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MATH114 Fall 2007 - Final Cheat Sheet

MATH114 Fall 2007 - Final Cheat Sheet - 13.5 Equations of...

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13.5 Equations of Lines and Planes Distance from a point to a plane: 1 1 1 2 2 2 ax by cz d D a b c + + + = + + 13.7 Cylindrical and Spherical Coordinates Rectangular ( , , ) x y z Cylindrical ( , , ) r z θ 2 2 2 r x y = + , tan x y θ = , z = z Cylindrical ( , , ) r z θ Rectangular ( , , ) x y z cos x r θ = , sin y r θ = , z z = Rectangular ( , , ) x y z Spherical ( , , ) ρ θ φ 2 2 2 2 x y z ρ = + + , cos z φ ρ = , cos sin x θ ρ φ = Spherical ( , , ) ρ θ φ Rectangular ( , , ) x y z sin cos x ρ φ θ = , sin sin y ρ φ θ = , cos z ρ φ = Spherical ( , , ) ρ θ φ Cylindrical ( , , ) r z θ sin r ρ φ = , θ θ = , cos z ρ φ = 14.3 Arc Length and Curvature Arc Length : [ ] [ ] [ ] 2 2 2 '( ) '( ) '( ) '( ) b b a a L f t g t h t dt r t dt = + + = Arc Length Function : ( ) '( ) t a s t r u du = , length of curve from a to t. Curvature : 3 '( ) '( ) ''( ) '( ) '( ) T t r t r t dT ds r t r t κ = = = Normal and Binormal Vectors: '( ) ( ) '( ) T t N t T t = , ( ) ( ) ( ) B t N t T t = Plane with T and N = Osculating Plane defined by B(t) Plane with N and B = Normal Plane defined by r’(t) 14.4 Motion in Space 0 ( cos ) x v t α = , 2 0 ( sin ) 0.5 y v t gt α = - '( ) ''( ) '( ) T r t r t a r t x = , '( ) ''( ) '( ) N r t r t a r t x = 15.2 Limits and Continuity To see if lim exists: x=0, y=0, x=y etc.
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