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MATH503 hw1

# MATH503 hw1 - x = u v and y =-2 u v • The electrostatic...

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Homeworks (Math 503) T1 : Div, grad, curl and all that , by Schey (4th edition) T2 : Calculus of variations , by Gelfand & Fomin T3 : Nonlinear dynamics and chaos , by Strogatz Note: Detail your work to receive full credit Homework#1: (due Wed Sep 22) Chap. II (T1): 4(b), 5(a), 23(b), 24 Evaluate Z Z S p 5 x 2 + 2 xy + 2 y 2 dxdy over the region S bounded by the ellipse 5 x 2 +2 xy +2 y 2 = 1 by making the change of variables
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Unformatted text preview: x = u + v and y =-2 u + v . • The electrostatic potential at (0 , ,-a ) of a charge of constant density ρ on the hemisphere S : x 2 + y 2 + z 2 = a 2 ,z ≥ 0 is U = Z Z S ρ p x 2 + y 2 + ( z + a ) 2 dS Use the spherical coordinates to evaluate U . • Prove ∇ · ( ∇ × F ) = 0. Verify by applying to F = x 2 yz i-x 3 y 3 j + xyz 2 k ....
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