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329fall08hw4 - ECE 329 Homework 4 Due 5PM 1 Curl and...

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ECE 329 Homework 4 Due: Sept 23, 2008, 5PM 1. Curl and divergence exercises: a) On a 25-point graph consisting of x and y coordinates having the integer values { - 2 , - 1 , 0 , 1 , 2 } sketch the vector field V = x ˆ x + y ˆ y and find ∇ × V (curl of V ) and · V (divergence of V ). b) Repeat (a) for V = - y ˆ x + x ˆ y . c) Based on above results choose the correct answer in the statements below: i. ∇ × V = 0 implies the field strength varies ( along or across ) the direction of the field. ii. · V = 0 implies the field strength varies ( along or across ) the direction of the field. 2. Verifying vector calculus identities , ∇ × ( f ) = 0 and · ( ∇ × A ) = 0 : a) The gradient of a scalar field f is defined as f f x ˆ x + f y ˆ y + f z ˆ z . Assuming that the order of di ff erentiation can be switched, show that ∇ × ( f ) = 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.
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