The Ultimate Math 192 Cheat Sheet

The Ultimate Math 192 Cheat Sheet - Partial Derivatives lim...

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Area of a Smooth Surface: ∫ ∫ × = d c b a v u dv du A r r Parametric Surface Integral: ∫ ∫ ∫ ∫ × = d c b a v u S dv du v u h v u g v u f G d z y x G r r )) , ( ), , ( ), , ( ( ) , , ( σ Partial Derivatives: ) , ( ) , ( ) , ( ) , ( 0 lim 2 2 2 0 0 0 0 ) , ( 0 0 b a f b a f dx f dy dydx f f dx f f h y x f y h x f h x f yx xy xy xx y x =  ∂ = = = - + = Implicit Differentiation: y x F F dx dy - = Gradient Field: k j i z f y f x f f + + = Directional Derivative: u u u = = f f D ds df P P 0 0 ) ( , 0 0 0 0 , , ) , , ( z y x P c x y x f = = Tangent Plane: 0 ) )( ( ) )( ( ) )( ( 0 0 0 0 0 0 = - + - + - z z P f y y P f x x P f z y x Normal Line: t P f z z t P f y y t P f x x z y x ) ( , ) ( , ) ( 0 0 0 0 0 0 + = + = + = )) , ( , , ( ) , ( 0 0 0 0 0 y x f y x P y x f z = = Tangent Plane: 0 ) ( ) )( ( ) )( ( 0 0 0 0 0 = - - - + - z z y y P f x x P f y x Differential: dy y x f dx y x f df y x ) , ( ) , ( 0 0 0 0 + = Estimating Change: ( 29 ds f df P = u 0 Coordinate Conversion Formulas Cylindrical to Spherical to Spherical to Rectangular Rectangular Cylindrical z z r y r x = = = θ θ sin cos φ ρ θ φ ρ θ φ ρ cos sin sin cos sin = = = z y x θ θ φ ρ φ ρ = = = cos sin z r θ φ ρ φ ρ θ d d d d dr r dz dz dy dx dV sin 2 = = = Work Over a Smooth Curve: = = = b t a t ds W T F Flow Integral, Circulation: Flow = b a ds T F Flux Across a Closed Curve: Flux = C ds n F Flux = - C dx N dy M Double Integrals Average Value: ∫ ∫ = R avg dA f R of area f 1 Mass:
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