Final%20exam%20formula

Final%20exam%20formula - rr cos θ = a • b /(|| a |||| b...

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Unformatted text preview: rr cos θ = a • b /(|| a |||| b ||) ˆ compb a =|| a || cos θ = a ⋅ b rˆˆ projb a = a ⋅ b b () Area of a parallelogram = ⎪⎢a × b ⎪⎢ Volume of a parallelepiped =⎮a • (b×c)⎮ Equation of a line : rr rr rr r = r2 + t (r2 − r1 ) = r2 + ta Equation of a plane : a x + b y + cz + d = 0 rr rr rr also : [(r2 − r1 ) × (r3 − r1 )]• (r − r1 ) = 0 r r r dT d 2r | T' | || r′(t ) × r′′(t ) || κ= =|| 2 ||= r = ds ds | r'| || r′(t ) ||3 r ˆ dv ˆ ˆ ˆ a(t ) = κv 2 N + T = a N N + aT T dt The Binormal ˆ = r′(t ) ˆ = dT / dt T N ˆˆˆ B = T× N || r′(t ) || || dT / dt || r r dr ( s ) dr ds = dt ds dt Length of a curve : s = ∫ 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 rr r equation of normal line to a surface : no × (r − ro ) = 0, no = ∇F at P rr r rr r no • (r − ro ) = 0, no = ∇F at P W = ∫ F • dr C b ˆˆ 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 rr r ˆ ∫ F • dr = ∫∫ (curl F ) • ndS C ~= x ∫ ∫∫ D xρ ( x, y, z )dV m r r ˆ ( F • n)dS = ∫∫∫ (div F )dV ∫ [Pdx + Qdy ] = ∫∫ ⎡ ∂Q − ∂P ⎤ dxdy ∫∫S D C R ⎢ ∂x ∂y ⎥ ⎣ ⎦ , m= ∫∫∫ D 2 2 ρ ( x, y, z )dV I x = ∫ ∫∫D ( y + z ) ρ ( x, y, z )dV ; x = r cos θ , y = r sin θ ; z = z; r = x 2 + y 2 , θ = tan -1 ( y / x) J (u, v) = x = ρ sin φ cos θ , y = ρ sin φ sin θ , z = ρ cos φ , dV = r dr dθ dz ∂ ( x, y ) ∂ (u , v) ρ = x 2 + y 2 + z 2 , θ = tan -1 ( y / x), φ = tan −1 ( x 2 + y 2 / z ) dV = ρ 2sinφ dρdφ dθ ...
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This note was uploaded on 05/08/2009 for the course AAE 580 taught by Professor Nnjkl during the Spring '09 term at Ohio State.

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