Elementary Analysis III (Math 23)
1st Semester AY 2026-2027. For questions, contact me via pjburgos@math.upd.edu.ph.
Unit 1
We begin with an introduction to functions of several variables, their use, and their methods of visualization. These functions can be used to model real-world quantities and describe many different curves and surfaces.
In order to study these features, we borrow the idea of derivatives from single-variable calculus and apply it to the multivariable setting. This, in turn, will allow us to study both the analytic (e.g. approximations) and geometric (e.g. tangent planes) properties of multivariable functions.
- Lec 1.1. Functions of Several Variables, Level Curves and Level Surfaces
- Lec 1.2. Limits and Continuity of Functions of Several Variables
- Lec 1.3. Partial Derivatives, Higher Order Derivatives
- Lec 1.4. Differentiability, Differentials, Local Linear Approximation
- Lec 1.5. Chain Rule, Implicit Differentiation
- Lec 1.6. Directional Derivatives, Gradients, Tangent Planes
Lecture Files
Unit 2
We continue with our discussion of analytic and geometric properties of multivariable functions. For the former, we focus on the optimization of multivariable functions by using (partial) derivatives. For the latter, we consider the construction and analysis of more general and exotic surfaces using functions of two variables.
We then introduce the definition of an integral for a function of two variables: the double integral. This new integral can be computed by evaluating two single-variable integrals. We end this unit with a few typical applications of double integrals.
- Lec 2.1. Relative Extrema and the Second Derivative Test
- Lec 2.2. Absolute Extrema and Lagrange Multipliers
- Lec 2.3. Parametric Surfaces and Surfaces of Revolution
- Lec 2.4. Volume as Double Integral in Rectangular Coordinates
- Lec 2.5. Double Integrals over General Regions
- Lec 2.6. Double Integrals in Polar Coordinates
- Lec 2.7. Applications of Double Integrals
Lecture Files
Unit 3
We further extend the notion of integrals to functions of three variables. To compute these new integrals, we need ways to accurately and efficiently describe regions in 3D space referred to as solids. To this end, two new coordinate systems for 3D space will be introduced: the cylindrical and spherical coordinate systems.
We then introduce the notion
- Lec 3.1. Mass as Triple Integrals in Rectangular Coordinates
- Lec 3.2. More on Triple Integrals
- Lec 3.3. Triple Integrals in Cylindrical Coordinates, Center of Mass
- Lec 3.4. Triple Integrals in Spherical Coordinates
- Lec 3.5. Vector Fields, Curl and Divergence
- Lec 3.6. Conservative Vector Fields
- Lec 3.7. Line Integrals of Scalar Fields and Applications