HW1_Solution

HW1_Solution - ECE 201A Problem set 1 solution 1. (a) ( AB...

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ECE 201A Problem set 1 solution 1. (a) 2 ()() ()( ) () AA A A A AB C B A C C ∇× ∇× =∇ ∇⋅ − ∇⋅ ∇ =∇∇ ⋅ − ↑↑↑↑ ↑↑ ×× = (b) ( ) ( ) cc EH E H ∇⋅ × =∇⋅ × +∇⋅ × This is the application of the chain rule. Subscript c means that in the differentiation the term with that subscript will be treated as a constant. Then since doesn't rule #1 rule #2 operate on ( ) ( ) c c E HE × =−∇⋅ × =−∇× =− ∇× since doesn't rule #1 operate on ( ) c H × =∇× ⋅ = ∇× Combining the two ( ) × − ∇× 2. (a) In rectangular coordinates 123 (, , ) (,,) uuu xyz = In cylindrical coordinates z ρφ = In spherical coordinates (, , ) ( ,,) r φθ = x y z r X Y Z φ θ ( x,y,z ) ( ρ , ,z) ( r, , ) (b) In rectangular coordinates ( , , ) (1,1,1) hhh = In cylindrical coordinates 1 ,, 1 ) = In spherical coordinates 1 ,,s i n) rr =
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(c) According to the definition of gradient () 1 11 1 1 hu φ ∂∂ ∇= = A s o , 123 2 2 33 1 ˆˆˆ aaa φφ ∂∂∂ + + In rectangular coordinates: x yz x + + In cylindrical coordinates: 1 ˆˆ ˆ z aa a z ρφ ρρ + + In spherical coordinates: r â r + ââ sin rr θ θθ + (d) lim 0 V V Fd S F ∇• = G G G v u 2 u 1 u 3 h 1 du 1 h 2 du 2 h 3 du 3 0 V Let ˆ Fa F ++ = JG Flux coming out of volume, V, has 6 components, which are:
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11 2 3 2 3 2 323 21 2 3 2 3 1 313 3 3 123 31 2 3 3 1 212 22 ,, du du Fd S F u uu uu hhd ud u du du Fuu u Fuu u h h d u d u du du F u u u F u u u h h du du ⎡⎤ ⎛⎞ ⋅= + ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ ++ G G v Note that h 1 , h 2 and h 3 are functions of u 1 ,u 2 and u 3 . So we should write: 2 3 1 1 23 2 3 2 3 , du du hhF u u u du du +− + …similar terms Dividing this by the volume and taking the limit as V Æ 0, for the first term we get: () 231 1 2 3 2 3 23 1 1 00 (, , ) , ) 1 lim lim 1 du du du du hhh du hhF hhh u →→ = Other terms are evaluated similarly and the result becomes: 231 13 2 12 3 1 2 3 1 Fh h F h h F h h F u u u ∂∂∂ ∇⋅ = + + G In rectangular coordinates: y x z F FF F x yz ∂∂ = + + G In cylindrical coordinates: ( ) 1 z z pF F F F F zz ρ φφ ρρ φ = + + = + + G Appropriate expressions can also be generated for the spherical coordinates.
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This note was uploaded on 12/02/2009 for the course ECE 000 taught by Professor O during the Spring '09 term at UCSB.

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HW1_Solution - ECE 201A Problem set 1 solution 1. (a) ( AB...

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