oo x( i), where {x U)} ~1 is a sequence in lR. n generated by Fixed Point Iteration: xU) = g(X(i-1)), i = 1,2, ... , starting with any initial value x(O). 15 The Contraction Mapping Theorem 807 The proof is word by word the same as in the case 9 : :IR. --+ :IR. considered in Chapter Fixed Points and Contmction Mappings. We repeat the proof for the convenience of the reader. Subtracting the equation x ek ) = g(X ek - 1)) from x ek+ 1) = g(x ek )), we get Xek+1) _ xek) = g(xek)) _ g(X ek - 1)), and using the Lipschitz continuity of g, we thus have Repeating this estimate, we find that and thus for j >i j-1 Ilx ei ) - x(j) II ~ L Ilx ek ) - x ek+1) 11 k=i Since L < 1, {xe i)} ~1 is a Cauchy sequence in :lR.

2 Paying Taxes . 3 Hiking...... 6 The Derivative of x 2 Is 2x . . . 7 The Derivative of x n Is nx n - 1 . . 9 The Derivative as a Function . . 11 Denoting the Derivative of f(x) by 1,; .. 12 The Derivative as a Limit of Difference Quotients . 13 How to Compute a Derivative? . . . . . 14 Uniform Differentiability on an Interval . . . 16 A Slightly Different Viewpoint . 17 Swedenborg . . . . . . 1 Introduction . . . . . 4 The Chain Rule . . . . 5 The Quotient Rule .

The Fundamental Theorem of Linear Algebra Change of Basis: Coordinates and Matriees Least Squares Methods . . . . . . 1 Eigenvalues and Eigenveetors . . . . . . 2 Basis of Eigenveetors . . . . . . . . . 4 Applying the Speetral Theorem to an IVP . . . 5 The General Spectral Theorem for Symmetrie Matriees . . . . . 6 The Norm of a Symmetrie Matrix. . . 7 Extension to Non-Symmetrie Real Matriees . . 1 Introduction . . . . . . 2 Direct Methods . . . .

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Applied Mathematics: Body and Soul: Calculus in Several Dimensions by Kenneth Eriksson, Donald Estep, Claes Johnson


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