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FactSheet2_Math20E

# FactSheet2_Math20E - Fact Sheet for Midterm 2 If f(x y is...

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Fact Sheet for Midterm 2 If f ( x, y ) is of class C 2 then 2 f ∂x∂y = 2 f ∂y∂x First order Taylor approximation of f : R n R at x 0 : l ( x ) = f ( x 0 ) + n i =1 h i ∂f ∂x i ( x 0 ), with h = x - x 0 . Second order Taylor approximation of f : R n R at x 0 : q ( x ) = f ( x 0 ) + n i =1 h i ∂f ∂x i ( x 0 ) + 1 2 n i,j =1 h i h j 2 f ∂x i ∂x j ( x 0 ), with h = x - x 0 . Differentiation of vector-valued functions is done component wise. If c ( t ) is a path, then its velocity is v ( t ) = c 0 ( t ), its speed is || c 0 ( t ) || , and its acceleration is a ( t ) = c 00 ( t ). The arc length of a path c ( t ) on the interval [ t 0 , t 1 ] is R t 1 t 0 || c 0 ( t ) || dt . A flow line of a vector field F in R n is a path c ( t ) in R n such that c 0 ( t ) = F ( c ( t )). Let f be a function from R n to R , then the gradient of f is f = ( ∂f ∂x 1 , · · · , ∂f ∂x n ). div F = ∇· F = ∂F 1 ∂x + ∂F 2 ∂y + ∂F 3 ∂z , where F = ( F 1 , F 2 , F 3 ) is a vector field. curl F = ∇ × F = i j k ∂x ∂y ∂z F 1 F 2 F 3 , where F = ( F 1 , F 2 , F 3 ) is a vector field in R 3 . • ∇ × ( f ) = 0 and ∇ · ( ∇ × F ) = 0 with f : R 3 R , and F a vector field in R 3 . Let T be a linear mapping from R 2 to R 2 such that T ( x ) = A x , where A
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