Lectures (Video)

- 1. Dot product
- 2. Determinants; cross product
- 3. Matrices; inverse matrices
- 4. Square systems; equations of planes
- 5. Parametric equations for lines and curves
- 6. Kepler's second law
- 7. Review
- 8. Partial derivatives
- 9. Least squares
- 10. Second derivative test
- 11. Differentials; chain rule
- 12. Directional derivative
- 13. Lagrange multipliers
- 14. Non-independent variables
- 15. Partial differential equations
- 16. Double integrals
- 17. Double integrals in polar coordinates
- 18. Change of variables
- 19. Vector fields and line integrals
- 20. Path independence and conservative fields
- 21. Gradient fields and potential functions
- 22. Green's theorem
- 23. Flux; normal form of Green's theorem
- 24. Simply connected regions
- 25. Triple integrals
- 26. Spherical coordinates
- 27. Surface integrals and flux
- 28. Divergence theorem
- 29. Divergence theorem II
- 30. Line integrals in space
- 31. Stokes' theorem
- 32. Stokes' theorem II
- 33. Maxwell's equations
- 34. Final Review
- 35. Final Review

## Calculus II

### Course Summary

This course is based on

This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space. The lectures were conducted by Prof. Denis Auroux at MIT.
*18.02 Multivariable Calculus, Fall 2007*made available by*Massachusetts Institute of Technology: MIT OpenCourseWare*under the*Creative Commons BY-NC-SA*license.### Reading Material

1.**Textbook (MIT 18.02): Multivariable Calculus. 6th ed.**

Edwards, Henry C., and David E. Penney. Multivariable Calculus. 6th ed. Lebanon, IN: Prentice Hall, 2002. ISBN: 9780130339676.

2.

**Supplementary Notes I**

Determinants

3.

**Supplementary Notes II**

Matrices and linear algebra

4.

**Supplementary Notes III**

Kepler's second law

5.

**Supplementary Notes IV**

The tangent approximation

6.

**Supplementary Notes V**

Second derivative test

7.

**Supplementary Notes VI**

Least squares interpolation

8.

**Supplementary Notes VII**

Non-independent variables

9.

**Supplementary Notes VIII**

Partial differential equations

10.

**Supplementary Notes IX**

Limits in iterated integrals

11.

**Supplementary Notes X**

Changing variables in multiple integrals

12.

**Supplementary Notes XI**

Gravitational attraction

13.

**Supplementary Notes XII**

Plane vector fields

14.

**Supplementary Notes XIII**

Gradient fields and exact differentials

15.

**Supplementary Notes XIV**

Two-dimensional flux

16.

**Supplementary Notes XV**

Green's theorem in normal form

17.

**Supplementary Notes XVI**

Simply-connected regions

18.

**Supplementary Notes XVII**

Multiply-connected regions; topology

19.

**Supplementary Notes XVIII**

Laplace's equation and harmonic functions

20.

**Supplementary Notes XIX**

Vector fields in space

21.

**Supplementary Notes XX**

Surface integrals

22.

**Supplementary Notes XXI**

The divergence theorem

23.

**Supplementary Notes XXII**

Line integrals in space

24.

**Supplementary Notes XXIII**

Gradient fields in space

25.

**Supplementary Notes XXIV**

Stokes' theorem

26.

**Supplementary Notes XXV**

Some topological questions

27.

**Supplementary Notes XXVI**

Relation to physics

### Course Material

1.**Assignments (MIT 18.02)**

Problem Set 1 (PDF)

Problem Set 2 (PDF)

Problem Set 3 (PDF)

Problem Set 4 (PDF)

Problem Set 5 (PDF)

Problem Set 6 (PDF)

Problem Set 7 (PDF)

Problem Set 8 (PDF)

Problem Set 9 (PDF)

Problem Set 10 (PDF)

Problem Set 11 (PDF)

Problem Set 12 (PDF)

2.

**Exams (MIT 18.02)**

EXAMS | SOLUTIONS |
---|---|

Practice exam 1A (PDF) | (PDF) |

Practice exam 1B (PDF) | (PDF) |

Practice exam 2A (PDF) | (PDF) |

Practice exam 2B (PDF) | (PDF) |

Practice exam 3A (PDF) | (PDF) |

Practice exam 3B (PDF) | (PDF) |

Practice exam 4A (PDF) | (PDF) |

Practice exam 4B (PDF) | (PDF) |

### Other Resources

1.**Calculus**

Gilbert Strang,

*Calculus*, Wellesley-Cambridge Press, 1991, ISBN 9780961408824.

This excellent book is written by Prof. Gilbert Strang of MIT and is a useful resource for both students and educators alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor's Manual and a student Study Guide availble. You can download the whole book in pdf format at this link (38.5 MB).

### Software

1.**Interactive math applets**

A series of specially written interactive math applets ("mathlets"), which let you visualize and experiment with various concepts in the course.

- Functions of Two Variables
- Lagrange Multipliers (Two Variables)
- Curves and Vector Fields
- Flux Across Circle
- Surfaces and Flux in Space