Explanation Given: 1) Need to prove c u r l ( f F ) = f c u r l ( F ) + ∇ f × F Formula used: 1) c u r l ( F ) = ∇ × F 2) ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F 1 F 2 F 3 | where F = F 1 i + F 2 j + F 3 k Calculation: Lets calculate c u r l ( f F ) c u r l ( f F ) = ∇ × ( f F ) [usingformula c u r l ( F ) = ∇ × F ] = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z f F 1 f F 2 f F 3 | [using formula ∇ × F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F 1 F 2 F 3 | ] = i [ ∂ ∂ y ( f F 3 ) − ∂ ∂ z ( f F 2 ) ] − j [ ∂ ∂ x ( f F 3 ) − ∂ ∂ z ( f F 1 ) ] + k [ ∂ ∂ x ( f F 2 ) − ∂ ∂ y ( f F 1 ) ] = i [ ∂ f ∂ y F 3 + f ∂ F 3 ∂ y − ∂ f ∂ z F 2 − f ∂ F 2 ∂ z ] − j [ ∂ f ∂ x F 3 + f ∂ F 3 ∂ x − ∂ f ∂ z F 1 − f ∂ F 1 ∂ z ] + k [ ∂ f ∂ x F 2 + f ∂ F 2 ∂ x − ∂ f ∂ y F 1 − f ∂ F 1 ∂ y ] = i [ ∂ f ∂ y F 3 − ∂ f ∂ z F 2 ] + i [ f ∂ F 3 ∂ y − f ∂ F 2 ∂ z ] − j [ ∂ f ∂ x F 3 − ∂ f ∂ z F 1 ] − j [ f ∂ F 3 ∂ x − f ∂ F 1 ∂ z ]

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Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18.2, Problem 33E

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Prove the given product rule.

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Chapter 18.2, Problem 32E

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Chapter 18 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 18.1 - Prob. 1PQ Ch. 18.1 - Prob. 2PQ Ch. 18.1 - Prob. 3PQ Ch. 18.1 - Prob. 4PQ Ch. 18.1 - Prob. 5PQ Ch. 18.1 - Prob. 1E Ch. 18.1 - Prob. 2E Ch. 18.1 - Prob. 3E Ch. 18.1 - Prob. 4E Ch. 18.1 - Prob. 5E

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Chapter 18 Solutions