# Introductory Courses

*Dusty Grundmeier*

2019 Fall (2 Credits)

**
Schedule: **
TR 0900 AM - 10:15 AM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
n/a

In his seminal work from 1945, How to Solve It, George Polya introduced principles of mathematical problem solving that are widely applicable to problems in science and engineering. This year-long class focuses on building a powerful and portable problem-solving and modeling tool kit while bridging the divide between mathematics and science courses. The second semester will be organized around projects in areas of student interest. Both Math ESPA and Math ESPB must be taken in the same academic year to receive credit.

Additional Course Attributes::

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

Full Year Course | Indivisible Course |

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Dusty Grundmeier*

2020 Spring (2 Credits)

**
Schedule: **
TR 09:00 AM - 10:15 AM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
n/a

In his seminal work from 1945, How to Solve It, George Polya introduced principles of mathematical problem solving that are widely applicable to problems in science and engineering. This year-long class focuses on building a powerful and portable problem-solving and modeling tool kit while bridging the divide between mathematics and science courses. The second semester will be organized around projects in areas of student interest. Both Math ESPA and Math ESPB must be taken in the same academic year to receive credit.

Additional Course Attributes::

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

Course Search Attributes | Display Only in Course Search |

All: Cross Reg Availability | Available for Harvard Cross Registration |

Full Year Course | Indivisible Course |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Brendan Kelly*

2019 Fall (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year-long reinforcement. Applications to biology and economics emphasized according to the interests of our students.

- Course Notes:
- This is a lecture course taught in small sections. In addition, participation in two one-hour workshops is required each week. This course, when taken together with Mathematics Mb, can be followed by Mathematics 1b. Mathematics Ma and Mb together cover all the material in Mathematics 1a (and more).

- Class Notes:
- The required first meeting in Fall: Tuesday, September 3, 8:15 am, Science Center C Fall Section Times: MWF 9:00, MWF 10:30, MWF 12, MWF 1:30, MWF 3 with sufficient enrollment Caroline Junkins, Kate Penner, Florence Orosz, Matthew Demers, David Freund, Jill Guerra, Voula Collins, Emily Braley, Carolyn Gardener-Thomas

Additional Course Attributes::

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Brendan Kelly*

2020 Spring (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Continued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b.

- Course Notes:
- This is a lecture course taught in small sections. In addition, participation in two one-hour workshops is required each week. This course, when taken together with Mathematics Ma, can be followed by Mathematics 1b. Mathematics Ma and Mathematics Mb together cover all the material in Mathematics 1a (and more).

- Class Notes:
- Spring Section Times: MWF 9:00, MWF 10:30, MWF 12, MWF 1:30, and MWF 3:00 with sufficient enrollment. Caroline Junkins, Kate Penner, David Freund, Matthew Demers, Hakim Walker, Emily Braley, Voula Collins, Florence Orosz, Carolyn Gardener-Thomas

- Requirements:
- Prerequisite: Mathematics MA

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Jill Guerra*

2019 Fall (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines.

- Course Notes:
- In the fall, Math 1a is taught in sections of 20-30 students. In the spring, Math 1a is taught in a larger lecture format. Participation in a weekly 90-minute workshop is required. Mathematics Ma and Mb together cover all of the material in Mathematics 1a (and more).

- Class Notes:
- The required first meeting in Fall: Tuesday, September 3, 7:45 am, Science Center C Fall Section Times: MWF 9:00, MWF 10:30, MWF 12:00, MWF 1:30, and MWF 3:00 with sufficient enrollment. Voula Collins, Elana Kalashnikov

- Recommended Prep:
- A solid background in precalculus.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Undergraduate Students |

*Oliver Knill*

2020 Spring (4 Credits)

**
Schedule: **
MWF 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines.

- Course Notes:
- In the fall, Math 1a is taught in sections of 20-30 students. In the spring, Math 1a is taught in a larger lecture format. Participation in a weekly 90-minute workshop is required. Mathematics Ma and Mb together cover all of the material in Mathematics 1a (and more).

- Class Notes:
- Spring Section Time: MWF 10:30 and a weekly lab section to be arranged.

- Recommended Prep:
- A solid background in precalculus.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Hakim Walker*

2019 Fall (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.

- Course Notes:
- This is a lecture taught in small sections.

- Class Notes:
- The required first meeting in Fall: Tuesday, September 3, 8:15 am, Science Center B Fall Section Times: MWF 9:00, MWF 10:30, MWF 12:00, MWF 1:30, and MWF 3:00 with sufficient enrollment. John Cain, Marius Lemm, Stepan Paul, Hakim Walker, Drew Zemke, and members of the Department

- Recommended Prep:
- Mathematics 1a or Ma and Mb; or 5 on the AB advanced placement test; or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

*Robin Gottlieb*

2020 Spring (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.

- Course Notes:
- This is a lecture taught in small sections.

- Class Notes:
- Spring Section Times: MWF 9:00, MWF 10:30, MWF 12:00, MWF 1:30, and MWF 3:00 with sufficient enrollment. Hakim Walker, Forence Orosz, Elana Kalashnikov, Voula Collins

- Recommended Prep:
- Mathematics 1a or Ma and Mb; or 5 on the AB advanced placement test; or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Undergraduate Students |

*Drew Zemke*

2019 Fall (4 Credits)

**
Schedule: **
MWF 01:30 PM - 02;45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Focus 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 a single variable and multivariable functions; modeling with derivatives. Covers topics from Math 21a most useful to social sciences.

- Course Notes:
- Mathematics 21b can be taken before or after Mathematics 18. Examples draw primarily from economics and the social sciences, though Mathematics 18 may be useful to students in certain natural sciences. Students whose main interests lie in the physical sciences, mathematics, or engineering should consider Math or Applied Math 21a.

- Recommended Prep:
- Mathematics 1b or equivalent, or a 5 on the BC Advanced Placement Examination in Mathematics.

- Requirements:
- Anti-requisite: Not to be taken in addition to Mathematics 21a or Applied Mathematics 21a

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Core Curriculum(old) | Quantitative Reasoning |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*John Cain*

2019 Fall (4 Credits)

**
Schedule: **
MWF 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Considers the construction and analysis of mathematical models that arise in the life sciences, ecology and environmental life science. Introduces mathematics that includes multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).

- Course Notes:
- This course is recommended over Math 21a for those planning to concentrate on the life sciences and ESPP. Can be taken with or without Mathematics 21a,b. Students with interests in the social sciences and economics might consider Mathematics 18. This course can be taken before or after Mathematics 18.

- Recommended Prep:
- A course in one variable calculus preferably at the level of Mathematics 1b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

*Stepan Paul*

2020 Spring (4 Credits)

**
Schedule: **
MWF 01:30 AM - 02:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Probability, statistics and linear algebra with applications to life sciences, chemistry, environmental sciences, economics, and social sciences. Students will learn to use computing software to perform relevant calculations on data sets coming from these areas of study. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.

- Course Notes:
- This course is recommended over Math 21b for those planning to concentrate on the life sciences and ESPP. Can be taken with Mathematics 21a. Students who have seen some multivariable calculus can take Math 19b before Math 19a.

- Recommended Prep:
- A course in one variable calculus preferably at the level of Mathematics 1b.

- Requirements:
- Not to be taken in addition to Mathematics 21b or Applied Mathematics 21b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

*Oliver Knill*

2019 Fall (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Green’s, Stokes’s, and Divergence Theorems.

- Course Notes:
- This is a lecture taught in small sections. May not be taken for credit by students who have passed Applied Mathematics 21a. Activities using computers to calculate and visualize applications of these ideas will not require programming experience.

- Class Notes:
- The required first meeting in Fall: Wednesday, September 4, 8:15 am, Science Center B Fall Section times: MWF 9:00, MWF 10:30, MWF 12, MWF 1:30, and MWF 3:00 with sufficient enrollment. Drew Zemke, Jameel Al-Aidroos, Stepan Paul, Florence Orosz, Arnav Tripathy, Fabian Gundlach

- Recommended Prep:
- Mathematics 1b or an equivalent background in mathematics.

- Requirements:
- Anti-requisite: Not to be taken in addition to AM21a.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

*Drew Zemke*

2020 Spring (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Green’s, Stokes’s, and Divergence Theorems.

- Course Notes:
- This is a lecture taught in small sections. May not be taken for credit by students who have passed Applied Mathematics 21a. Activities using computers to calculate and visualize applications of these ideas will not require programming experience

- Class Notes:
- Spring Section Times: MWF 9:00, MWF 10:30, MWF 12, MWF 1:30, and MWF 3:00 with sufficient enrollment. Stepan Paul, Jill Guerra, Elden Elmato, Matthew Demers

- Recommended Prep:
- Mathematics 1b or an equivalent background in mathematics.

- Requirements:
- Anti-requisite: Not to be taken in addition to AM21a.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

*Matthew Demers*

2019 Fall (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series.

- Course Notes:
- This is a lecture taught in small sections. May not be taken by students who have passed Applied Mathematics 21b.

- Class Notes:
- The required first meeting in Fall: Tuesday, September 3, 8:15 am, Science Center D Fall Section Times: MWF 9:00, MWF 10:30, MWF 12:00, MWF 1:30 with sufficient enrollment, MWF 3:00 with sufficient enrollment.

- Recommended Prep:
- Mathematics 1b or an equivalent background in mathematics. Mathematics 21a is commonly taken before Mathematics 21b, but is not a prerequisite, although familiarity with partial derivatives is useful.

- Requirements:
- Anti-requisite: Not to be taken in addition to Math 19b or AM 21b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

*Janet Chen*

2020 Spring (4 Credits)

**
Schedule: **
TBD

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series.

- Course Notes:
- This is a lecture taught in small sections. May not be taken by students who have passed Applied Mathematics 21b.

- Class Notes:
- Spring Section Times: MWF 9:00, MWF 10:30, MWF 12:00, MWF 1:30, MWF 3:00 Jill Guerra, Jameel Al-Aidroos David Freund, Brendan Kelly, Dori Bejleiri, Chris Gerig

- Recommended Prep:
- Mathematics 1b or an equivalent background in mathematics. Mathematics 21a is commonly taken before Mathematics 21b, but is not a prerequisite, although familiarity with partial derivatives is useful.

- Requirements:
- Anti-requisite: Not to be taken in addition to Math 19b or AM 21b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Dusty Grundmeier*

2019 Fall (4 Credits)

**
Schedule: **
MWF 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

This course covers multivariable calculus and linear algebra for students interested in mathematical sciences. It covers the same topics as Mathematics 21, but does so with more rigor. Students are taught techniques of proof and mathematical reasoning. The workload and content is comparable with the Mathematics 21 sequence. But unlike the latter, the linear algebra and calculus are more interlinked.

- Course Notes:
- Students in Mathematics 22 are required to participate in a weekly class length workshop dedicated to proof aspects (the schedule of workshop times will be finalized after the first class meeting).

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Dusty Grundmeier*

2020 Spring (4 Credits)

**
Schedule: **
MWF 12:00 PM - 01:15 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
50

A continuation of Mathematics 22a

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

FAS: General Education | Empirical and Mathematical Reasoning |

FAS: Final Assessment Category | Three-hour Exam |

*Paul Bamberg*

2019 Fall (4 Credits)

**
Schedule: **
R 03:00 PM - 05:45 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
45

Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Single variable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series, and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse, and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl. Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results.

- Course Notes:
- Students are expected to watch videos of the lectures from Fall 2015 before attending class. Weekly two-hour classes will consist of a one-hour seminar in which students present key definitions and proofs and a one-hour activity-based session in which students work in small groups to solve problems. Students are expected to continue in either Mathematics 23b (recommended for students who are thinking of concentrating in mathematics, the physical sciences, or engineering) or Mathematics 23c (recommended for students who are not sure of their concentration, or who are thinking about a concentration in the social sciences, economics, computer science, life sciences or data science). Either alternative will provide a solid foundation for a concentration in mathematics or any field that uses mathematics.

- Class Notes:
- The required first meeting in Fall: Tuesday, September 3, 8:15 am, Science Center A

- Recommended Prep:
- Mathematics 1b or a grade of 4 or 5 on the Calculus BC Advanced Placement Examination, plus an interest both in proving mathematical results and in using them. No background in linear algebra, real analysis, or multivariable calculus is assumed.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Paul Bamberg*

2019 Fall (4 Credits)

**
Schedule: **
R 03:00 PM - 05:45 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
96

Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Singlevariable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl. Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results.

- Course Notes:
- Students are expected to watch videos of the lectures from Fall 2015 before attending class. Weekly two-hour classes will consist of a one-hour seminar in which students present key definitions and proofs and a one-hour activity-based session in which students work in small groups to solve problems. Students are expected to continue in either Mathematics 23b (recommended for students who are thinking of concentrating in mathematics, the physical sciences, or engineering) or Mathematics 23c (recommended for students who are not sure of their concentration, or who are thinking about a concentration in the social sciences, economics, computer science, life sciences or data science). Either alternative will provide a solid foundation for a concentration in mathematics or any field that uses mathematics.

- Class Notes:
- Required first meeting: Tuesday, September 4, 8:15 am, Science Center A.

- Recommended Prep:
- Mathematics 1b or a grade of 4 or 5 on the Calculus BC Advanced Placement Examination, plus an interest both in proving mathematical results and in using them. No background in linear algebra, real analysis, or multivariable calculus is assumed.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Paul Bamberg*

2019 Fall (4 Credits)

**
Schedule: **
R 03:00 PM - 05:45 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
30

dimension, eigenvectors, and eigenvalues, including an introduction to the R scripting language. Single variable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series, and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse, and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl. Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results.

- Course Notes:
- Students are expected to watch videos of the lectures from Fall 2015 before attending class. Weekly two-hour classes will consist of a one-hour seminar in which students present key definitions and proofs and a one-hour activity-based session in which students work in small groups to solve problems. Students are expected to continue in either Mathematics 23b (recommended for students who are thinking of concentrating in mathematics, the physical sciences, or engineering) or Mathematics 23c (recommended for students who are not sure of their concentration, or who are thinking about a concentration in the social sciences, economics, computer science, life sciences or data science). Either alternative will provide a solid foundation for a concentration in mathematics or any field that uses mathematics.

- Recommended Prep:
- Mathematics 1b or a grade of 4 or 5 on the Calculus BC Advanced Placement Examination, plus an interest both in proving mathematical results and in using them. No background in linear algebra, real analysis, or multivariable calculus is assumed.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Paul Bamberg*

2019 Fall (4 Credits)

**
Schedule: **
F 12:00 PM - 02:45 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
30

Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Single variable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series, and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse, and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl. Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results.

- Course Notes:

- Recommended Prep:

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | Primarily for Undergraduate Students |

*Paul Bamberg*

2019 Fall (4 Credits)

**
Schedule: **
F 12:00 PM - 02:45 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
30

Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Singlevariable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl. Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results.

- Course Notes:

- Recommended Prep:

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Cross Registration |

*Paul Bamberg*

2019 Fall (4 Credits)

**
Schedule: **
F 12:00 PM - 02:45 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
30

Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Single variable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series, and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse, and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl. Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results.

- Course Notes:

- Recommended Prep:

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Paul Bamberg*

2019 Fall (4 Credits)

**
Schedule: **
F 12:00 PM - 02:45 PM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
30

- Course Notes:

- Recommended Prep:

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Cross Registration |

*Paul Bamberg*

2020 Spring (4 Credits)

**
Schedule: **
F 12:00 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A rigorous, integrated treatment of linear algebra and multivariable calculus. Topics: Riemann and Lebesgue integration, determinants, change of variables, the volume of manifolds, differential forms, and exterior derivative. Stokes’s theorem is presented both in the language of vector analysis (div, grad, and curl) and in the language of differential forms.

- Course Notes:
- Mathematics 23b is a sequel to Mathematics 23a, recommended for students who are thinking of concentrating in mathematics, the physical sciences, or engineering. Students are expected to watch videos of the lectures from spring 2016 before attending class. Weekly two-hour classes will consist of a one-hour seminar in which students present key definitions and proofs and a one-hour activity-based session in which students work in small groups to solve problems.

- Class Note:
- Required first meeting: Monday, January 28, 8:15 am, Science Center E

- Recommended Prep:
- Mathematics 23a.

- Requirements:
- Prerequisite: MATH 23A OR (MATH 21A AND MATH 21B) AND (Not to be taken in addition to MATH 23C)

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Paul Bamberg*

2020 Spring (4 Credits)

**
Schedule: **
TR 01:30 AM - 02:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Proof strategies and logic. Sets, countability, sigma fields, and axiomatic foundations of probability. Summation of series and evaluation of multiple integrals, with emphasis on calculation of expectation and variance. Abstract vector spaces and inner product spaces, with applications to the analysis of large datasets. Key functions and theorems of mathematical statistics. A brief introduction to classical vector calculus as used in electromagnetic theory. Students will learn to use some of the statistical and graphical display tools in the R scripting language.

- Course Notes:
- This course is a sequel to Mathematics 23a, recommended for students who are not sure of their concentration or who are thinking about a concentration in the social sciences, economics, computer science, life sciences or data science. Graduate students wishing to take this course for credit should speak with Dr. Bamberg to arrange enrollment in Mathematics 370 instead.

- Class Notes:
- Required first meeting: Monday, January 28, 8:15 am, Science Center E

- Recommended Prep:
- Mathematics 23a or Mathematics 21a and 21b. The latter option is for seniors who are preparing for graduate programs in statistics, computer science, or data science.

- Requirements:
- Prerequisite: MATH 23A OR (MATH 21A AND MATH 21B) AND (Not to be taken in addition to MATH 23B)

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

Quantitative Reasoning with Data | Yes |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*John Cain*

2019 Fall (4 Credits)

**
Schedule: **
MWF 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A rigorous treatment of linear algebra. Topics include Construction of number systems; fields, vector spaces, and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness, and connectedness.

- Course Notes:
- Expect to spend a lot of time doing mathematics.

- Class Notes:
- Should not be taken for credit after the freshman year without permission from the Mathematics Director of Undergraduate Studies.

- Recommended Prep:
- 5 on the Calculus BC Advanced Placement Examination and some familiarity with writing proofs, or the equivalent as determined by the instructor.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*John Cain*

2020 Spring (4 Credits)

**
Schedule: **
MWF 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

A rigorous treatment of basic analysis. Topics include convergence, continuity, differentiation, the Riemann integral, uniform convergence, the Stone-Weierstrass theorem, Fourier series, differentiation in several variables. Additional topics, including the classical results of vector calculus in two and three dimensions, as time allows.

- Course Notes:
- Expect to spend a lot time doing mathematics.

- Class Notes:
- Should not be taken for credit after the freshman year without permission from the Mathematics Director of Undergraduate Studies.

- Requirements:
- Prerequisite: Mathematics 25A OR Mathematics 55A

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | Primarily for Undergraduate Students |

*Joseph D. Harris*

2019 Fall (4 Credits)

**
Schedule: **
MWF 10:30 AM - 11:45 AM

**
Instructor Permission: **
Instructor

**
Enrollment Cap: **
n/a

A rigorous introduction to abstract algebra, including group theory and linear algebra. This course covers the equivalent of Mathematics 25a and Mathematics 122 and prepares students for Mathematics 123 and other advanced courses in number theory and algebra. (A course in analysis such as Mathematics 25b or 55b is recommended for Spring semester.)

- Course Notes:
- Mathematics 55a is an intensive course for students who are comfortable with abstract mathematics. (Students without this background will gain it and learn the material from Math 55a,b in other courses by continuing into the Mathematics Concentration as sophomores.) Students can switch between Mathematics 55a and either Mathematics 25a, 23a, 22a, 21a during the first three weeks without penalty.

- Recommended Prep:
- Familiarity with proofs and abstract reasoning; and commitment to a fast-moving course.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Joseph D. Harris*

2020 Spring (4 Credits)

**
Schedule: **
MWF 10:30 AM - 11:45 AM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
n/a

A rigorous introduction to real and complex analysis. This course covers the equivalent of Mathematics 25b and Mathematics 113, and prepares students for Mathematics 114 and other advanced courses in analysis.

- Course Notes:
- Mathematics 55b is an intensive course for students having significant experience with abstract mathematics.

- Requirements:
- Prerequisite: Mathematics 55A

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |