Introductory Courses & Sequences

Math 1120 focuses on differential calculus, where the goal is to measure a function’s instantaneous rate of change (the derivative). First, we define limits, which allow us to talk rigorously about instantaneous changes. Then, we define the derivative and find many rules that simplify its computation. Next, we apply the derivative to better understand function behavior (allowing us to minimize cost or maximize profit in economic models). Finally, we develop strategies for reversing the derivative process to find antiderivative functions. These antiderivatives have surprising applications, like computing areas and modeling population growth.

Math 1120 requires the placement exam. Students typically take the course after completing pre-calculus.

Another option is the Math 1100/1110 sequence, a two-semester course that integrates pre-calculus and differential calculus topics. Successful completion of Math 1100 and 1110 is equivalent to Math 1120 in that it satisfies the same major and professional-school (e.g. pre-med) requirements; however a student completing Math 1100 and 1110 receives two course credits and two QR credits.  

Math 1100/1110 requires the placement exam. Some prior knowledge of pre-calculus is helpful. 

Math 1150 investigates the mathematics of accumulation. With integral calculus, we use information about instantaneous rates (like the velocity of a moving object) to compute net changes (like change in position). Series convergence tests and Taylor series help us understand how calculators work, and various other topics (solids of revolution, parametric equations, polar coordinates) allow us to apply the tools of single variable calculus in broader contexts. Throughout the semester, we consider both exact and approximate solutions to problems and investigate the role of error.

Math 1150 and 1160 both cover integral calculus, and both can be used as pre-requisites for any course that requires Math 1150. Math 1160 places emphasis on applications to biology, and is particularly suitable for biology and pre-medical students.

Math 1150 and 1160 require the placement exam, or completion of Math 1110 or 1120 at Yale. Typically, students place into Math 1150 after completing a differential calculus course such as AP Calculus AB. 

Math 1200 extends skills and knowledge you gain in single variable calculus to two and three variables. We will study how to graph surfaces and solids in three dimensions, differentiate and integrate functions of several variables, optimize functions of several variables (for example when maximizing profit in economics),  we will learn how to integrate along curves and surfaces (for example when calculating how much energy a solar panel will generate while the sun is shining on it), and many other topics.

Math 1200 and 1210 both cover multivariable calculus, and both can be used as pre-requisites for any course that requires Math 1200. Math 1210 places emphasis on applications to biology, and is particularly suitable for biology and pre-medical students. [Note that Math 1210 is typically only offered every other year.]

Math 1200 and 1210 require the placement exam or completion of Math 1150 or 1160 at Yale. Typically, students place into Math 1200 after completing an integral calculus course, such as AP Calculus BC. 

Math 2220 is a linear algebra course best suited for students who wish to focus on applications and practical solving problem practice, rather than abstract mathematics and mathematical proofs. It is often taken by students majoring in engineering, technology, science, social sciences, and economics. Mathematics majors, as well as students majoring in Math + CPSC or Math + Econ, need to complete Math 2250 or 2260, rather than 2220.

Math 2250 is a linear algebra course, and a part of our introductory sequence into the math major. In linear algebra, you will learn key language and concepts used throughout pure mathematics as well as in a wide variety of applications.  Linear algebra starts by studying systems of equations with many variables, and it builds a detailed understanding of how to work in abstract n-dimensional space. This version of linear algebra focuses on concepts, and provides an introduction to writing mathematical proofs.

Math 2250 usually has a zoom town hall meeting in early August: it will be listed under the “Summer / Fall event schedule” toward the top of this page. 

We also offer an intensive version of this course, Math 2260, for students who are looking for an extra challenge. Math 2260 will teach the same main topics as Math 225, but it may go into more depth, ask more challenging problems on homework and exams, or cover optional side topics. Both courses prepare students equally well for subsequent mathematics courses, or for using proofs, problem solving skills, and linear algebra in other fields. 

If you start in Math 2260, and find that it is too time consuming, you can move to Math 2250 until Midterm. You can read more about how moving between courses work in our Calculus FAQ. 

Math 2220, 2250, and 2260 do not require the placement exam or preference selection, you can simply sign up for them when registration opens. They can be taken directly after integral calculus, though most students complete multivariable calculus first. For advice about whether linear algebra may be the best place for you to start, please see the section below on choosing your first math course(s) at Yale (particularly if you have not yet taken multivariable calculus, and it may be required by your prospective major or other courses you plan to take). 

Math 2550 is an introduction to real analysis, and a part of our introductory sequence into the math major. The topics are similar to differential and integral calculus, but it’s done in a rigorous way that focuses on proofs and details of the concepts that happen behind the scenes in calculus.  Said differently, our calculus sequence teaches you how to use the tools of limits, derivatives, and integrals to solve a variety of problems, whereas this course teaches you how to build all of those tools from scratch.

We also offer an intensive version of this course, Math 2560, for students who are looking for an extra challenge. Math 2560 will teach the same main topics as Math 2550, but it may go into more depth, ask more challenging problems on homework and exams, or cover some optional side topics. Both courses prepare student equally well for subsequent math courses, or for using real analysis in other fields. 

Math 2550 and 2560 require prior completion of Math 2250 or 2260.

Math 2550 and 2560 do not use the placement exam or preference selection. It requires Math 2250 or 2260. If you have taken a proof-based linear algebra course elsewhere, and are considering Math 2550 or 2560 as your first math course, please contact the Math DUS at math.dus@yale.edu to assist you with placement, as well as with requirements of your potential major (if it requires linear algebra). For more information, see section on choosing your first math course(s) below. 

Math 3020 is a rigorous and exciting course that explores some of the beautiful connections between linear algebra and real analysis. For this reason, it requires completion of 2250/2260 and 2550/2560. 

This course is an introduction to manifolds, which are low-dimensional shapes living in a higher dimensional universe (e.g., a one-dimensional coat hanger or a two-dimensional eggshell each living in a three-dimensional world), and it teaches how to do calculus in this abstract setting.  Although the focus of this course will be on abstract mathematical objects and their properties, the material in this course also has applications to a broad range of topics including theoretical physics and machine learning.

Many students who have not completed multivariable calculus in high school would benefit from taking Math 1200. However, prospective math majors, and students excited about rigorous proof-based math should consider taking Math 2250/2260, Math 2550/2560 and Math 3020 instead.  Math 3020 is also appropriate for students who have already completed Math 1200 and are curious to learn multivariable analysis rigorously. (Note that Math 3020 can only be taken after Math 2250 and 2550.) 

Math 3020 does not use the placement exam or preference selection. It requires both proof-based linear algebra, and real analysis (Math 2250 and 2550). If you have taken similar courses elsewhere and wish to start at Yale with Math 3020, please be sure to contact the Math DUS during the summer before your first year, at math.dus@yale.edu , so we can help you as far as course placement as well as requirements of your potential major. For more information, please see the section on choosing your first math course(s) below. 

1060 content TK

The math department offers other Quantitative Reasoning (QR) courses that assume no calculus experience:

In Math 1070 (Mathematics in the real world), students use mathematical ideas to solve real world problems. Topics include compound interest, population growth, probability and its applications to games of chance, mortgage payments, false positives in drug testing, computer security, and other questions. 

Math 1080 (Estimation and error) leads students through a problem-based investigation of basic mathematical principles and techniques that help make sense of the world. Applications include geology, ecology, finance, and other fields. 

Math 2410 and 2420: probability and statistics teach us how to collect, analyze, and interpret data. That being so, it is directly useful in almost any field where experiments, surveys, or other means for data collection are employed.

Math 2440: discrete mathematics includes many topics of interest in computer science.

Math 2460: differential equations can be used for modeling complex systems in biology, physics, economics, and other fields. 

Math 2700: set theory is at the foundation of mathematics and often taken by philosophy students.