# Introductory Math Courses

## Calculus Preparation

**Math 18- ***Foundations for Calculus (2 units, S/NC, Fall only)* covers the mathematical background and fundamental skills necessary for success in calculus and other college-level quantitative work. Topics include ratios, unit conversions, functions and graphs, polynomials and rational functions, exponential and logarithm, trigonometry and the unit circle, and word problems. Class sessions are a mix of lecture and worksheets.

## Single Variable Calculus

### The 20-Series:

This series covers differential calculus, integral calculus, and power series in one variable. It can be started at any point in the sequence for those with sufficient background. See the detailed list of topics.

**Math 19-** *Calculus (3 units)* covers properties and applications of limits, continuous functions, and derivatives. Calculations involve trigonometric functions, exponentials, and logarithms, and applications include max/min problems and curve-sketching.

**Math 20-** *Calculus (3 units)* covers properties and applications of integration, including the Fundamental Theorem of Calculus and computations of volumes, areas, and arc length of parametric curves. An introduction to some basic notions related to differential equations (such as exponential growth/decay and separable equations) is also given.

**Math 21-** *Calculus (4 units)* covers limits at infinity and unbounded functions in the context of integration as well as infinite sums, including convergence/divergence tests and power series. Taylor series and applications are also covered.

Math 21 is an **enforced requirement** of *all* majors in the School of Engineering (including CS and MS&E) and Chemistry and Symbolic Systems, and is required knowledge for Data Science (BS), Geophysics, and Physics. You cannot do university-level quantitative work (including probability) without this material. The placement diagnostic is only advisory (no exam security), and Math 21 content is not in Math 51, **so placement diagnostic guidance or passing Math 51 doesn't waive Math 21 requirements.** For more details, click on the red button above.

## Courses in Multivariable Mathematics

The department offers 3 sequences in multivariable mathematics. If you took multivariable calculus elsewhere, please click the button below:

Already took some multivariable calculus?

**The 50-Series:**

**Math 51- ***Linear Algebra, Multivariable Calculus, and Modern Applications (5 units)* covers linear algebra and multivariable differential calculus in a unified manner alongside applications related to many quantitative fields. This material includes the basic geometry and algebra of vectors, matrices, and linear transformations, as well as optimization techniques in any number of variables (involving partial derivatives and Lagrange multipliers).

The unified treatment of both linear algebra (beyond dimension 3 and including eigenvalues) and multivariable optimization is not covered in a single course accessible to non-majors anywhere else. Many students who learn some multivariable calculus before arriving at Stanford find Math 51 to be instructive to take due to its broad scope and synthesis of concepts. *If you want transfer credit to substitute for Math 51 then you will likely need two courses (one on multivariable calculus, one on linear algebra).*

**Math 52- ***Integral Calculus of Several Variables (5 units)*** **covers multivariable integration, and in particular Green’s Theorem and Stokes’ Theorem. This uses both linear algebra and matrix derivative material from Math 51.

**Math 53- ***Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications (5 units)* develops core concepts, examples, and results for ordinary differential equations, and covers important partial differential equations and Fourier techniques for solving them. This uses both linear algebra and matrix derivative material from Math 51.

The unified treatment of both ordinary and partial differential equations (PDE’s) along with Fourier methods is not covered in a single course accessible to non-majors anywhere else. For those who studied differential equations before arriving at Stanford, Math 53 will strengthen your skills and broaden your knowledge via its use of linear algebra and its coverage of PDE's and Fourier methods. *If you want to transfer credit to substitute for Math 53 then you will likely need two courses (one on ordinary differential equations using linear algebra, and one on PDE/Fourier material).*

**Math 52 and Math 53 can be taken in either order.

This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.

Math 51 Textbook Math 53 Textbook

The table of contents and page of applications near the start of the course texts provide more information; these course texts are freely available to anyone with an SUNetId.

For those with a strong interest in math and a preference for more conceptual and theoretical understanding we recommend the following two sequences:

**The 60CM-Series**

**Math 61CM-62CM-63CM- **Modern Mathematics: Continuous Methods** **(5 units each) This proof-oriented three-quarter sequence covers the material of 51, 52, 53, and additional advanced calculus, higher-dimensional geometry, and ordinary and partial differential equations. This provides a unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know single-variable calculus very well and have an interest in a theoretical approach to the subject.

This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.

**The 60DM-Series**

This proof-oriented three-quarter sequence covers the same linear algebra and multivariable optimization material as the 60CM-series but draws its motivation from topics in discrete math rather than from the more analytic topics as in the 60CM-series. Its discrete math coverage includes combinatorics, probability, some basic group theory, number theory, and graph theory. Students should have an interest in a theoretical approach to the subject.

This series provides the necessary mathematical background for majors in Computer Science, Economics, Mathematics, Mathematical and Computational Science, and most Natural Sciences and some Engineering majors. Those who plan to major in Physics or in Engineering majors requiring Math 50’s beyond Math 51 are recommended to take Math 60CM.

## Learning Proof Writing

Many 100-level mathematics courses assume familiarity with writing proofs, and if you plan to be a Math major then you should learn proof writing as soon as possible. Here is a list of courses to begin learning proof-writing:

- Math 56
- 61CM or 61DM
- Math 108
- Math 110
- Math 113
- Math 115

Math 104 also provides an introduction to proof-writing, but not at the same level as the above courses (a variety of proofs are covered, but students are not expected to write proofs of their own at the same level as some of those shown in class).

Please note: Math 104 and Math 113 do not assume background beyond the linear algebra covered in Math 51.

## More Information

For more information about these courses see ExploreCourses for course descriptions and schedule. If you have further questions about which course to take, contact your academic advisor, or our Director of Undergraduate Studies.