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:
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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]
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Chapter 10
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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]
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Chapter 12
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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]
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Chapter 13
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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]
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Sections 14.1-14.6
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Week 5 : Maximum and Minimum, Lagrange Multipliers. Midterm
Videos of lectures:
[Minimum and Maximum]
[Lagrange Multipliers]
[Discussion]
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Sections 14.7-14.8
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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]
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Chapter 15
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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]
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Chapter 16
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Preparation Materials:
Math 53 [Worksheet]
Archive of exams and midterms on [Tau Beta Pi]