor.Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as elasticity, fluid mechanics, and general relativity. Tensors were first conceived by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus . The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.[1] Cauchy stress tenso r , a second-order tensor. The tensor's components, in a three-dimensional Cartesian coordinate system, form the matrix whose columns are the stresses (forces per unit area) acting on the e 1, e 2, and e 3 faces of the cube.HistoryThe concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century.[2]The word "tensor" itself was introduced in 1846 by William Rowan Hamilton[3] to describe something different from what is now meant by a tensor.[Note 1] The contemporary usage was brought in by Woldemar V oigt in 1898.[4]Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892.[5] It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications).[6]In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann.[7]Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect:I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.—Albert Einstein, The Italian Mathematicians of Relativity[8]Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics. From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem).[citation needed] Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. Multilinear algebra can be developed in greater generality than for scalars coming from a field, but the theory is then certainly less geometric, and computations more technical and less algorithmic.[clarification needed]Tensors are generalized within category theory bymeans of the concept of monoidal category, from the 1960s.DefinitionThere are several approaches to defining tensors. Although seemingly different, the approaches just describe the same geometric concept using different languages and at different levels of abstraction.As multidimensional arraysJust as a scalar is described by a single number, and a vector with respect to a given basis is described by an array of one dimension, any tensor with respect to a basis is described by a multidimensional array. The numbers in the array are known as the scalar components of the tensor or simply its components.They are denoted by indices giving their position in the array, in subscript and superscript, after the symbolic name of the tensor. The total number of indices required to uniquely select each component is equal to the dimension of the array, and is called the order or the rank of the tensor.[Note 2]For example, the entries of an order 2 tensor T would be denoted T ij, where i and j are indices running from 1 to the dimension of the related vector space.[Note 3]Just as the components of a vector change when we change the basis of the vector space, the entries of a tensor also change under such a transformation. Each tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see covariance and contravariance of vectors),where the new basis vectors are expressed in terms of the old basis vectors as,where R i j is a matrix and in the second expression the summation sign was suppressed (a notational convenience introduced by Einstein that will be used throughout this article). The components, v i, of a regular (or column) vector, v, transform with the inverse of the matrix R,where the hat denotes the components in the new basis. While the components, w i, of a covector (or row vector), w transform with the matrix R itself,The components of a tensor transform in a similar manner with a transformation matrix for each index. If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index. The transformation law for an order-m tensor with n contravariant indices and m−n covariant indices is thus given as,Such a tensor is said to be of order or type (n,m−n).[Note 4] This discussion motivates the following formal definition:[9]Definition. A tensor of type (n, m−n) is an assignment of a multidimensional arrayto each basis f = (e1,...,e N) such that, if we apply the change of basisthen the multidimensional array obeys the transformation lawThe definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.[1]Nowadays, this definition is still used in some physics and engineering text books.[10][11]Tensor fieldsMain article: Tensor fieldIn many applications, especially in differential geometry and physics, it is natural to consider a tensor with components which are functions. This was, in fact, the setting of Ricci's original work. In modern mathematical terminology such an object is called a tensor field, but they are often simply referred to as tensors themselves.[1]In this context the defining transformation law takes a different form. The "basis" for the tensor field is determined by the coordinates of the underlying space, and thedefining transformation law is expressed in terms of partial derivatives of thecoordinate functions, , defining a coordinate transformation,[1]As multilinear mapsA downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach is to define a tensor as a multilinear map. In that approach a type (n,m) tensor T is defined as a map,where V is a vector space and V* is the corresponding dual space of covectors, which is linear in each of its arguments.By applying a multilinear map T of type (n,m) to a basis {e j} for V and a canonical cobasis {εi} for V*,an n+m dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realised as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.Using tensor productsMain article: Tensor (intrinsic definition)For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property. A type (n,m) tensor is defined in this context as an element of the tensor product of vectorspaces,[12]If v i is a basis of V and w j is a basis of W, then the tensor product has anatural basis . The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {e i} for V and its dual {εj}, i.e.Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (m,n) tensor. Moreover, the universal property of the tensor product gives a 1-to-1 correspondence between tensors defined in this way and tensors defined as multilinear maps.OperationsThere are a number of basic operations that may be conducted on tensors that again produce a tensor. The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the scaling of a vector. On components, these operations are simply performed component for component. These operations do not change the type of the tensor, however there also exist operations that change the type of the tensors.Raising or lowering an indexMain article: Raising and lowering indicesWhen a vector space is equipped with an inner product (or metric as it is often called in this context), operations can be defined that convert a contravariant (upper) index into a covariant (lower) index and vice versa. A metric itself is a (symmetric) (0,2)-tensor, it is thus possible to contract an upper index of a tensor with one of lower indices of the metric. This produces a new tensor with the same index structure as the previous, but with lower index in the position of the contracted upper index. This operation is quite graphically known as lowering an index.Conversely the matrix inverse of the metric can be defined, which behaves as a (2,0)-tensor. This inverse metric can be contracted with a lower index to produce an upper index. This operation is called raising an index.ApplicationsContinuum mechanicsImportant examples are provided by continuum mechanics. The stresses inside a solid body or fluid are described by a tensor. The stress tensor and strain tensor are both second order tensors, and are related in a general linear elastic material by a fourth-order elasticity tensor. In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently represented as a 3×3 array. The three faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number. Thus, 3×3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment. Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, a second order tensor is needed.If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2,0), in linear elasticity, or more precisely by a tensor field of type (2,0), since the stresses may vary from point to point.Other examples from physicsCommon applications include∙Electromagnetic tensor(or Faraday's tensor) in electromagnetism∙Finite deformation tensors for describing deformations and strain tensor for strain in continuum mechanics∙Permittivity and electric susceptibility are tensors in anisotropic media∙Four-tensorsin general relativity (e.g. stress-energy tensor), used to represent momentum fluxes∙Spherical tensor operators are the eigen functions of the quantum angular momentum operator in spherical coordinates∙Diffusion tensors, the basis of Diffusion Tensor Imaging, represent rates of diffusion in biologic environments∙Quantum Mechanicsand Quantum Computing utilise tensor products for combination of quantum statesApplications of tensors of order > 2The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix.The field of nonlinear optics studies the changes to material polarization density underextreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:Here is the linear susceptibility, gives the Pockels effect and secondharmonic generation, and gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.Generalizations[edit]Tensors in infinite dimensionsThe notion of a tensor can be generalized in a variety of ways to infinite dimensions. One, for instance, is via the tensor product of Hilbert spaces.[15]Another way of generalizing the idea of tensor, common in nonlinear analysis, is via the multilinear maps definition where instead of using finite-dimensional vector spaces and their algebraic duals, one uses infinite-dimensional Banach spaces and their continuous dual.[16] Tensors thus live naturally on Banach manifolds.[17]Tensor densitiesMain article: Tensor densityIt is also possible for a tensor field to have a "density". A tensor with density r transforms as an ordinary tensor under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian to the r th power.[18] Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle such as the (1-dimensional) space of n-forms (where n is the dimension of the space), as opposed to taking their values in just R. Higher "weights" then just correspond to taking additional tensor products with this space in the range.In the language of vector bundles, the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times. While locally the more general transformation law can indeed be used to recognise these tensors, there is aglobal question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value. Non-integral powers of the (positive) transition functions of the bundle of densities make sense, so that the weight of a density, in that sense, is not restricted to integer values.Restricting to changes of coordinates with positive Jacobian determinant is possible on orientable manifolds, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of n-forms are distinct. For more on the intrinsic meaning, see density on a manifold.SpinorsMain article: SpinorStarting with an orthonormal coordinate system, a tensor transforms in a certain way when a rotation is applied. However, there is additional structure to the group of rotations that is not exhibited by the transformation law for tensors: see orientation entanglementand plate trick. Mathematically, the rotation group is not simply connected. Spinors are mathematical objects that generalize the transformation law for tensors in a way that is sensitive to this fact.Einstein summation conventionThe Einstein summation convention dispenses with writing summation signs, leaving the summation implicit. Any repeated index symbol is summed over: if the index i is used twice in a given term of a tensor expression, it means that the term is to be summed for all i. Several distinct pairs of indices may be summed this way.。

张量分析书籍附详尽易懂

n个
称为n维仿射空间。E n 中旳每一种元素称为点。
记：
o (0, ,0),
x (x1,, xn ) ,
(x1, , xn )
且分别称为放射空间旳原点、位置矢量和负矢量。
对于n维仿射空间，全部旳位置矢量构成一种集合：
V0 x (x1,, xn ) xi , xi F,1 i n
(1 t)(1,1) t(1,1) a t b
(1 2t,1 2t) a t b
当 t b 时：
(2t 1,2t 1) (1,1)
当 t a 时：
(2t 1,2t 1) (1,1)
由此可得 a 0 ，b 1 。显然 r1 等 r2 价。
r1 与 r5 ： (取 s b5 b1 )
域上旳矢量空间。且仍记为V0 。
数域上旳矢量空间V0 具有如下性质：x, y, z V0 ,、 F
（1）
x yyx
（2）
(x y) z x ( y z)
（3）V0中存在称为有关加法旳单位元素o，使得：
xo x
x V0
（4）V0中每一种元素x都存在唯一旳(-x )，使得：
x (x) o
当t=b时：位置矢量标
定b点。即：
S
(4b 2,3 2b) (2,1)
由此拟定b=1 。
x2
当t=a时：位置矢量标
3
2
定a点。即：
1
(4a 2,3 2a) (1,1.5 )
由此拟定a=0.75 。
图中画出了计算成果。
x2 3
2 u ab
1
2 (a)
u xy
x1
4
6
u xy u ab
1
2
。 Vx空间中旳矢量称为约束矢量。

张量分析总结[范文]

张量分析总结[范文]第一篇：张量分析总结[范文]中国矿业大学《张量分析》课程总结报告第 1 页一、知识总结张量概念1.1 指标记法哑标和自由指标的定义及性质自由指标：在每一项中只出现一次，一个公式中必须相同。

性质：在表达式或方程中自由指标可以出现多次，但不得在同项内重复出现两次。

哑标：一个单项式内，在上标（向量指标）和下标（余向量指标）中各出现且仅出现一次的指标。

性质：哑标可以把多项式缩写成一项；自由指标可以把多个方程缩写成一个方程。

例：A11x1+A12x2+A13x3=B1A21x1+A22x2+A23x3=B2 A31x1+A32x2+A33x3=B3式(1.1)可简单的表示为下式：(1.1)Aijxj=Bi(1.2)其中：i为自由指标，j为哑标。

特别区分，自由指标在同一项中最多出现一次，表示许多方程写成一个方程；而哑标j则在同项中可出现两次，表示遍历求和。

在表达式或者方程中自由指标可以出现多次，但不得在同项中出现两次。

1.2 Kronecker符号定义δij为：δij=⎨⎧1,i=j0,i≠j⎩(1.3)δij的矩阵形式为：⎡100⎤⎥δij=⎢010⎢⎥⎢⎣001⎥⎦(1.4)可知δijδij=δii=δjj=3。

δ符号的两指标中有一个与同项中其它因子的指标相同时，可把该因子的重指标换成δ的另一个指标，而δ符号消失。

如：δijδjk=δikδijδjkδkl=δil(1.5)中国矿业大学《张量分析》课程总结报告第 2 页δij的作用：更换指标、选择求和。

1.3 Ricci符号为了运算的方便，定义Ricci符号或称置换符号：⎧1,i,j,k为偶排列⎪lijk=⎨-1,i,j,k为奇排列⎪0,其余情况⎩(1.6)图1.1 i,j,k排列图lijk的值中，有3个为1，3个为-1，其余为0。

Ricci符号（置换符号）是与任何坐标系都无关的一个符号，它不是张量。

1.4 坐标转换图1.2 坐标转换如上图所示，设旧坐标系的基矢为ei，新坐标系的基矢为ei'。

张量分析陈国荣徐芝纶

为了使张量在每个具体坐标系里能取得具有相同的物理量纲的分量在正交曲线坐标系取切于坐标曲线的无量纲单位矢量作为基矢量即cossinsincoscossinsinsincoscoscoscossinsincossinsin由此得到曲线坐标系的hamilton算子比较式a712与式a715得到bca并考虑到得到在正交曲线坐标系关于指标j和k是反称的因为共有6个为011332232333113132233232221121211331311122121只有212张量的梯度为了方便以后仍然把物理基记为16四圆柱坐标系张量的导数公式221212rzzzrrzr
8
gi j ,k k ( gi g j ) k gi g j k g j gi
g j k ,i i ( g j gk ) i g j gk i gk g j
（ａ）（ｂ）（ｃ）
gk i, j j ( gk gi ) j gk gi j gi gk
2
g i j 称为度量张量
r r ds dr.dr . dxi dxj gij dxi dxj xi x j
2
例1
求圆柱坐标系的自然基 gi 和度量张量g i j
空间任意点的向径为
r r cos e1 r sin e 2 ze 3 r g1 cos e1 sin e 2 r r g2 r sin e1 r cos e 2 r g3 e3 z
（b）＋（c）－（a），并考虑到
k gi g j i gk g j
得到
1 i g j g k ( g j k ,i g k i , j g i j ,k ) 2
9
1 1 1 i j k [ ( g j k ,i g k i , j g i j ,k ) g j j ( )g jk ] xi g j j gii g j j g k k 2

张量分析答案完整版.

1.1 求证： u × (v × w ) = ( u • w) v − ( u • v) w
黄克智版张量分析课后习题答案完整版
并问： u × ( v × w ) 与（u × v） × w 是否相等？ u 、v、w 为矢量证明：因为 u= (u x , u y , u z ) ; v= ( vx , vy , vz ) ; w= ( wx , wy , wz ) ;
左边= u × (v × w ) = (u x , u y , u z ) × [ ( vx , vy , vz ) × ( wx , wy , wz ) ]
g11g1 + g 12g 2 + g 13g 3 = 2g1 + g 2 + g 3
= j + k = g1
1.11 根据上题结果验算公式： g j = g jig i 1 1 1 由上题结果： g = 2 , g1 = ( −i + j + k ) , g 2 = (i − j + k ) , g3 = ( i + j − k ) 2 2 2
=[
⎧2 g rs = ⎨ ⎩1
及： g1 = g11g1 + g12g 2 + g13g3
所以 u × (v × w) ≠（u × v） ×w
第一章
同理; g 21g1 + g 22g 2 + g 23g3 = g1 + 2g 2 + g 3 当r=s 当r ≠ s
=
=
, u y ( vx wy − wx v y ) − u z ( wxv z − v x wz ) u x ( wx vz − vx wz ) − u y ( v y wz − w yv z ) ] 所以： u × (v × w ) = (u • w)v − ( u • v) w 同理可证： ( u × v ) × w = ( u • w) v − ( v • w) u

张量分析-第1讲LJ

a2 F3 a3 F2 a c b1 a b c1 a3 F1 a1 F3 a c b2 a b c2 a1 F2 a2 F1 a c b3 a b c3
所以有： a b c a c b a b c
g1和g 2
g1和g 2 不是单位矢量，即它们有量纲的，一般地说，
其长度也不为单位长度。此外它们也并不正交。矢量F可以在 g1和g 2 上分解:
F F g1 F g 2
1 2
(平行四边形法则)
则有: F g 1 F 1g 1 g 1 F 2 g 2 g 1
F g 1 F 1g 1 g 1 F 2 g 2 g 1
e2 b2 c2
e3
e3 b3 b2 c3 b3 c2 e 1 b3 c1 b1c3 e 2 b1c2 b2 c1 e 3 c3
b3 a 2 F3 a3 F2 e 1 a3 F1 a1 F3 e 2 a1 F2 a 2 F1 e 3 F3
j 1
F2 ' e 2 ' e1 F1 e 2 ' e 2 F2 e 2 ' e 3 F3 2 ' j F j
j 1 3
3
F3' e 3' e1 F1 e 3' e 2 F2 e 3' e 3 F3 3' j F j
j 1
矢量场函数的散度: 矢量场函数的旋度:
i F x Fx j y Fy
Fx Fy Fz F z y x
k Fz Fy Fx Fz Fy Fx i k j y z y z z x x Fz

张量分析第5章

1 2
对称性：

度量张量的行列式
a d e t( a ) a11 a 21 a12 a 22
2
a11 a 22 a12 a 21 a11a 22 a12

度量张量的逆变分量
a
11
a 22 a
a
22
a11 a
a
12
a a
12
a
21
a12 a a 21 a

d d

♣ 曲面的第二基本形式：
n ρ

2
d d

b d d
bαβ 是外蕴的，表达了切平面外的信息。
曲面的第二基本张量
b n
ρ

2
n
ρ

。
( , )
1

2
b1 b2 b1b2 b2 b1 0
1 2 1 2 1 2
1 2
(1 ) ( 2 ) b 1 b 2 b t r b (1 ) ( 2 ) b 1 b 2 b 2 b 1 d e t b
2
其中， ρ
ρ

ρ
ρ b n

对比欧氏空间的基矢量求偏导数：
gi x
j
gi x
j
ij g k

k
ij g k
k
ρ

ρ b n
。
仍在欧氏空间内
曲面内+法向
曲面的第二基本张量

张量分析-第10讲LJ

i Lagrange坐标系中, 物理量求物质导数仅需保持坐标不变, 而对
时间 t 求导. Euler 坐标系中求物质导数时, 不仅要考虑物理自身随时间的变化, 而且要考虑由于质点运动而引起的位置坐标 x i 的变化. Lagrange 坐标系一般是曲线坐标系, 而Euler坐标系可以取直角坐标系,一般在推导时采用Lagrange坐标系, 然后转换到Euler坐标系中进行计算. 5
ˆi dg ˆ v) g ˆ i ( ˆi E g ˆi ω g dt
ˆi dg ˆ ) g ˆ i ( v ˆ i (E - Ω) g ˆi E g ˆ i Ω g ˆi E g ˆi ω g dt
13
4.3 欧拉坐标系基矢量的物质导数
r r ( x i (t )),
2. Lagrange坐标
r ( x j (t )) j gi g ( x (t )) i i x
ξ3
拉格朗月坐标是嵌在质点上, 随物体一起运动和变形, 又称随体坐标或嵌入坐标: i ξ3 变形前后的同一个质点坐标值不 B g ξ2 改变, 但是两质点的距离在变形前 A 后发生了变化。
ˆij dT
j ˆij d T ˆ ˆ d g dT d ˆ i d g j j i j i ˆ j i g ˆ j g ˆ ig ˆ T ˆ T ˆi ˆ ig ˆ ) (T j g g dt dt dt dt dt
ˆ r ˆ ˆi dr ( i ) t d i d i g ˆ r ˆ ˆ i ( j , t ) g i ( i )t g
ξ3
B A
3 g
ξ3
ˆ3 g
B'
ˆ2 g
A'

第二章张量分析

P也可用另外三个变量 x，1' x，2' 来x 表3' 示，即
P P x1' , x 2' , x 3' P x i'
这种坐标系记为 xi。' 这两组变量
x1 , 和x 2 , x 3 表示x同1' 一, x空2' , x 3'
间点的位置。两者由下列坐标变换联系起来：
xi xi xi'
aigi a jg j aia j gi g j
aiaj 0
令
gij gi g j
g ij g i g j
gi j gi g j g j gi
它们分别称为协变度量张量、逆变度量张量和混合度量张量
考虑到矢量a的任意性 g ji gi g j i j
可知：基矢量 g与i 是g i 正交的，它们称为互逆基矢量互逆基矢量间具有下列关系：
gig j g k gi g j g k gig j g k eijk
这两个量定义为爱丁顿(Eddington)张量并分别记为和ijk 。ijk
由此定义可知
123 g1g 2g3
123 g1g 2g3
对于矢量 a 0 ，则有
a 2 a a a i g i a j g j a i a j g i g j ai gi a j g j ai a j gi g j
(ii) x 2' (常数 C)为2 通过z轴的平面； (iii) x3' (常z 数 C)3为垂直于z轴的平面；
和坐标曲线：
(i) x1' 和r C1 x的2' 交线 (zC线2 )是直线； (ii) x 2' 和 C2 的x3'交线z (r线C3)是直线；

张量分析Huang_Introduction to Tensor

矢量和张量所代表的物理量本身并不依赖于坐标系而存在，但要对该物理量进行数值上的描述和分析，则常常需要引入一个适当的坐标系。在三维物理空间中一个矢量具有三个分量，一个二阶张量则有九个分量。一般地，三维空间中的一个n 阶张量则有 3n 个分量。而标量和矢量可分别看作为零阶和一阶张量
1. 矢量和张量的表述
自然界中的物理量，有的只需要用一个实数来描述，如温度、气压、时间、质量、能量等。这样的物理量称为标量。有些物理量包含了大小和方向等要素，需要用矢量(向量)来描述，如力、力矩、位移、速度、动量等。
自然界中还有一些物理量，包含了比矢量更多的要素，如连续体中一点处的应力、应变等。这样的物理量称为张量。
òijk A1i A2 j A3k i òijk Ai1 Aj 2 Ak 3 i
（4）
i1 i 2 i 3 òijk det j1 j 2 j 3 k 1 k 2 k 3
（5）
蝌 ijk
pqr
ip iq ir det jp jq jr kp kq kr
Aij
D Dijkl ei e j ek el
自由下标
Dijkl
(i, j, k , l {1, 2,3})
取遍1、2、3
张量的指标表述是依赖于坐标系的
2. Einstein求和约定:
c a jbj
j 1 3

c a jb j

j 1 3 3
3
Aij x j bi
HOHAI UNIVERSITY
弹性力学 Engineering Elasticity
黄文雄

张量分析答案完整版

= T J• T ii • 2
=
tr(T
•T
)
=T
•T
•G
•
T T = •m •a am
• •
JT 3
=T •T •T •G •
=T T T •m •p •a a mp
对于 S ：
得证。
JT 1
=T jj
J• T
•2
= tr(T T
•TT) = TT
TT
•
•G
•
= T T J m •a • T •a m • 3
2δ
i j
[u
v
w
]
+
2δ
i j
[u
v
w]
[ = T⋅ii δ
i j
u
v
w ]=T⋅ii [u
v
w ]= φ1T [u
v
w ]，命题得证。
（2）式左边
[ ] [ ] [ ] = T⋅ija jgi
T
a ⋅b
b
b
g
a
c cgc
+ adgd
T ⋅ijb jgi
T⋅ab cb g a + T⋅ija jgi
∂v m
'
∂x n '
−
∂vn' ∂x m'
∂xm = ∂xm'
∂x n ∂xn '
(
∂vm ∂xn
−
∂vn ∂x m
)
即T(m' .n' )
=
β m' m'
β n' n'
(
∂vm ∂xn
−
∂vn ∂x m

第1章张量分析(清华大学张量分析,你值得拥有)

jigggg一般来说张量的对称化与反对称化若四阶张量满足ijgggg则称张量t对其12指标是对称张量用来表示其转置张量ijjiklkl若四阶张量满足ijgggg则称张量t对其12指标是反对称张量用ijjiklkl张量的对称化与反对称化可立即得出反对称张量的对角分量均为零同为对称结构加任意载荷均可分为对称和反对称
g1 1
gi xi g2 ix1 sin x2 jx1 cos x2
g2 r
曲线坐标系：斜角直线坐标系的延伸
※三维球坐标系
(x, y, z) (x1, x2, x3)
(r, , ) (x1, x2 , x3 )
x3
r
gr g
g
x2
r x1i x2 j x3k xigi x1
新、老坐标之间的变换和逆变换： xi = xi xi'
gidxi gi' dx i'
dxi
=
xi xi'
dx i '
dx i
i'
i'
→
gi
i i'
dx
i'
g i
'
dx
i'
再由：
gidxi
g i
'
dx
i
'
→
gi'
i i'
gi
dxi'
=
xi' xi
dxi
i i
'
dx
i
→
gidxi
gi'
i i
'
dx
定义式（实体形式，几何表达）：
u v u v cos
v cos

tensor单词记忆

tensor单词记忆1. 定义与释义- 单词：tensor- 1.1词性：名词- 1.2中文释义：张量，是一个在多个线性代数、物理和工程等领域广泛使用的数学概念，是向量和矩阵概念的推广。

- 1.3英文释义：In mathematics and physics, a tensor is a geometric object that maps in a multi - linear fashion.- 1.4相关词汇：tensor product（张量积，近义词）、tensor field（张量场，派生词）。

2. 起源与背景- 2.1词源：“tensor”源于拉丁语“tensus”，有拉伸的意思。

最初在数学和物理学的发展过程中，为了描述具有多个线性关系的量而被定义。

- 2.2趣闻：在爱因斯坦的广义相对论中，张量被广泛使用来描述时空的弯曲等复杂概念。

如果没有张量这个数学工具，爱因斯坦很难准确地用数学公式表达他关于时空和引力的伟大理论。

3. 常用搭配与短语- 3.1短语：- tensor analysis：张量分析。

例句：Tensor analysis is very important in modern theoretical physics.翻译：张量分析在现代理论物理中非常重要。

- tensor algebra：张量代数。

例句：He is studying tensor algebra to solve the complex problems in his research.翻译：他正在学习张量代数以解决他研究中的复杂问题。

- second - order tensor：二阶张量。

例句：The stress in a material can be represented by a second - order tensor.翻译：材料中的应力可以用二阶张量来表示。

4. 实用片段- （1）. “I'm having a hard time understanding this tensor concept in my math class. It seems so abstract.” John complained to his friend. His friend replied, “Well, you can start with the basic examples of vectors and matrices, since tensors are just a generalization of them.”翻译：“我在数学课上理解张量这个概念好难啊，它看起来太抽象了。