简介
用mathematica计算简单的张量,方便查找.
基于mathematica自身的张量计算
mathematica中的张量基础运算
单位四维张量($\delta_i^k$):
\[\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)\]DiagonalMatrix[{1, 1, 1, 1}]
或者:
IdentityMatrix[4]
度规张量($g^{i,k}$):
\[\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{array} \right)\]DiagonalMatrix[{1, -1, -1, -1}]
全反单位对称张量($e^{i,k,l,m}$):
LeviCivitaTensor[4]
任意张量$a^{i,j},i,j=0,1,2,3$
\[\left( \begin{array}{cccc} a(1,1) & a(1,2) & a(1,3) & a(1,4) \\ a(2,1) & a(2,2) & a(2,3) & a(2,4) \\ a(3,1) & a(3,2) & a(3,3) & a(3,4) \\ a(4,1) & a(4,2) & a(4,3) & a(4,4) \\ \end{array} \right)\]Array[a, {4, 4}]
张量积运算($\otimes$)
\[\{a(1),a(2),a(3),a(4)\}\otimes\{b(1),b(2),b(3),b(4)\}=\\ \left( \begin{array}{cccc} a(1) b(1) & a(1) b(2) & a(1) b(3) & a(1) b(4) \\ a(2) b(1) & a(2) b(2) & a(2) b(3) & a(2) b(4) \\ a(3) b(1) & a(3) b(2) & a(3) b(3) & a(3) b(4) \\ a(4) b(1) & a(4) b(2) & a(4) b(3) & a(4) b(4) \\ \end{array} \right)\]Array[a,{4}]\[TensorProduct]Array[b,{4}]
张量缩并
$a^ib^i=a^0b^0+a^1b^1+a^2b^2+a^3b^3$
TensorContract[Array[a,{4}]\[TensorProduct]Array[b,{4}],1]
简单算例
$a_i a^i={a_0}^2-{a_1}^2-{a_2}^2-{a_3}^2$
dg = DiagonalMatrix[{1, -1, -1, -1}];
TensorContract[
Array[a, {4}]
\[TensorProduct]
dg
\[TensorProduct]
Array[b, {4}]
, { {1, 2}, {3, 4} }
]
$e^{i,j,l,m}e_{i,j,l,m}=-24$
dg = DiagonalMatrix[{1, -1, -1, -1}];
TensorContract[
TensorContract[LeviCivitaTensor[4]\[TensorProduct](dg\[TensorProduct]dg\[TensorProduct]dg\[TensorProduct]dg), { {1, 5}, {2, 7}, {3, 9}, {4, 11} }]\[TensorProduct]LeviCivitaTensor[4],
{ {1, 5}, {2, 6}, {3, 7}, {4, 8} }
]
注意
这种方式计算张量实际上相当于张量全部展开,所以对于高阶张量,计算较慢.
基于mathematica张量计算包的计算
待定
