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### Advanced Calculus Lecture Notes for Mathematics by James S. Muldowney

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Advanced Calculus Lecture Notes for Mathematics written by James S. Muldowney cover the following topics.

• The Real Number system & Finite Dimensional Cartesian Space
The Real Number System R (Fields, Ordered Fields), Complete Ordered Field, Properties of R, Cartesian Spaces (Functions, Convexity), Topology

• Limits, Continuity, and Differentiation
Sequences, Continuity, Global Properties of Continuous Functions, Uniform Continuity, Limits, Differentiation of real valued functions of a real variable

• Riemann Integration
Content and the Riemann Integral (Partition of I, Riemann Sums, Exercises), Cauchy Criteria and Properties of Integrals, Evaluation of Integrals (Real valued functions of a real variable, Real valued functions on R^2, Real valued functions on R^n)

• Differentiation 0f Functions of Several Variables
Preliminaries (Linear Functions, Straight Lines and Curves), The Directional Derivative and the Differential, Differentiation Rules, Partial Derivatives of Higher Order, Min-Max Theory, Local Properties of C, Implicit Function Theorem (Dimension, Application: Lagrange Multipliers)

• Further Topics in Integration
Changes of Variables in Integrals, Integration on Curves and Surfaces, Curves (1-surfaces), Surfaces (2-surfaces)

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