MATH 53

Multivariable Calculus

Peter Koroteev, UC Berkeley

Summer Course Format:

Eight hours of lecture and two hours of discussion per week.

Description: In this course we will study functions in several variables and operations on them, such as integration and partial differentiation, as well as their myriad applications.

Textbook: Stewart, Multiple Variable Calculus (8th edition)

Outline of the Course:


Week 1 :Parametric equation curves, Calculus with parametric equations, Polar coordinates, Areas and lengths in Polar coordinates, Conic sections.

Videos of lectures: [Introduction, Dimensions, Parametric Curves] [Derivatives, Tangent Lines] [Arc Length, Areas of Rotational Surfaces] [Areas, Polar Coordinates, Conical Sections] [Discussion]
iPad lecture notes: [Week 1]

Chapter 10

Week 2 : Three-dimensional coordinate systems, vectors, the dot products, the cross products, equations of lines and planes, cylinders and quadric surfaces, cylindrical and spherical coordinates.

Videos of lectures: [3D Coordinate Systems, Vectors] [Dot Product, Cross Product] [Triple Product] [Applications, Quadratic Surfaces]
iPad lecture notes: [Week 2]
[Homework 1] [Quiz 1]

Chapter 12

Week 3 : Derivatives and integrals of vector functions, Arclength, Curvature, Motion in space.

Videos of lectures: [Derivatives of Vector Functions] [Curvature, Torsion, Frenet Triple, Motion of Paricles] [Kepler Laws, Review] [Discussion, Review] [Discussion]
iPad lecture notes: [Week 3]
[Homework 2] [Quiz 2]

Chapter 13

Week 4 : Functions of several variables, Limits, Partial derivatives, Tangent lines Chain Rule, Gradient vector, Maximum and Minimum.

Videos of lectures: [Functions of Several Variables, Limits and Continuity] [Partial Derivatives, Tangent Planes, Differentiability, Chain Rule] [Implicit Differentiation, Directional Derivatives, Gradient] [Review] [Discussion]
iPad lecture notes: [Week 4]
[Homework 3] [Quiz 3]

Sections 14.1-14.6

Week 5 : Maximum and Minimum, Lagrange Multipliers. Midterm

Videos of lectures: [Minimum and Maximum] [Lagrange Multipliers] [Discussion]
iPad lecture notes: [Week 5]
[Homework 4] [Midterm]

Sections 14.7-14.8

Week 6 : Double Integrals, Surface Areas, Triple Intetgrals, Volumes.

Videos of lectures: [Double Integrals] [Double Integrals in polar coordinates, Applications] [Physics applications, Triple Integrals, Cylindrical coordinates] [Discussion]
iPad lecture notes: [Week 6]
[Homework 5] [Quiz 4]

Chapter 15

Weeks 7,8 : Line integrals of vector fields, Green's theorem, differential forms, Parametric Surfaces, Areas, Orientation. Final 8/14

Videos of lectures: [Vector Fields, Line Integrals] [Line Integrals of Vector Fields] [Green's Theorem, Curl and Divergence] [Green's Theorem in Differential Form] [Divergence Theorem, Orientation]
iPad lecture notes: [Week 7] [Week 8]
[Homework 6] [Quiz 5]

Chapter 16

Preparation Materials:
Math 53 [Worksheet]
Archive of exams and midterms on [Tau Beta Pi]