multivariable calculus

May 12th, 2015
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Multivariable Calculus1. The DerivativeDefinition: Derivative at a PointLet U be an open subset of Rm , let F : U Rn , and let a U . We say that F isdifferentiable at a if there exists an n m matrix DFa satisfyingF (a + h) F (a) DFa (h)= 0.h0khklimIn this case, DFa is called the derivative of F at a.A few notes about this definition:1. The matrix DFa is uniquely determined. Specifically, if A and B are two matrices satisfying the definition of DFa , then subtracting the limits givesAh Bh= 0,h0khklimwhich implies that A = B.2. Though we have stated the definition using the variable h, we could alternativelyuse the variable x = a + h. In this case, the limit definition of DFa becomesF (x) F (a) DFa (x a)= 0.xakx aklim3. The limit definition for DFa is equivalent to the following condition: for every > 0, there exists a > 0 so thatkhk < kF (a + h) F (a) DFa (h)k khk.Multivariable Calculus2If we use x = a + h instead, this implication becomeskx ak < kF (x) F (a) DFa (x a)k kx ak.4. The definition of the derivative is closely related to a linear approximationformula. Specifically, the definition states that the error in the approximationF (a + h) F (a) + DFa (h)is much smaller than h as h 0. This is sometimes writtenF (a + h) = F (a) + DFa (h) + o(h),where o(h) denotes an arbitrary function satisfyingo(h)= 0.h0 khk

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