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# a1ho1 - M2A1 Double Taylor Series the Jacobian Matrix 1...

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1 Double Taylor Series Recall the single Taylor expansion of a function F ( x ) about a point x 0 . Let x x 0 = h , then F ( x ) = F ( x 0 ) + h dF ( x 0 ) dx + h 2 2! d 2 F ( x 0 ) dx 2 + ... (1) where we are using the notation 1 dF ( x 0 ) dx = dF dx ± ± ± ± ± x = x 0 d 2 F ( x 0 ) dx 2 = d 2 F dx 2 ± ± ± ± ± x = x 0 . (2) For the double Taylor expansion, consider the function F ( x,y ) which we desire to expand about the point ( x 0 , y 0 ). Let x x 0 = h and y y 0 = k. (3) Lemma 1 The double Taylor Series expansion for F ( x,y ) about the point ( x 0 , y 0 ) is F ( x,y ) = F ( x 0 ,y 0 ) + h ∂F ( x 0 ,y 0 ) ∂x + k ∂F ( x 0 ,y 0 ) ∂y + 1 2! ² h 2 2 F ( x 0 ,y 0 ) ∂x 2 + 2 hk 2 F ( x 0 ,y 0 ) ∂x∂y + k 2 2 F ( x 0 ,y 0 ) ∂y 2 ³ + ... (4) where we are using the notation ∂F ( x 0 ,y 0 ) ∂x = F ∂x ± ± ± ± ± x = x 0 ,y = y 0 2 F ( x 0 ,y 0 ) ∂x 2 = 2 F ∂x 2 ± ± ± ± ± x = x 0 ,y = y 0 etc (5) Proof: Consider the single Taylor expansion of F ( x 0 + h,y 0 + k ) in terms of h at ﬁxed y 0 + k F ( x 0 + h,y

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a1ho1 - M2A1 Double Taylor Series the Jacobian Matrix 1...

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