Explanation The vector operator ∇ (read as Del) is defined by ∇ = i ∂ ∂ x + j ∂ ∂ y + k ∂ ∂ z ; Where ∂ ∂ x , ∂ ∂ y , ∂ ∂ z are partial derivatives with respect to x , y , z . Given: Formula used: Let F = F 1 i + F 2 j + F 3 k Divergence of continuously differentiable vector point function F is d i v F = ∇ ⋅ F = ( i ∂ ∂ x + j ∂ ∂ y + k ∂ ∂ z ) ⋅ ( F 1 i + F 2 j + F 3 k ) d i v F = ∂ F 1 ∂ x + ∂ F 2 ∂ y + ∂ F 3 ∂ z ---------------------------- (i) Curl of a continuously differentiable vector point function F is c u r l F = ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F 1 F 2 F 3 | -------------------------- (ii) Calculation: To verify d i v ( c u r l F ) = 0 for F = ( x z , y e x , y z ) The components of the vector F are F 1 = x z , F 2 = y e x , F 3 = y z using (ii) c u r l F = ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z x z y e x y z | , by expanding 3 × 3 determinant = i ( ∂ ∂ y ( y z ) − ∂ ∂ z ( y e x ) ) − j ( ∂ ∂ x ( y z ) − ∂ ∂ z ( x z ) ) + k ( ∂ ∂ x ( y e x ) − ∂ ∂ y ( x z ) ) = i ( z − 0 ) − j ( 0 − x ) + k ( y e x − 0 ) = i z + j x + k y e x c u r l F = i z + j x + k y e x , [using (i)] d i v ( c u r l F ) = ∇ ⋅ ( c u r l F ) = ( i ∂ ∂ x + j ∂ ∂ y + k ∂ ∂ z ) ⋅ ( i z +

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17, Problem 17CRE

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$d i v (c u r l F) = 0$

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Chapter 17, Problem 16CRE

Chapter 17, Problem 18CRE

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Chapter 17 Solutions

CALCULUS (CLOTH)

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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d i v ( c u r l F ) = 0

Solution Summary: The author explains that d i v ( c u r l F ) = 0 is verified for the given vectors F and G.

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Chapter 17 Solutions

$d i v (c u r l F) = 0$