final_formula

final_formula - ECE311 Potentially Useful Formulas Fall...

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Unformatted text preview: ECE311 Potentially Useful Formulas - Fall 2007 A. Simple Integrals integraldisplay dx √ x 2 + c 2 = ln parenleftBig x + radicalbig x 2 + c 2 parenrightBig integraldisplay dx x 2 + c 2 = 1 c tan − 1 x c integraldisplay dx ( x 2 + c 2 ) 3 / 2 = 1 c 2 x √ x 2 + c 2 integraldisplay xdx √ x 2 + c 2 = radicalbig x 2 + c 2 integraldisplay xdx x 2 + c 2 = 1 2 ln ( x 2 + c 2 ) integraldisplay xdx ( x 2 + c 2 ) 3 / 2 = − 1 √ x 2 + c 2 integraldisplay dx ( a + bx ) 2 = − 1 b ( a + bx ) B. Coordinate Transformations Cartesian to Cylindrical : hatwide a x hatwide a y hatwide a z hatwide a ρ cos φ sin φ hatwide a φ − sin φ cos φ hatwide a z 1 Cartesian to Spherical : hatwide a x hatwide a y hatwide a z hatwide a r sin θ cos φ sin θ sin φ cos θ hatwide a θ cos θ cos φ cos θ sin φ − sin θ hatwide a φ − sin φ cos φ Cylindrical to Spherical : hatwide a ρ hatwide a φ hatwide a z hatwide a r sin θ cos θ hatwide a θ cos θ − sin θ hatwide a φ 1 1 Cartesian Cylindrical Spherical x = ρ cos φ = r sin θ cos φ y = ρ sin φ = r sin θ sin φ z = z = r cos θ Cylindrical Cartesian Spherical ρ = radicalbig x 2 + y 2 = r sin θ φ = arctan( y/x ) = φ z = z = r cos θ Spherical Cartesian Cylindrical r = radicalbig x 2 + y 2 + z 2 = radicalbig ρ 2 + z 2 θ = arctan( radicalbig x 2 + y 2 /z ) = arctan( ρ/z ) φ = arctan( y/x ) = φ C. Differential elements in several coordinate systems Cartesian coordinates ¯ dl = ¯ a x dx + ¯ a y dy + ¯ a z dz ds x = dydz ds y = dxdz ds z = dxdy dv = dxdydz Cylindrical coordinates ¯ dl = ¯ a ρ dρ + ¯ a φ ρdφ + ¯ a z dz ds ρ = ρdφdz ds φ = dρdz ds z = ρdρdφ dv = ρdρdφdz Spherical coordinates ¯ dl = ¯ a r dr + ¯ a θ rdθ + ¯ a φ r sin θdφ ds r = r 2 sin θdθdφ ds θ = r sin θdrdφ ds φ = rdrdθ dv = r 2 sin θdrdθdφ 2 D. Grad, Div, Curl and Laplacian 1. Cartesian ( x,y,z ) ∇ Φ = ¯ a x ∂ Φ ∂x + ¯ a y ∂ Φ ∂y + ¯ a z ∂ Φ ∂z ∇ · ¯ A = ∂A x ∂x + ∂A y ∂y + ∂A z ∂z ∇ × ¯ A = ¯ a x parenleftbigg ∂A z ∂y − ∂A y ∂z parenrightbigg + ¯ a y parenleftbigg ∂A x ∂z − ∂A z ∂x parenrightbigg + ¯ a z parenleftbigg ∂A y ∂x − ∂A x ∂y parenrightbigg ∇ 2 Φ = ∂ 2 Φ ∂x 2 + ∂ 2 Φ ∂y 2 + ∂ 2 Φ ∂z 2 2. Cylindrical ( ρ,φ,z ) ∇ Φ = ¯ a ρ ∂ Φ ∂ρ + ¯ a φ 1 ρ ∂ Φ ∂φ + ¯ a z ∂ Φ ∂z ∇ · ¯ A = 1 ρ ∂ ∂ρ ( ρA ρ ) + 1 ρ ∂A φ ∂φ + ∂A z ∂z ∇ × ¯ A = ¯ a ρ bracketleftbigg 1 ρ ∂A z ∂φ − ∂A φ ∂z bracketrightbigg + ¯ a φ bracketleftbigg ∂A ρ...
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final_formula - ECE311 Potentially Useful Formulas Fall...

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