Multivariable Calculus

Fall 2007

Screenshot of Mathlet from the d'Arbeloff Interactive Math Project.

Lagrange multipliers with two variables Mathlet from the d'Arbeloff Interactive Math Project. (Image courtesy of Jean-Michel Claus.)

Course Description

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.

MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the Fall and Spring terms at MIT, and is a required subject for all MIT undergraduates.

Special Features

Technical Requirements

Special software is required to use some of the files in this course: .jar.

Important Notes : -

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Click Below to Download the files :-

LEC #	TOPICS	LECTURE NOTES
I. Vectors and matrices
0	Vectors	Week 1 summary (PDF)
1	Dot product
2	Determinants; cross product
3	Matrices; inverse matrices	Week 2 summary (PDF)
4	Square systems; equations of planes
5	Parametric equations for lines and curves
6	Velocity, acceleration Kepler's second law	Week 3 summary (PDF)
7	Review
II. Partial derivatives
8	Level curves; partial derivatives; tangent plane approximation	Week 4 summary (PDF)
9	Max-min problems; least squares
10	Second derivative test; boundaries and infinity
11	Differentials; chain rule	Week 5 summary (PDF)
12	Gradient; directional derivative; tangent plane
13	Lagrange multipliers
14	Non-independent variables	Week 6 summary (PDF)
15	Partial differential equations; review
III. Double integrals and line integrals in the plane
16	Double integrals	Week 7 summary (PDF)
17	Double integrals in polar coordinates; applications
18	Change of variables	Week 8 summary (PDF)
19	Vector fields and line integrals in the plane
20	Path independence and conservative fields
21	Gradient fields and potential functions	Week 9 summary (PDF)
22	Green's theorem
23	Flux; normal form of Green's theorem
24	Simply connected regions; review	Week 10 summary (PDF)
IV. Triple integrals and surface integrals in 3-space
25	Triple integrals in rectangular and cylindrical coordinates	Week 10 summary (PDF)
26	Spherical coordinates; surface area	Week 11 summary (PDF)
27	Vector fields in 3D; surface integrals and flux
28	Divergence theorem
29	Divergence theorem (cont.): applications and proof	Week 12 summary (PDF)
30	Line integrals in space, curl, exactness and potentials	Week 13 summary (PDF)
31	Stokes' theorem
32	Stokes' theorem (cont.); review
33	Topological considerations Maxwell's equations	Week 14 summary (PDF)
34	Final review
35	Final review (cont.)

LEC #	TOPICS
I. Vectors and matrices
0	Vectors (Note: Video is not available for this topic.)
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	Velocity, acceleration - Kepler's second law
7	Review
II. Partial derivatives
8	Level curves; partial derivatives; tangent plane approximation
9	Max-min problems; least squares
10	Second derivative test; boundaries and infinity
11	Differentials; chain rule
12	Gradient; directional derivative; tangent plane
13	Lagrange multipliers
14	Non-independent variables
15	Partial differential equations; review
III. Double integrals and line integrals in the plane
16	Double integrals
17	Double integrals in polar coordinates; applications
18	Change of variables
19	Vector fields and line integrals in the plane
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; review
IV. Triple integrals and surface integrals in 3-space
25	Triple integrals in rectangular and cylindrical coordinates
26	Spherical coordinates; surface area
27	Vector fields in 3D; surface integrals and flux
28	Divergence theorem
29	Divergence theorem (cont.): applications and proof
30	Line integrals in space, curl, exactness and potentials
31	Stokes' theorem
32	Stokes' theorem (cont.); review
33	Topological considerations - Maxwell's equations
34	Final review
35	Final review (cont.)

Lecture Notes
The notes below represent summaries of the lectures as written by Professor Auroux to the recitation instructors.

LEC # TOPICS LECTURE NOTES
I. Vectors and matrices
0 Vectors Week 1 summary (PDF)
1 Dot product
2 Determinants; cross product
3 Matrices; inverse matrices Week 2 summary (PDF)
4 Square systems; equations of planes
5 Parametric equations for lines and curves
6
Velocity, acceleration
Kepler's second law
Week 3 summary (PDF)
7 Review
II. Partial derivatives
8 Level curves; partial derivatives; tangent plane approximation Week 4 summary (PDF)
9 Max-min problems; least squares
10 Second derivative test; boundaries and infinity
11 Differentials; chain rule Week 5 summary (PDF)
12 Gradient; directional derivative; tangent plane
13 Lagrange multipliers
14 Non-independent variables Week 6 summary (PDF)
15 Partial differential equations; review
III. Double integrals and line integrals in the plane
16 Double integrals Week 7 summary (PDF)
17 Double integrals in polar coordinates; applications
18 Change of variables Week 8 summary (PDF)
19 Vector fields and line integrals in the plane
20 Path independence and conservative fields
21 Gradient fields and potential functions Week 9 summary (PDF)
22 Green's theorem
23 Flux; normal form of Green's theorem
24 Simply connected regions; review Week 10 summary (PDF)
IV. Triple integrals and surface integrals in 3-space
25 Triple integrals in rectangular and cylindrical coordinates Week 10 summary (PDF)
26 Spherical coordinates; surface area Week 11 summary (PDF)
27 Vector fields in 3D; surface integrals and flux
28 Divergence theorem
29 Divergence theorem (cont.): applications and proof Week 12 summary (PDF)
30 Line integrals in space, curl, exactness and potentials Week 13 summary (PDF)
31 Stokes' theorem
32 Stokes' theorem (cont.); review
33
Topological considerations
Maxwell's equations
Week 14 summary (PDF)
34 Final review
35 Final review (cont.)

Video Lectures

LEC # TOPICS
I. Vectors and matrices
0 Vectors (Note: Video is not available for this topic.)
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
Velocity, acceleration - Kepler's second law
7 Review
II. Partial derivatives
8 Level curves; partial derivatives; tangent plane approximation
9 Max-min problems; least squares
10 Second derivative test; boundaries and infinity
11 Differentials; chain rule
12 Gradient; directional derivative; tangent plane
13 Lagrange multipliers
14 Non-independent variables
15 Partial differential equations; review
III. Double integrals and line integrals in the plane
16 Double integrals
17 Double integrals in polar coordinates; applications
18 Change of variables
19 Vector fields and line integrals in the plane
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; review
IV. Triple integrals and surface integrals in 3-space
25 Triple integrals in rectangular and cylindrical coordinates
26 Spherical coordinates; surface area
27 Vector fields in 3D; surface integrals and flux
28 Divergence theorem
29 Divergence theorem (cont.): applications and proof
30 Line integrals in space, curl, exactness and potentials
31 Stokes' theorem
32 Stokes' theorem (cont.); review
33
Topological considerations - Maxwell's equations
34 Final review
35 Final review (cont.)

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