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 cos sin hatwide a sin cos hatwide a z 1 Cartesian to Spherical : hatwide a x hatwide a y hatwide a z hatwide a r sin cos sin sin cos hatwide a cos cos cos sin sin hatwide a sin cos Cylindrical to Spherical : hatwide a hatwide a hatwide a z hatwide a r sin cos hatwide a cos sin hatwide a 1 1 Cartesian Cylindrical Spherical x = cos = r sin cos y = sin = r sin sin z = z = r cos Cylindrical Cartesian Spherical = radicalbig x 2 + y 2 = r sin = arctan( y/x ) = z = z = r cos Spherical Cartesian Cylindrical r = radicalbig x 2 + y 2 + z 2 = radicalbig 2 + z 2 = arctan( radicalbig x 2 + y 2 /z ) = arctan( /z ) = arctan( y/x ) = 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 d + a d + a z dz ds = ddz ds = ddz ds z = dd dv = dddz Spherical coordinates dl = a r dr + a rd + a r sin d ds r = r 2 sin dd ds = r sin drd ds = rdrd dv = r 2 sin drdd 2 D. Grad, Div, Curl and Laplacian 1. Cartesian ( x,y,z ) = a x x + a y y + a z 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 = 2 x 2 + 2 y 2 + 2 z 2 2. Cylindrical ( ,,z ) = a + a 1 + a z z A = 1 ( A ) + 1 A + A z z A = a bracketleftbigg 1 A z A z bracketrightbigg + a bracketleftbigg A...
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final_formula - ECE311 Potentially Useful Formulas - Fall...

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