18.024 | Spring 2011 | Undergraduate
Multivariable Calculus with Theory


Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

Recitations: 2 sessions / week, 1 hour / session


Apostol, Tom M. Calculus 1, One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, Mass: Blaisdell, 1967. ISBN: 9780471000051.

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


18.01 Single Variable Calculus or equivalent and permission of the instructor.


We will cover the same material as 18.02 Multivariable Calculus but with an emphasis on proofs and conceptual understanding rather than computation. In addition, we will cover many topics from linear algebra that are not considered in 18.02.

Recitation Assignments

In addition to the problem sets, there will be problems assigned for each recitation. Students should be prepared to participate in the discussion and present their solutions during the specified recitation.

Problem Sets

Problem sets are assigned weekly, and due the following week. There will be 12 problem sets during the semester. At the end of the course, the lowest pset grade will be dropped.


There will be two in-class, one-hour midterm exams, and one three-hour comprehensive final exam.


Problem Sets 20%
Midterms 40%
Final Exam 30%
Participation 10%


1 Linear spaces and subspaces  
2 Dependence, basis, dimension  
3 Linear transformations and invertibility  
4 Gauss-Jordan elimination, matrices Pset 1 due
5 Matrix of a transformation, Linear systems  
6 Matrix inverses, determinant  
7 Cross product, Lines and planes Pset 2 due
8 Vector valued functions, tangency  
9 Velocity/Acceleration, arclength  
10 Curvature, Polar coordinates Pset 3 due
11 Planetary motion, scalar and vector fields  
12 Total derivative, gradient  
13 Level sets, tangent planes, derivative of vector fields Pset 4 due
14 Exam 1 Midterm Exam 1
15 Chain rule  
16 Implicit differentiation, inverse functions Pset 5 due
17 Hessian matrix, maxima, minima, saddle points  
18 Second derivative test, Taylor’s Formula  
19 Implicit function theorem  
20 Extreme Values, Lagrange Multipliers Pset 6 due
21 Line integrals  
22 Fundamental theorem of line integrals Pset 7 due
23 Gradient fields  
24 Potential functions, conservation  
25 Double integrals over rectangles Pset 8 due
26 Existence and Fubini’s Theorem  
27 Double integrals over more general regions  
28 Applications of multiple integrals  
29 Exam 2 Midterm Exam 2
30 Green’s Theorem  
31 Applications Pset 9 due
32 Change of variables  
33 Cylindrical and spherical coordinates Pset 10 due
34 Parameterized surfaces  
35 Area, surface integrals  
36 Stokes’s Theorem Pset 11 due
37 Stokes’s Theorem (cont.)  
38 Divergence Theorem  
39 Minimal Surfaces (PDF)  
40 Final Exam  

Course Info
As Taught In
Spring 2011
Learning Resource Types
grading Exams with Solutions
notes Lecture Notes
assignment_turned_in Problem Sets with Solutions