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# Multivariable Calculus

### Overview

This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics. The materials have been organized to support independent study. The website includes all of the materials you will need to understand the concepts covered in this subject. The materials in this course include: - **Lecture Videos** recorded on the MIT campus - **Recitation Videos** with problem-solving tips - **Examples** of solutions to sample problems - **Problems** for you to solve, with solutions - **Exams** with solutions - **Interactive Java Applets** ("Mathlets") to reinforce key concepts Content Development Denis AurouxÂ  Arthur MattuckÂ  Jeremy OrloffÂ  John Lewis Heidi BurgielÂ  Christine BreinerÂ  David JordanÂ  Joel Lewis

### Syllabus

• Session 1 Clip: Vectors
• Session 2 Clip: Dot Products
• Session 3 Clip: Lengths and Angles
• Session 4 Clip: Vector Components
• Session 5 Clip: Area and Determinants in 2D
• Session 6 Clip: Volumes and Determinants in Space
• Session 7 Clip 1: Cross Products
• Session 7 Clip 2: More on Cross Products
• Session 8 Clip: Equations of Planes
• Session 9 Clip: Matrix Multiplication
• Session 10 Clip: Meaning of Matrix Multiplication
• Session 11 Clip: Matrix Inverses
• Session 12 Clip: Equations of Planes
• Session 13 Clip: Linear Systems and Planes
• Session 14 Clip: Solutions to Square Systems
• Session 15 Clip: Equations of Lines
• Session 16 Clip: Intersection of a Line and a Plane
• Session 17 Clip: General Parametric Equations and the Cycloid
• Session 18 Clip: Point (Cusp) of a Cycloid
• Session 19 Clip: Velocity and Acceleration
• Session 20 Clip: Velocity and Arc Length
• Session 21 Clip: Kepler's Second Law
• Session 22 Clip: Review of Topics
• Session 23 Clip: Review of Problems
• Session 24 Clip: Functions of Two Variables: Graphs
• Session 25 Clip: Level Curves and Contour Plots
• Session 26 Clip: Partial Derivatives
• Session 27 Clip: Approximation Formula
• Session 28 Clip: Optimization Problem
• Session 29 Clip: Least Squares
• Session 30 Clip 1: Quadratic Example
• Session 30 Clip 2: Second Derivative Test
• Session 31 Clip: Example
• Session 32 Clip: Total Differentials and Chain Rule
• Session 33 Clip: Examples
• Session 34 Clip: Chain Rule with More Variables
• Session 35 Clip: Definition, Perpendicular to Level Curves
• Session 36 Clip: Proof
• Session 37 Clip: Example
• Session 38 Clip: Directional Derivatives
• Session 39 Clip: Lagrange Multipliers by Example
• Session 40 Clip: Proof of Lagrange Multipliers
• Session 41 Clip: Advanced Example
• Session 42 Clip: Constrained Differentials
• Session 43 Clip: Clearer Notation
• Session 44 Clip: Example
• Session 47 Clip: Definition of Double Integration
• Session 48 Clip: Examples of Double Integration
• Session 49 Clip: Exchanging the Order of Integration
• Session 50 Clip: Double Integrals in Polar Coordinates
• Session 51 Clip: Applications: Mass and Average Value
• Session 52 Clip: Applications: Moment of Inertia
• Session 53 Clip: Change of Variables
• Session 54 Clip: Polar Coordinates
• Session 55 Clip: Example
• Session 56 Clip: Vector Fields
• Session 57 Clip: Work and Line Integrals
• Session 58 Clip: Geometric Approach
• Session 59 Clip: Line Integrals
• Session 60 Clip: An Important Non-Conservative Field
• Session 60 Clip: Fundamental Theorem
• Session 61 Clip: Conservative Fields, Path Independence, Exact
• Session 62 Clip: Gradient Fields
• Session 63 Clip: Potential Functions
• Session 64 Clip: Curl
• Session 65 Clip: Green's Theorem
• Session 66 Clip: Curl(F)=0 Implies Conservative
• Session 67 Clip: Proof of Green's Theorem
• Session 68 Clip: Planimeter: Green's Theorem and Area
• Session 69 Clip: Calculating Flux
• Session 69 Clip: Flux Across a Curve
• Session 70 Clip: Introduction and Definition of Divergence
• Session 71 Clip: Extended Green's Theorem
• Session 72 Clip: Simply Connected Regions and Conservative Fields
• Session 73 Clip: Exam Review
• Session 74 Clip: Cylindrical Coordinates
• Session 74 Clip: Triple Integrals
• Session 75 Clip: Applications and Examples
• Session 76 Clip: Spherical Coordinates
• Session 77 Clip: Triple Integrals in Spherical Coordinates
• Session 78 Clip: Gravitational Attraction
• Session 79 Clip: Vector Fields in Space
• Session 80 Clip: Flux Through a Surface
• Session 81 Clip: Calculating Flux, Finding ndS
• Session 82 Clip: ndS for a Surface z=f(x,y)
• Session 83 Clip: Other Ways to Find ndS
• Session 84 Clip: Divergence Theorem
• Session 85 Clip: Physical Meaning of Flux, Del Notation
• Session 86 Clip: Proof of the Divergence Theorem
• Session 87 Clip: Diffusion Equation
• Session 88 Clip: Line Integrals in Space
• Session 89 Clip: Gradient Fields and Potential Functions
• Session 90 Clip: Curl in 3D
• Session 91 Clip: Stokes' Theorem
• Session 92 Clip: Proof of Stokes' Theorem
• Session 93 Clip: Example
• Session 94 Clip: Simply Connected Regions, Topology
• Session 95 Clip: Stokes' Theorem and Surface Independence
• Session 96 Clip: Summary of Multiple Integration
• Session 97 Clip: Curl and Physics
• Session 98 Clip: Introduction
• Session 99 Clip 1: Vectors, Equations of Lines/Curves/Planes
• Session 99 Clip 2: Matrices, Determinants, Linear Systems
• Session 100 Clip 1: Functions of 2 Variables, Partial Derivatives
• Session 100 Clip 2: Max-Min Problems, Lagrange Multipliers
• Session 101 Clip 1: Double Integrals, Jacobian, Triple Integrals
• Session 101 Clip 2: Applications
• Session 102 Clip 1: Work, Line Integrals, Curl
• Session 102 Clip 2: Stokes, Green's, Divergence Theorems

### Taught by

Prof. Denis Auroux

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