Transcription of Multivariable Calculus - Duke University
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Multivariable CalculusJerry ShurmanReed CollegeContentsPreface..ix1 Results from One-Variable Calculus .. The Real Number System .. Foundational and Basic Theorems .. Taylor s Theorem .. 6 Part I Multivariable Differential Calculus2 Euclidean Space.. Algebra: Vectors .. Geometry: Length and Angle .. Analysis: Continuous Mappings .. Topology: Compact Sets and Continuity .. Review of the One-Variable Derivative .. Summary .. 603 Linear Mappings and Their Matrices.. Linear Mappings .. Operations on Matrices .. The Inverse of a Linear Mapping .. Inhomogeneous Linear Equations .. The Determinant: Characterizing Properties and TheirConsequences .. The Determinant: Uniqueness and Existence .. An Explicit Formula for the Inverse .. Geometry of the Determinant: Volume .. Geometry of the Determinant: Orientation .. The Cross Product, Lines, and Planes inR3.. Summary .. 130viContents4 The Derivative.. Trying to Extend the Symbol-Pattern: Immediate, IrreparableCatastrophe.
the variables x1 through xn, it can be viewed as optimizing over an (n−1)-dimensional spaceinside ndimensions. The problem mayappearunmotivated, but its solution leads quickly to a generalization of the arithmetic-geometric mean inequality √ ab≤(a+b)/2 for all nonnegative aand b, ae1 1 ···a en n ≤e1a1 +···+enan for all nonnegative ...
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