n_15046 - Formulas for Final Polar coordinates: x = r cos ,...

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Formulas for Final Polar coordinates: x = r cos θ,y = r sin θ,r 2 = x 2 + y 2 , tan θ = y/x Identity satis±ed by angle between two vectors: a · b = | a || b | cos θ Vector projection of b onto a: proj a ( b )= ³ b · a | a | ´ a | a | The component of b in the direction of a: comp a ( b )= b · a | a | Cross product: h a 1 ,a 2 ,a 3 i×h b 1 ,b 2 ,b 3 i =( a 2 b 3 a 3 b 2 ) i +( a 3 b 1 a 1 b 3 ) j +( a 1 b 2 a 2 b 1 ) k The area of the parallelogram with a and b as edges is | a × b | Arclength: of the curve deFned by vector function r ( t ), a t b ,is L = R b a | r 0 ( t ) | dt Gradient: f ( x,y )= h f x ( x,y ) ,f y ( x,y ) i Chain Rule: Case 1 : d dt F ( x ( t ) ,y ( t )) = ∂F ∂x dx dt + ∂F ∂y dy dt Case 2: If z = f ( x,y ) where x = g ( s,t ) and y = h ( s,t ), then ∂z ∂s = ∂z ∂x ∂x ∂s + ∂z ∂y ∂y ∂s ∂z ∂t = ∂z ∂x ∂x ∂t + ∂z ∂y ∂y ∂t Directional Derivative: In direction of unit vector u = h a,b i : D u f ( x,y )= f x ( x,y ) a + f y ( x,y ) b Tangent plane: Tangent plane to surface z = f ( x,y )atpo int( x 0 ,y 0 ,z 0 )is z z 0 = f x ( x 0 ,y 0 )( x x 0 )+ f y ( x 0 ,y 0 )( y y 0 ) . The value of D: (used in the second derivative test) D = f xx f yy f 2
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