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18.3 Ex 26 - 32

# 18.3 Ex 26 - 32 - 1260 C H A P T E R 18 F U N D A M E N TA...

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1260 C H A P T E R 18 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS (ET CHAPTER 17) 25. Prove that div ( f × ∇ g ) = 0. SOLUTION We compute the cross product: f × ∇ g = f x , f y , f z × g x , g y , g z = i j k f x f y f z g x g y g z = f y g z f z g y , f z g x f x g z , f x g y f y g x We now compute the divergence of this vector. Using the Product Rule for scalar functions and the equality of the mixed partials, we obtain div ( f × ∇ g ) = x ( f y g z f z g y ) + y ( f z g x f x g z ) + z ( f x g y f y g x ) = f yx g z + f y g zx f zx g y f z g yx + f zy g x + f z g xy f xy g z f x g zy + f xz g y + f x g yz f yz g x f y g xz = ( f yx f xy ) g z + ( g zx g xz ) f y + ( f xz f zx ) g y + ( g xy g yx ) f z + ( f zy f yz ) g x + ( g yz g zy ) f x = 0 In Exercises 26–28, let denote the Laplace operator defined by ϕ = ϕ 2 x 2 + ϕ 2 y 2 + ϕ 2 z 2 26. Prove the identity curl ( curl ( F )) = ∇ ( div ( F )) F where F denotes F 1 , F 2 , F 3 . SOLUTION We compute the left-hand side of the identity. We have curl ( F ) = i j k x y z F 1 F 2 F 3 = F 3 y F 2 z , F 1 z F 3 x , F 2 x F 1 y Hence, curl ( curl ( F )) = y F 2 x F 1 y z F 1 z F 3 x , z F 3 y F 2 z x F 2 x F 1 y , x F 1 z F 3 x y F 3 y F 2 z = 2 F 2 y x 2 F 1 y 2 2 F 1 z 2 + 2 F 3 z x , 2 F 3 z y 2 F 2 z 2 2 F 2 x 2 + 2 F 1 x y , 2 F 1 x z 2 F 3 x 2 2 F 3 y 2 + 2 F 2 y z (1) We now compute the right-hand side of the given identity: ( div ( F )) = ∇ F 1 x + F 2 y + F 3 z = x F 1 x + F 2 y + F 3 z , y F 1 x + F 2 y + F 3 z , z F 1 x + F 2 y + F 3 z = 2 F 1 x 2 + 2 F 2 x y + 2 F 3 x z , 2 F 1 y x + 2 F 2 y 2 + 2 F 3 y z , 2 F 1 z x + 2 F 2 z y + 2 F 3 z 2 Therefore, ( div ( F )) F = ∇ ( div ( F )) 2 F 1 x 2 + 2 F 1 y 2 + 2 F 1 z 2 , 2 F 2 x 2 + 2 F 2 y 2 + 2 F 2 z 2 , 2 F 3 x 2 + 2 F 3 y 2 + 2 F 3 z 2

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S E C T I O N 18.3 Divergence Theorem (ET Section 17.3) 1261 = 2 F 2 x y + 2 F 3 x z 2 F 1 y 2 2 F 1 z 2 , 2 F 1 y x + 2 F 3 y z 2 F 2 x 2 2 F 2 z 2 , 2 F 1 z x + 2 F 2 z y 2 F 3 x 2 2 F 3 y 2 (2) Since the mixed partials are equal, the expressions obtained in (1) and (2) are the same. This proves the desired identity.
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18.3 Ex 26 - 32 - 1260 C H A P T E R 18 F U N D A M E N TA...

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