Formula_sheet - r r cos = a b(|| a |||| b || ^ compb a =||...

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Unformatted text preview: r r cos = a b /(|| a |||| b ||) ^ compb a =|| a || cos = a b r ^ ^ projb a = a b b ( ) Area of a parallelogram = a b Volume of a parallelepiped =a (bc) Equation of a li : E i f line r r r r r r r = r2 + t (r2 - r1 ) = r2 + ta Equation of a plane : a x + b y + cz + d = 0 E i f l r r r r r r also : [(r2 - r1 ) (r3 - r1 )] (r - r1 ) = 0 r r r dT d 2r | T' | || r(t ) r(t ) || = =|| 2 ||= r = ds ds | r'| || r(t ) ||3 r dv ^ d ^ ^ ^ a(t ) = v 2 N + T = a N N + aT T dt The Binormal (t dT ^ = r(t ) ^ = dT / dt T N ^ ^ ^ B = T N || r(t ) || || dT / dt || r r dr ( s ) dr ds = dt ds dt Length of a curve : s = g t2 t1 r | r ' ( t ) | dt aT = || v a || dv v a = & a N = kv 2 = || v || dt || v || =i + j +k x y z f f u f v f f u f v = + = & + x u x v x y u y v y Equation of Tangent Plane: r r r r equation of normal line to a surface : no (r - ro ) = 0, no = F at P r r r r r r no (r - ro ) = 0, no = F at P W = F dr C b ^ ^ Du ( F ) = F u, u = unit vector C F ( x, y )ds = F ( f (t ), g (t ) ) [ f ' ] + [ g ' ] dt = F ( x, f ( x ) ) 1 + [ f ' ]2 dx b 2 2 a a S r r r ^ F dr = (curl F ) ndS C ~= x D x ( x, y, z )dV m r r ^ ( F n)dS = (div F )dV [Pd + Qd ] = Q - P d d Pdx Qdy dxdy S D C R x y , m= D 2 2 ( x, y, z )dV I x = D ( y + z ) ( x, y, z )dV ; x = r cos , y = r sin ; z = z; r = x 2 + y 2 , = tan -1 ( y / x) J (u, v) = x = sin cos , y = sin sin , z = cos , dV = r dr d dz ( x, y ) (u , v) = x 2 + y 2 + z 2 , = tan -1 ( y / x), = tan -1 ( x 2 + y 2 / z ) dV = 2sin dd d ...
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This note was uploaded on 05/09/2010 for the course ENGR 233 taught by Professor S.samuelli during the Winter '10 term at Concordia Canada.

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