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# 329sp09hw5 - ECE 329 Homework 5 Due Thursday 5PM = 0 and 1...

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ECE 329 Homework 5 Due: Thursday, Feb 24, 2009, 5PM 1. Verifying vector calculus identities , ∇ × ( Φ ) = 0 and · ( ∇ × A ) = 0 : a) The gradient of a scalar field Φ is defined as Φ Φ x ˆ x + Φ y ˆ y + Φ z ˆ z . Assuming that the order of di ff erentiation can be switched, show that ∇ × ( Φ ) = 0 . Consequently, any curl-free vector field can be expressed as the gradient of some scalar field — important in the definition of electrostatic potential studied in Chapter 6. b) Given any di ff erentiable vector field A = ˆ xA x + ˆ yA y + ˆ zA z , show that · ( ∇ × A ) = 0 by first expanding ∇ × A in terms of partial derivatives (e.g., A x x , A y x etc.) of the components of A . Consequently, any divergence-free vector field can be expressed as the curl of some other vector field — important in the definition of vector potential studied in Chapter 6. 2. Another important vector identity is ∇ × ( ∇ × A ) = ( · A ) - ∇ 2 A , where 2 A ( 2 x

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329sp09hw5 - ECE 329 Homework 5 Due Thursday 5PM = 0 and 1...

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