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ELEN_411_Homework_1

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

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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: g1777 ∙ g1777 = A g2870 + B g2870 − 2AB cos g2016] 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) ∇g1858 = 2g1876g2183 g2206 + 3g1877 g2870 g2183 g2207 + 4g1878 g2871 g2183 g2208 , (b) ∇g1859 = 2g1876g1877 g2871 g1878 g2872 g2183 g2206 + 3g1876 g2870 g1877 g2870 g1878 g2872 g2183 g2207 + 4g1876 g2870 g1877 g2871 g1878 g2871 g2183 g2208 , (c) ∇ℎ = e g3051 sing1877 ln g1878 g2183 g2206 + e g3051 cos g1877 ln g1878 g2183 g2207 + e g3051 sing1877 g2869 g3053 g2183 g2182 ] 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) ∇ ∙ g1775 = 0 , (b) ∇ ∙ g1776 = 3g1876 + g1877 + 2g1878 , (c) ∇ ∙ g1777 = 2g1876 + 2g1877 ] Problem 4:

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

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