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sol4 - Math 264 Advanced Calculus Assignment 4 Winter 2008...

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Math 264: Advanced Calculus Winter 2008 Assignment 4 Solutions Every problem is worth 5 points. Due to time constraints, some problems may not be marked. In Problems 1 and 2, D is a domain in R 3 satisfying the conditions of divergence theorem, S is the boundary of D , and N is the outward unit normal to S . The functions φ and ψ are smooth on D S . Finally, for any smooth f , the normal derivative of f , ∂f/∂n , is defined by ∂f/∂n = f N . Problem 1 (Adams, § 16.4 # 23). If F is a smooth vector field on a domain D , show that Z Z Z D [ φ (div F ) + φ F ] dV = Z Z S φ F N dS. Solution. By Adams, Theorem 3b (section 16.2, p. 859), we have div( φ F ) = ( φ ) F + φ (div F ) . So, the left-hand side is equal to R R R div( φ F ) dV . The result then follows by the Divergence Theorem. Problem 2 (Adams, § 16.4 # 28). Recall that Δ f denotes the Laplacian of f . Verify that Z Z Z D [ φ ψ ) - ψ φ )] dV = Z Z S φ ∂ψ ∂n - ψ ∂φ ∂n dS. Solution. Recall that Δ f = div grad f . By Adams, Theorem 3b (section 16.2, p. 859), we have div( φ ψ - ψ φ ) = φ
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