18.024 | Spring 2011 | Undergraduate
Multivariable Calculus with Theory

Readings

Unless noted otherwise, all textbook readings are from:

Apostol, Tom M. Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability. Wiley, 1969. ISBN: 9780471000075.

Additional course notes by James Raymond Munkres, Professor of Mathematics, Emeritus, are also provided.

SES # TOPICS TEXTBOOK READINGS COURSE NOTES READINGS
1 Linear spaces and subspaces 1.1-1.6 Course Notes A (A1-A6)
2 Dependence, basis, dimension 1.7-1.8, 1.11-1.12 Course Notes A (A7-A15)
3 Linear transformations and invertibility 2.1-2.7  
4 Gauss-Jordan elimination, matrices  

Course Notes A (A17-A23)

Course Notes B (B1-B5)

5 Matrix of a transformation, Linear systems 2.10-2.11 Course Notes B (B6-B16)

6 Matrix inverses, determinant   Course Notes B (B25-B51)
7 Cross product, Lines and planes  

Course Notes B (B53-B56)

Course Notes A (A25-A33)

Course Notes B (B18-B23)

8 Vector valued functions, tangency (14.1-14.5) (Apostol Vol. I)  
9 Velocity/Acceleration, arclength (14.6-14.12) (Apostol Vol. I)  
10 Curvature, Polar coordinates (14.114-14.16) (Apostol Vol. I) Course Notes B (B57-B63)
11 Planetary motion, scalar and vector fields (14.17, 14.20), 8.1-8.5 (Apostol Vol. I)  
12 Total derivative, gradient 8.6-8.8, 8.10-8.13  
13 Level sets, tangent planes, derivative of vector fields 8.15-8.17  
14 Exam 1    
15 Chain rule 8.18-8.21  
16 Implicit differentiation, inverse functions 9.6-9.7 Course Notes C (C10-C21)
17 Hessian matrix, maxima, minima, saddle points 9.9  
18 Second derivative test, Taylor’s Formula 9.9-9.12 Course Notes C (C22-C27)
19 Implicit function theorem    
20 Extreme Values, Lagrange Multipliers 9.14 Course Notes C (C28-C33)
21 Line integrals 10.1-10.7  
22 Fundamental theorem of line integrals 10.10-10.14  
23 Gradient fields 10.15-10.16  
24 Potential functions, conservation 10.17, 10.21  
25 Double integrals over rectangles 11.1-11.8  
26 Existence and Fubini’s Theorem 11.10-11.11 Course Notes D (D1-D17)

27 Double integrals over more general regions 11.12-11.14 Course Notes D (D17-D25)
28 Applications of multiple integrals 11.16, 11.17, 11.31  
29 Exam 2    
30 Green’s Theorem 11.19-11.23  
31 Applications   Course Notes E (E1-E22)

32 Change of variables 11.26-11.31 Course Notes E (E23-E33)
33 Cylindrical and spherical coordinates 11.32, 11.33  
34 Parameterized surfaces 12.1-12.5  
35 Area, surface integrals 12.7-12.9  
36 Stokes’s Theorem 12.11-12.12, 12.18  
37 Stokes’s Theorem (cont.)   Course Notes F (F1-F5)
38 Divergence Theorem 12.19  
39 Minimal Surfaces   Course Notes F (F7-F16)
40 Final Exam    
Course Info
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Spring 2011
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grading Exams with Solutions
notes Lecture Notes
assignment_turned_in Problem Sets with Solutions