ELEN_411_Homework_1

# ELEN_411_Homework_1 - (a) A = x 2 ȃ x + 3 xz 2 ȃ y – 2...

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ELEN 411 Homework 1 Due 3/16/11 Problem 1: Using the figure below, compute C · C and verify that the result is law of cosines. [ Answer: g ∙ g = A G + B G − 2AB cos ±] Problem 2: Find the gradients of the following functions: (a) f ( x, y, z ) = x 2 + y 3 + z 4 (b) g ( x , y , z ) = x 2 y 3 z 4 (c) h ( x , y , z ) = e x sin( y ) ln( z ) [ Answers: (a) ∇² = 2³´ µ + 3¶ G ´ · + 4¸ ¹ ´ º , (b) ∇» = 2³¶ ¹ ¸ ¼ ´ µ + 3³ G G ¸ ¼ ´ · + 4³ G ¹ ¸ ¹ ´ º , (c) ∇ℎ = e ½ sin¶ ln ¸ ´ µ + e ½ cos ¶ ln ¸ ´ · + e ½ sin¶ ¾ ¿ ´ À ] Problem 3: Calculate the divergence of the following vector functions: (a) A = x 2 ȃ x + 3 xz 2 ȃ y – 2 xz ȃ z (b) B = xy ȃ x + 2 yz ȃ y + 3 xz ȃ z (c) C = y 2 ȃ x + (2 xy + z 2 ) ȃ y + 2 yz ȃ z [ Answers: (a) ∇ ∙ Á = 0 , (b) ∇ ∙ Â = 3³ + ¶ + 2¸ , (c) ∇ ∙ g = 2³ + 2¶ ] Problem 4: Calculate the curls of the following vector functions:

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Unformatted text preview: (a) A = x 2 ȃ x + 3 xz 2 ȃ y – 2 xz ȃ z (b) B = xy ȃ x + 2 yz ȃ y + 3 xz ȃ z (c) C = y 2 ȃ x + (2 xy + z 2 ) ȃ y + 2 yz ȃ z [ Answers: (a) ∇ × Á = 2¸´ · + 3¸ G ´ º , (b) ∇ × Â = −2¶´ µ − 3¸´ · − ³´ º , (c) ∇ × g = 0 ] Problem 5: Find formulas for r , θ , φ in terms of x , y , z (the inverse of the relationships given in class). [ Answer: Ã = Ä³ G + ¶ G + ¸ G , ± = tan Å¾ Ä½ Æ ÇÈ Æ ¿ ,É = tan Å¾ È ½ ] Problem 6: Use surface integrals to derive the equation for the surface area of a cylinder of radius R with height h . [ Answer: g = 2G± ² + 2G±ℎ ]...
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## This note was uploaded on 09/26/2011 for the course ECE 123 taught by Professor Crank during the Spring '11 term at LSU.

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ELEN_411_Homework_1 - (a) A = x 2 ȃ x + 3 xz 2 ȃ y – 2...

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