张量分析基础.pdf

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P = P * A P * = PA线性变换
™ 线性变换：矢量长度不变⇒3个独立矩阵元
P * P * = PA PA = PA A P = P P
A A = I AA −1 = I
⎡a11 a12 a13 ⎤ ⎡a11 a 21 a31 ⎤ ⎡1 0 0⎤
A = A −1 A = ± 1 ⎢a a a ⎥ ⎢a a a ⎥ = ⎢0 1 0⎥
⎢ 21 22 23 ⎥ ⎢ 12 22 32 ⎥ ⎢ ⎥
⎢a a a ⎥ ⎢a a a ⎥ ⎢0 0 1⎥
™ 矢量变换 ⎣ 31 32 33 ⎦ ⎣ 13 23 33 ⎦ ⎣ ⎦
⎡a11 a 21 a31 ⎤
P * = PA −1 = P A []P * P * P * = []P P P ⎢a a a ⎥
1 2 3 1 2 3 ⎢ 12 22 32 ⎥
⎣⎢a13 a 23 a33 ⎦⎥
*
* ⎡P1 ⎤ ⎡a11 a12 a13 ⎤ ⎡ P1 ⎤
P = P A = A P = A P ⎢ * ⎥
P = ⎢a a a ⎥ ⎢P ⎥
* ⎢ 2 ⎥ ⎢ 21 22 23 ⎥ ⎢ 2 ⎥
P = P P = AP ⎢ * ⎥ ⎢ ⎥ ⎢ ⎥
⎣P3 ⎦ ⎣a31 a32 a33 ⎦ ⎣ P3 ⎦置换矩阵
™ 置换矩阵
⎧1 i = j
ei ⋅e j = δ ij δ ij = ⎨
⎩0 i ≠ j δ ijTjl = Til
⎡1 0 0⎤
δ P = P
[]δ = ⎢0 1 0⎥ ij j i
ij ⎢ ⎥ δ T = T
δ ij Pi = Pj il jl ji
⎣⎢0 0 1⎦⎥
™ 反对称三重积
⎧ 1 i, j,k顺序轮换
⎪
ei ×e j = εijke j εijk = ⎨−1 i, j,k反序轮换
⎪
⎩ 0 两个以上角标同二阶张量的变换
* *
P ⇔ P → Q ⇔ Q P* = ATAQ*
P* = AP P* = a P * *
i ik k Pi = aikTkl a jlQ j
P = TQ P = T Q * * * * * *
k k