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Multivariable Calculus

Massachusetts Institute of Technology via MIT OpenCourseWare

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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 Auroux Arthur Mattuck Jeremy Orloff John Lewis Heidi Burgiel Christine Breiner David Jordan 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

Tags

united states

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