Ex3Ch8 - f ∇ g-g ∇ f to obtain ZZ ∂W f ∇ g-g ∇ f...

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1 Review Example 3, Chapter 8. Example 8.R.4 For a region W in space with boundary ∂W , unit outward normal n and functions f and g defined on W and ∂W , prove Green’s identi- ties: ZZ ∂W ( f g - g f ) · n dS = ZZZ W ( f 2 g - g 2 f ) dx dy dz, where 2 f = 2 f ∂x 2 + 2 f ∂y 2 + 2 f ∂z 2 is the Laplacian of f . Solution. To prove this identity, we apply Gauss’ theorem to the vector field
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Unformatted text preview: f ∇ g-g ∇ f to obtain ZZ ∂W ( f ∇ g-g ∇ f ) · n dS = ZZZ W ∇ · ( f ∇ g-g ∇ f ) dV = ZZZ W ∇ f · ∇ g + f ∇ 2 g- ∇ g · ∇ f-g ∇ 2 f dV = ZZZ W f ∇ 2 g-g ∇ 2 f dV....
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