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Browse materialsMIT 18.02SC · Fall 2010 · OCW Scholar

Multivariable Calculus

Subjects: Algebra · Calculus · Linear Algebra · Mathematics

Art: Max Brückner (1860 – 1934), Brückner Vielflache Tafel 07 · Public domain

Instructor
Prof. Denis Auroux
Term
Fall 2010
Level
Undergraduate
What's inside
182 videos · 29 problem sets · 26 exams · 119 solutions · 361 other files · 119 web pages
Materials open
Videos here · files on MIT OCW
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About this course

Source: MIT OpenCourseWare 18.02SC Multivariable Calculus, Fall 2010, Prof. Denis Auroux — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own units, each a lecture sequence in MIT's listing order; no single course-wide order across units is claimed. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 22, 2026. How RedLectern is built.

Learn one topic

101 concept pages, one per MIT session: the clips, notes and practice for that topic. MIT solutions for 91 of 101 topics.

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

Before you start

Prerequisites are not listed here. Check MIT's own syllabus for this course.

How this course was organized

What's here, and what isn't

  • This course has 178 videos, each embedded on its own MIT page, across 102 sections. They're shown in MIT's listing order within each section. A verified course-wide order is not claimed for this course.

Not included here

3 items from the source could not be safely included:

  • MIT's page format for this item wasn't recognized clearly enough to place it safely. (3 items)

Units

Each unit is offered as its own lecture sequence; there is no single course-wide order across units.

  1. 1. Vectors and Matrices40 activities · lecture sequence, in MIT's listing order
  2. 2. Partial Derivatives39 activities · lecture sequence, in MIT's listing order
  3. 3. Double Integrals and Line Integrals in the Plane46 activities · lecture sequence, in MIT's listing order
  4. 4. Triple Integrals and Surface Integrals in 3-Space45 activities · lecture sequence, in MIT's listing order
  5. Final Exam8 activities · lecture sequence, in MIT's listing order

Materials by section

Grouped by section, as MIT organized them. Not sequenced or ordered.

Syllabus

1. Vectors and Matrices

184 items in this section

Part A: Vectors, Determinants and Planes

Session 1: Vectors
7 files

4 images

Session 2: Dot Products
Session 3: Uses of the Dot Product: Lengths and Angles

3 files

5 images

Session 4: Vector Components
Session 5: Area and Determinants in 2D

3 files

4 images

Session 6: Volumes and Determinants in Space

6 files

2 images

Session 7: Cross Products

3 files

7 images
Session 8: Equations of Planes
Problem Set 1

Part B: Matrices and Systems of Equations

.Session 9: Matrix Multiplication

3 files

4 images

Session 10: Meaning of Matrix Multiplication
Session 11: Matrix Inverses

3 files

4 images

Session 12: Equations of Planes II

6 files

5 images

Session 13: Linear Systems and Planes
Session 14: Solutions to Square Systems

4 files

4 images

Problem Set 2

Part C: Parametric Equations for Curves

Session 15: Equations of Lines

3 files

4 images

Session 16: Intersection of a Line and a Plane

3 files

4 images

Session 17: General Parametric Equations; the Cycloid

3 files

6 images

Session 18: Point (Cusp) on Cycloid
Session 19: Velocity and Acceleration

4 files

4 images

Session 20: Velocity and Arc Length

4 files

4 images

Session 21: Kepler's Second Law

3 files

6 images

Problem Set 3

Exam 1

Practice Exam
Session 22: Review of Topics
Session 23: Review of Problems
10 images
Exam

2. Partial Derivatives

216 items in this section

Part A: Functions of Two Variables, Tangent Approximation and Opt

Session 24: Functions of Two Variables: Graphs

4 files

6 images

Session 25: Level Curves and Contour Plots

3 files

4 images

Session 26: Partial Derivatives

3 files

5 images

Session 27: Approximation Formula

5 files

5 images

Session 28: Optimization Problems

3 files

6 images

Session 29: Least Squares

4 files

7 images
Session 30: Second Derivative Test

4 files

15 images
Session 31: Example
Problem Set 4

Part B: Chain Rule, Gradient and Directional Derivatives

Session 32: Total Differentials and the Chain Rule

3 files

8 images
Session 33: Examples

2 files

5 images

Session 34: The Chain Rule with More Variables

4 files

4 images

Session 35: Gradient: Definition, Perpendicular to Level Curves

3 files

5 images

Session 36: Proof

3 files

4 images

Session 37: Example

3 files

4 images

Session 38: Directional Derivatives

3 files

8 images
Problem Set 5

Part C: Lagrange Multipliers and Constrained Differentials

Session 39: Statement of Lagrange Multipliers and Example

3 files

8 images
Session 40: Proof of Lagrange Multipliers
Session 41: Advanced Example

2 files

5 images

Session 42: Constrained Differentials

3 files

5 images

Session 43: Clearer Notation

4 files

3 images

Session 44: Example

2 files

10 images
Problem Set 6

Exam 2

Practice Exam
Session 45: Review of Topics

1 file

14 images
Session 46: Review of Problems
Exam Materials

3. Double Integrals and Line Integrals in the Plane

262 items in this section

Part A: Double Integrals

Session 47: Definition of Double Integration

3 files

5 images

Session 48: Examples of Double Integration

4 files

7 images
Session 49: Exchanging the Order of Integration
Session 50: Double Integrals in Polar Coordinates

3 files

5 images

Session 51: Applications: Mass and Average Value

3 files

4 images

Session 52: Applications: Moment of Inertia

3 files

8 images
Session 53: Change of Variables

3 files

10 images
Session 54: Example: Polar Coordinates
Session 55: Example

3 files

4 images

Problem Set 7

Part B: Vector Fields and Line Integrals

Session 56: Vector Fields

3 files

4 images

Session 57: Work and Line Integrals

3 files

2 videos

8 images
Session 58: Geometric Approach

3 files

4 images

Session 59: Example: Line Integrals for Work

2 files

8 images
Session 60: Fundamental Theorem for Line Integrals

5 files

9 images
Session 61: Conservative Fields, Path Independence, Exact Differentials
Session 62: Gradient Fields

3 files

7 images
Session 63: Potential Functions

3 files

10 images
Session 64: Curl

3 files

6 images

Problem Set 8

Part C: Green's Theorem

Session 65: Green's Theorem

4 files

7 images
Session 66: Curl(F) = 0 Implies Conservative

3 files

4 images

Session 67: Proof of Green's Theorem

1 file

8 images
Session 68: Planimeter: Green's Theorem and Area
Session 69: Flux in 2D

3 files

12 images
Session 70: Normal Form of Green's Theorem

4 files

7 images
Session 71: Extended Green's Theorem: Boundaries with Multiple Pieces

3 files

4 images

Session 72: Simply Connected Regions and Conservative Fields
Problem Set 9

Exam 3

Practice Exam
Session 73: Exam Review
12 images
Exam

4. Triple Integrals and Surface Integrals in 3-Space

221 items in this section

Part A: Triple Integrals

Session 74: Triple Integrals: Rectangular and Cylindrical Coordinates

4 files

8 images
Session 75: Applications and Examples

2 files

9 images
Session 76: Spherical Coordinates

3 files

5 images

Session 77: Triple Integrals in Spherical Coordinates

6 files

7 images
Session 78: Applications: Gravitational Attraction

3 files

6 images

Problem Set 10

Part B: Flux and the Divergence Theorem

Session 79: Vector Fields in Space

3 files

4 images

Session 80: Flux Through a Surface

3 files

5 images

Session 81: Calculating Flux; Finding ndS

3 files

7 images
Session 82: ndS for a Surface z = f(x, y)

3 files

6 images

Session 83: Other Ways to Find ndS

3 files

6 images

Session 84: Divergence Theorem
Session 85: Physical Meaning of Flux; Del Notation
Session 86: Proof of the Divergence Theorem

1 file

6 images

Session 87: Diffusion Equation

3 files

6 images

Problem Set 11

Part C: Line Integrals and Stokes' Theorem

Session 88: Line Integrals in Space

3 files

8 images
Session 89: Gradient Fields and Potential Functions

3 files

8 images
Session 90: Curl in 3D

5 files

5 images

Session 91: Stokes' Theorem

3 files

8 images
Session 92: Proof of Stokes' Theorem
Session 93: Example
Session 94: Simply Connected Regions; Topology

3 files

6 images

Session 95: Stokes' Theorem and Surface Independence
Session 96: Summary of Multiple Integration
Problem Set 12

Exam 4

Practice Exam
Exam

Physics Applications

Session 97: Curl and Physics
Session 98: Maxwell's Equations

Final Exam

36 items in this section

Practice Final Exam

Review

Session 99: Unit 1 Review
8 images
Session 100: Unit 2 Review
7 images
Session 101: Unit 3 Review
Session 102: Unit 4 Review
11 images

Final Exam

Other materials from this course

98 files
71 video transcripts
592 images

MIT OpenCourseWare, Prof. Denis Auroux, 18.02SC Multivariable Calculus, Fall 2010.

License: CC BY-NC-SA 4.0 · Edition: Fall 2010 · Open the original course on OCW ↗

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