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ENGR233-_formula_sheet

ENGR233-_formula_sheet - cos = a b(| a | b | compb a =| a |...

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||) |||| /(|| cos b a b a = θ ˆ cos || || comp b a a a b = = θ ( 29 b b a a b ˆ ˆ proj = Volume of a parallelepiped = a (b × c) Area of a parallelogram =  a × b  a t r r r t r r + = - + = 2 1 2 2 ) ( : line a of Equation [ ] 0 ) ( ) ( ) ( : also 0 z y x : plane a of Equation 1 1 3 1 2 = - - × - = + + + r r r r r r d c b a dt ds ds r d dt s r d = ) ( = 2 1 | ) ( ' | : curve a of Length t t dt t r s || ) ( || ) ( ˆ t t r r T = 3 2 2 || ) ( || || ) ( ) ( || | ' | | ' | || || t t t ds d ds d r r r r T r T × = = = = κ || / || / ˆ dt d dt d T T N = T N T N a ˆ ˆ ˆ ˆ ) ( 2 T N a a dt dv v t + = + = κ || || || || & || || 2 v a v v a v × = = = = kv a dt dv a N T N T B ˆ ˆ ˆ × = The Binormal z y x + + = k j i r unit vecto ˆ , ˆ ) ( = = u u F F D u Equation of Tangent Plane: ( 29 P at , 0 F n r r n o o o = = - ( 29 P at , 0 : surface a to line normal of equation F n r r n o o o = = - × & y v v f y u u f y f x v v f x u u f x f + = + = = C r d F W ( 29 ( 29 + = + = b a b a C dx f x f x F dt g f t g t f F ds y x F 2 2 2 ] ' [ 1 ) ( , ] ' [ ] ' [ ) ( ), ( ) , ( ) , ( ) , ( ) , ( v u y x v u J = ∫ ∫ = S C dS n F curl r d F ˆ ) ( dV F div dS n F D S ) ( ) ˆ ( ∫ ∫ ∫ ∫ ∫ =
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