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!!!!
高微 證明 line & surface integrals
2011-04-29 8:42 am
回答 (2)
2011-04-30 12:46 am
✔ 最佳答案
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
2011-04-30 1:33 am
圖片參考：http://imgcld.yimg.com/8/n/AC06918685/o/101104290033613869669250.jpg
圖片參考：http://imgcld.yimg.com/8/n/AC06918685/o/101104290033613869669261.jpg
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.
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