六个平面x+y+z=±1,-x+2y+3z=±2,2x-y+5z=±3所围平行六面体的体积
1个回答

用换元积分.

考虑R^3→R^3的线性变换F:(x,y,z)→(x+y+z,-x+2y+3z,2x-y+5z).

其Jacobi矩阵为常矩阵

1 1 1

-1 2 3

2 -1 5

行列式|dF| = 21.

由换元积分公式,

体积V = ∫∫∫{|x+y+z| ≤ 1,|-x+2y+3z| ≤ 2,|2x-y+5z| ≤ 3} 1 dxdydz

= ∫∫∫{|F1(x,y,z)| ≤ 1,|F2(x,y,z)| ≤ 2,|F3(x,y,z)| ≤ 3} 1 dxdydz

= ∫∫∫{|u| ≤ 1,|v| ≤ 2,|w| ≤ 3} |dF|^(-1) dudvdw

= |dF|^(-1)·∫∫∫{|u| ≤ 1,|v| ≤ 2,|w| ≤ 3} 1 dudvdw

= |dF|^(-1)·2·4·6

= 16/7.

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