The following is the first main text for the course. Introduction to Analysis in Several Variables.

We refer to this text as “Intro to Analysis.” The following provides useful additional (and more basic) background in analysis.

Introduction to Analysis in One Variable.

Multivariable calculus makes use of linear algebra. Sufficient background in linear algebra can be found here.

Linear Algebra

Section 2.2 of Intro to Analysis treats the fundamental Inverse Function Theorem, for a smooth function of n variables (mapping into n-dimensional Euclidean space). The following set of notes presents an alternative treatment, using Newton’s method.

The Inverse Function Theorem via Newton’s Method

Section 2.3 of Intro to Analysis deals with existence of solutions to systems of (nonlinear) ODE. Compare the treatment in Section 1 of the following.

Nonlinear ODE

See the following for background on linear systems of ODE, with emphasis on the matrix exponential. This makes use of the Linear Algebra notes mentioned above.

Linear ODE

Section 1.1 of Intro to Analysis develops the Riemann integral for functions on an interval in the real line. A similar treatment is given in the following, which is Chapter 1 of Measure Theory and Integration.

Chapter 1. The Riemann Integral

The multivariable integral covered in Intro to Analysis is also the Riemann integral. However, this course will emphasize an introduction to the Lebesgue integral. The following material (Chapters 2-4 of Measure Theory and Integration) will cover this.

Chapter 2. Lebesgue Measure on the Line

Chapter 3. Integration on Measure Spaces

This course is followed by a course on complex analysis, Math 656, covered in the following manuscript. Parts of these notes will also be useful for Math 653.

Introduction to Complex Analysis.

Click here for figures used in the text.