By Kenneth Eriksson, Donald Estep, Claes Johnson
Applied arithmetic: physique & Soul is a arithmetic schooling reform venture constructed at Chalmers collage of know-how and incorporates a sequence of volumes and software program. this system is stimulated by way of the pc revolution establishing new possibilitites of computational mathematical modeling in arithmetic, technological know-how and engineering. It contains a synthesis of Mathematical research (Soul), Numerical Computation (Body) and alertness. Volumes I-III current a contemporary model of Calculus and Linear Algebra, together with constructive/numerical suggestions and purposes meant for undergraduate courses in engineering and technological know-how. additional volumes current themes resembling Dynamical structures, Fluid Dynamics, sturdy Mechanics and Electro-Magnetics on a complicated undergraduate/graduate point.
The authors are best researchers in Computational arithmetic who've written a number of winning books.
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Extra info for Applied Mathematics: Body and Soul: Calculus in Several Dimensions
1 Introduction . . . . . . 2 An Analytieal Solution Formula . 3 Construction of the Solution . . 1 Introduction............... 2 An Analytical Solution Formula . . 4 A Generalization . . . . . 1 Introduction............ 2 Determinism and Materialism . 4 Construction of the Solution. . 5 Computational Work . . . . 7 Numerical Methods . . . . . . . . . 1 Introduction . . 2 Rational Numbers . . . . . 3 Real Numbers. 6 Derivatives . . . . . . 7 Differentiation Rules .
Matrix Calculus . . . . . . The Transpose of a Linear Transformation. Matrix Norms . . . . . . . . . The Lipsehitz Constant of a Linear Transformation Volume in IRn: Determinants and Permutations Definition of the Volume V(al, ... ,an) The Volume V(al, a2) in IR 2 . . The Volume V(al, a2, a3) in IR 3 . The Volume V(al, a2, a3, a4) in IR 4 The Volume V(al, ... , an) in IRn . The Determinant of a Triangular Matrix Using the Column Echelon Form to Compute det A . The Magie Formula det AB = det A det B .
The function f: [0,1] x [0,1]---* jR2 defined by f(Xl,X2) = (Xl + X2, XIX2), is Lipschitz continuous with Lipschitz constant L = 2. To show this, we note that h (Xl, X2) = Xl + X2 is Lipschitz continuous on [0,1] x [0,1] with Lipschitz constant LI = V2 because Ih(Xl,X2)h(Yl,Y2)1 :::; lXI - Yll + IX2 - Y21 :::; V2l1x - Yll by Cauchy's inequality. 8. The function f : jRn ---* jRn defined by is Lipschitz continuous with Lipschitz constant L = 1. 9. A linear transformation f : jRn ---* jRm given by an m x n matrix A = (aij), with f(x) = Ax and X a n-column vector, is Lipschitz continuous with Lipschitz constant L = IIAII.
Applied Mathematics: Body and Soul: Calculus in Several Dimensions by Kenneth Eriksson, Donald Estep, Claes Johnson