If you have completed the Math 1a/1b sequence at Harvard or if you
have had the equivalent material elsewhere, you may be wondering which
course is for you. The mathematics department provides a variety of
options which you should consider based on your academic interests and
your background. With exceedingly rare exceptions, students in your
position are advised to take one (or more) of Math 18, 19a, 19b, 21a, 21b,
23a, 25a, 55a, or 101. (The School of Engineering and Applied
Sciences also offers Applied Math 21a,b which covers selected topics
from Math 21.) This pamphlet describes the Mathematics Department's
offerings and should help you decide which course is for you.

Math 18 focuses on concepts and techniques of multivariable calculus most useful
to those studying the social sciences, particularly economics: functions of several variables:
partial derivatives; directional derivatives and the gradient: constrained and unconstrained
optimization, including the method of Lagrange multipliers. Covers linear and polynomial
approximation and integrals for single variable and multivariable functions: modeling with derivatives.

Math 19a and 19b are courses that are designed for
students concentrating in the life sciences. (These courses are recommended over Math 21a,b
by the various life science concentrations.) Math 19a is taught in the fall and repeated in the
spring; it focuses on
differential equations, related techniques and modeling. Math 19b
teaches linear algebra, probability and statistics; it is offered only
in the spring. Both courses focus
on applications and examples from the life sciences. If you passed
Mathematics 1b (or have the permission of the instructor), you can
take Mathematics 19a,b.

Math 21a,b is the standard second-year calculus and linear
algebra sequence. It is normally taken by those students who intend to
concentrate in the physical sciences or mathematics and who have had a
solid first year calculus course. Math 21 emphasizes computational
techniques and applications. It seeks to develop tools and intuition
rather than spend time proving the results used. Math 21 is given in
semester-long halves which may be taken in either order or
concurrently. Math 101 can be taken concurrently with either
Math 21a or 21b. The material in Math 21a/b is presented, where
feasible, in correlation with Physics 15/16.

Math 21a covers multivariable calculus, while Math 21b is a
one-semester introduction to linear algebra and differential
equations. First-year students who had an equivalent of Math 21a in
high school often take this course in the fall of their freshman year.
The students with such background who intend to major in math or
theoretical physics should also look into Math 23, Math 25, or Math
55. Those who are considering a concentration in mathematics may want
to take Math 101 concurrently with either Math 21a or b.

(over)

Math 101 is a one-semester introduction to the three main
branches of modern mathematics (algebra, analysis, and geometry) and
to the methodology used in higher mathematics. It has no official
prerequisites. In this course students learn to write rigorous proofs
and encounter fundamental concepts which are further developed in
other 100-level courses. Math 101 is intended both for those who wish
to concentrate in mathematics and for those in other fields (related
or not) who have an interest in learning what higher math is all
about. Students often take it concurrently with or right after Math
21. Those who are taking or have taken Math 23, 25, or 55 should not
take 101. In 2015-16, Math 101 will be given in the spring, but not in
the fall semester.

Math 23 is an advanced version of the 21 sequence designed for
students with strong math interest. This course develops theories of
functions of several variables and linear algebra. Students in this
course will learn to write rigorous proofs and encounter some of the
beauty and elegance of modern mathematics. Math 23 offers a
theoretical understanding of the mathematical concepts which are
taught in Math 21. Please note that Math 23 may not correlate
with the Physics 15/16 sequence. Also note that all 100 level Math
courses which accept Math 25 or 55 as a prerequisite also accept Math
23. Math 23 should not require an unusual out of class time
commitment.

Math 25 and 55 are both full-year advanced courses designed for
students with a very strong interest in theoretical mathematics. Each
covers multivariable calculus, linear algebra, and some additional
topics from a rigorous and advanced point of view. The students in
these courses are frequently committed to concentrating in mathematics
and are asked to put in extensive work outside the classroom. Many
have had more than one year of college mathematics while in high
school or have participated in various summer math programs. However,
it is not necessary to have had multivariable calculus before taking
25 or 55. Although the syllabus of Math 25 is similar to that of Math
23, students will usually have had more preparation in math.

Math 55 is a faster paced course and covers topics more deeply. It is
designed for students who arrive at Harvard with an extensive
background in college level math. Math 25 and 55 differ from Math 23
in the level of outside work required: homework assignments in Math 25
and 55 are typically very time consuming. Math 23, 25 or 55 all
provide an excellent foundation for further study of mathematics.

Skipping Math 25 and 55: Every year a few freshmen want
to skip the Math twenty/fifty level all together and start with a 100-
or 200- level course. The Department, based on many years of
experience, strongly discourages this. You may learn more
advanced material in higher level courses, but never at the same speed
and intensity as in Math 25 or 55. Moreover, you are learning more
than just a body of mathematics in these courses. You are also
learning how to `be' a research mathematician (as opposed to one who
only does well in Math courses). If, in spite of this warning, you
think that taking a higher level course as a freshman would best serve
your needs, you should speak to the Director of Undergraduate Studies
of the Mathematics Department, Professor Jacob Lurie
(lurie@math).