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# 高微 證明 line & surface integrals

1.Prove the following integration-by-parts formula for triple integrals:

∫∫∫(範圍為R) f*(∂g/∂x)dV =

- ∫∫∫(範圍為R) g*(∂f/∂x)dV + ∫∫(範圍為∂R) f*g*n(下標x)dA

where n(下標x) is the x-component of the unit outward normal to ∂R.

(Of course, similar formulas also hold with x replaced by y and z)

麻煩高手們!!! 謝謝!!!!

### 2 Answers

- mathmanliuLv 710 years agoFavorite Answer
考慮向量函數(fg, 0, 0), 由divergence theorem

∫∫_∂R (fg, 0, 0)∙n dA = ∫∫∫_R div(fg, 0, 0) dV

得 ∫∫_∂R f*g*nx dA = ∫∫∫_R ∂(fg)/∂x dV

即 ∫∫_∂R f*g*nx dA= ∫∫∫_R f(∂g/∂x)+g(∂f/∂x) dV

移項,故得證

- SamLv 610 years ago
圖片參考：http://imgcld.yimg.com/8/n/AC06918685/o/1011042900...

圖片參考：http://imgcld.yimg.com/8/n/AC06918685/o/1011042900...

Please draw a cube and a sphere and other bodys, maybe you will understand what my mean.

2011-04-29 17:45:31 補充：

The Answer of 煩惱即是菩提 ( 知識長 ) is better than mine.