Working Analysis is for a two semester course in advanced calculus. It develops the basic ideas of calculus rigorously but with an eye to showing how mathematics connects with other areas of science and engineering. In particular, effective numerical computation is developed as an important aspect of mathematical analysis.
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Sequences of Real Numbers
Higher Derivatives and Polynomial
Solving Equations in One Dimension
The Derivative in Rn
Solving Systems of Equations
Quadratic Approximation and Optimization
Applications of Integration to Differential
Convergence and Continuity in
assume calculate Cauchy sequence chain rule change of variable Chapter compact compute constant constraint continuous function converges uniformly convex coordinates critical point curve deduce defined denote Df(x differential equation error estimate example exercise exists fixed point formula function f(x functional iterates given graph Hence implicit function theorem implies improper integral intermediate value theorem inverse function theorem Jordan domain Lagrange equations least upper bound Lemma Let f(x Let G limit linear approximation Lipschitz mapping maximizer mean value theorem minimizer minimum monotone Newton's method norm one-to-one open interval open set parameters partition positive definite power series problem quadratic rational numbers real numbers rectangle result satisfies Section sequence xn Show Simpson's rule solution steepest descent subset Suppose tangent Taylor polynomial tends to zero trapezoid rule uniformly continuous unique vector