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exam 1 spring 2006-1 - FORMULA SHEET n dS = Q D S encl E r...

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FORMULA SHEET ˆ encl S D ndS Q = Ñ ( 29 ( 29 2 0 ' ˆ ' 4 v V r E r R dV R ρ πε = F qE = dq i dt = ˆ S i J n dS = v J v ρ = B AB A V E dr = ( 29 ( 29 PE r q r C = Φ + dS d dz dV d d dz ρ φ ρ ρ φ = = 2 2 sin sin dS r d d dV r dr d d θ θ φ θ θ φ = = ˆ ˆ ˆ ˆ ˆ ˆ sin dr z dz d d dr r dr r d r d ρ ρ φ ρ φ φ θ φ θ θ = + + = + + 1
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TABLE OF INTEGRALS ( 29 ( 29 2 2 1/ 2 2 2 ln dx x x a x a = + + + ( 29 3/ 2 2 2 2 2 2 dx x a x a x a = + + ( 29 3/ 2 2 2 2 2 1 x dx x a x a - = + + ( 29 2 2 1/ 2 2 2 x dx x a x a = + + ( 29 ( 29 2 2 2 3/ 2 2 2 2 2 ln x dx x x x a x a x a - = + + + + + ( 29 ( 29 2 2 2 2 2 2 1/ 2 2 2 ln 2 2 x dx x x a a x x a x a + = - + + + ( 29 ( 29 3/ 2 5/ 2 2 2 2 2 1 5 x x a dx x a + = + ( 29 ( 29 ( 29 3/ 2 2 2 2 2 2 3/ 2 2 2 4 2 2 3 3 ln 4 8 8 x x a a x x a x a dx a x x a + + + = + + + + 2
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TABLE OF COORDINATE SYSTEM FORMULAS cos x ρ φ = ˆ ˆ sin cos / x r θ φ = = r x sin y ρ φ = ˆ ˆ sin sin / y r θ φ = = r y z z = ˆ ˆ cos / z r θ = = r z sin cos x r θ φ = sin sin y r θ φ = cos z r θ = 2 2 x y ρ = + ( 29 1 tan / y x φ - = z z = sin r ρ θ = cos z r θ = φ φ = 2 2 2 r x y z = + + ( 29 1 2 2
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