# Mathematics and Statistics Courses 2023-24

## Mathematics Requirement

The mathematics requirements for departmental and School of Engineering majors are delineated by major in the detailed “Majors & Minors” section. In general, each program requires a number of specific and elective courses from the list of approved courses shown below. Stanford requires all incoming students to take the Math Placement Diagnostic to help find the right course to start with; this is a prerequisite to enroll in introductory math courses (MATH 19 through 51). Students who have already taken a math course at Stanford can continue in the sequence without taking the placement diagnostic. **Placement Note:** The Math diagnostic exam is a tool you can use to self-assess your placement into a Stanford math course. However, It does not allow you to skip earning credit for Math 19-20-21 in whole or part. However, earning a grade of C or better in an approved more advanced math course allows you to petition to waive any such missing credits. Enrollment in Math 51 or CME 100 without credit from Math 21 is only permitted via strong performance on a written exam administered by the Math department (only during NSO and only to new undergraduates). Students who take this written exam (which will be proctored, and replace the use of the oral exam during 2023-24) will be required to affirm their knowledge that no course credit for Math 21 is earned via the exam. Moreover, taking this NSO exam will be invitation-only, requiring the student to provide some basic evidence of serious prior study of the Math 21 material despite the absence of any credit for it.

**ACE****:** ACE is a set of companion courses to Stanford's Math, CS, and CME foundational courses.These 1-unit courses are a popular and effective way to increase understanding of course material in a small group setting. Note that (since 2018-19) ACE units do not count toward major program unit minimums, though they do count toward the 180-unit minimum needed for graduation.

Here is an overview of the math courses taken by most students interested in engineering majors at Stanford.

**MATH 19, 20, and 21** present single variable calculus. Students who have 10 units of AP BC calculus credit and placement into MATH 51/CME 100 may skip these (students who score 4 on the BC test are eligible to petition to waive out of MATH 21 once they have successfully completed MATH 53 or CME 102 with a grade of C or better). If you took AB-level AP or IB calculus, take Math 21 before proceeding to multivariable calculus; the content of Math 21 is used everywhere in engineering fields (see Math Dept/Intro Math Courses), and its conceptual level is valuable experience prior to taking further mathematics courses. **Placement Note:** The Math diagnostic exam is a tool you can use to self-assess your placement into a Stanford math course. However, it does not allow you to skip the Math 20 series in whole or part except via petition after successful completion of more advanced courses (Math 51/CME 100 to waive Math 19 and/or 20; Math 53 to waive Math 21) with a grade of C or better; petition approval is not automatic but takes into account the individual record and circumstances.

** MATH 18** (2 units, Autumn) develops fundamental skills to prepare for success in calculus (the Math 20 series) and other courses at Stanford that rely on quantitative methods. While it cannot be counted for unit credit in any of the engineering majors, it is very helpful for achieving success in math and subsequent engineering courses if placement is recommended.

**MATH 51, 52, and 53** may be taken by students who have fulfilled the single-variable calculus requirement. They present multivariable calculus using serious linear algebra right from the start in Math 51, which uses high-dimensional linear algebra to solve optimization problems in many variables; the free e-textbook illustrates the practical relevance of the content across many contemporary engineering and scientific fields. Math 52 covers multivariable integration (needed in both probability and physical science), and Math 53 covers differential equations and introductory Fourier theory (using insights from linear algebra); these latter two courses can be taken in either order. To get experience with MATLAB, take the 1-unit 4-week CME 192 (offered Fall and Winter).

** CME 100, 102, and 104 or 106** (same as ENGR 154, 155A, 155B, and 155C): These courses are offered by the Institute for Computational and Mathematical Engineering. CME 100 presents multivariable optimization and integration with engineering applications, and introduces MATLAB (which is incorporated throughout the CME series and is useful in many later engineering and science courses). CME 102 covers ordinary differential equations (without linear algebra), CME 104 covers linear algebra and partial differential equations, and CME 106 covers probability and statistics for engineering – all with an emphasis on engineering examples and topics. CME 102 has single-variable calculus as a prerequisite (with CME 100 also recommended), CME 104 has CME 102 and CME 100 as prerequisites, and CME 106 has as prerequisites either CME 100 or MATH 52.

The content in CME 100 and Math 51 differ significantly (so these are not in any sense equivalent courses). CME 100 focuses on engineering examples and topics, while Math 51 provides a more comprehensive introduction to linear algebra and differential vector calculus (with applications illustrated across engineering and scientific fields in the course text). More detailed coverage of linear algebra in the CME series is deferred to CME 104. (The course ENGR 108 (formerly CME 103) on applied linear algebra is not a substitute, since its coverage of linear algebra concepts is only a portion of that in Math 51 and CME 104 in terms of topics relevant to many engineering fields.) Coverage of integral vector calculus in the Math series is deferred to Math 52. If this may make a difference to your future plans, please speak with your academic advisor. Students who take both MATH 51 and CME 100 will be able to apply only 8 units of credit toward their major program due to duplication of material.

*Note: The MATH department courses do not include MATLAB, which is important in many engineering fundamental and upper-division engineering courses. However, the advantage of the MATH series is that serious linear algebra is included right from the start. To get MATLAB experience, take the 1-unit CME 192 (4-week course offered A, W).

**ACE:** ACE units do not count toward major program unit minimums, though they do count toward the 180-unit minimum needed for graduation. Check Explore Courses to determine which quarter the ACE section is available for any given math or science course.

MATH 19 (A,W,Sum) 20 (A,W,S, Sum), 21 (A,W,S, Sum) | Calculus of a Single Variable (or AP BC Math Credit, or equivalent). If AP, IB, or transfer credit is used, a maximum 10 units of credit can be used toward the School of Engineering major minimum unit requirement for Math 19/20/21. | 3, 3, 4 |

CME 100** (A,S) | Vector Calculus for Engineers (same as ENGR 154)(Note: taking ACE is encouraged, but the 1-unit credit will not count toward major unit minimum requirements) | 5 |

CME 102 (A,W,Sum) | Ordinary Differential Equations for Engineers (same as ENGR 155A) | 5 |

ENGR 108 (A,W) | Introduction to Matrix Methods (formerly CME/EE 103) | 5 |

CME 104 (S) | Linear Algebra and Partial Differential Equations for Engineers (same as ENGR 155B) | 5 |

CME 108 (W,Sum) | Introduction to Scientific Computing | 3 |

CME 192 | Introduction to MATLAB (4 weeks; A,W,S) | 1 |

MATH 51** (A,W,S,Sum) | Linear Algebra & Differential Calculus of Several Variables (Note: taking ACE is encouraged, but the 1-unit credit will not count toward major unit minimum requirements)
| 5 |

MATH 52 (A,W,S,Sum) | Integral Calculus of Several Variables | 5 |

MATH 53 (W,S,Sum) | Ordinary Differential Equations with Linear Algebra | 5 |

MATH 61-63CM OR DM | Modern Mathematics: Continuous Methods (replaces 50H series) | 5, 5, 5 |

CEE 101D/201D (A) | Computations in CEE | 3 |

CS 103 (A,W,S,Sum) | Mathematical Foundations of Computing (Pre- or co-req: CS 106B/X) | 5 |

ENGR 62 (MS&E 111) (W,Sum) | Introduction to Optimization (Prereq: CME 100 or MATH 51) | 4 |

ENGR 62X (MS&E 111X) (S) | Introduction to Optimization Accelerated (Prereq: CME 100 or MATH 51) | 3-4 |

MATH 104 (A,W,S) | Applied Matrix Theory | 4 |

MATH 106 (W) | Functions of a Complex Variable | 4 |

MATH 109 (A) | Applied Group Theory | 4 |

MATH 110 (A) | Applied Number Theory & Field Theory | 4 |

MATH 113 (A,W,S) | Linear Algebra and Matrix Theory | 4 |

MATH 115 (A,S) | Functions of a Real Variable | 4 |

MATH 120 (A,S) | Groups and Rings | 4 |

MATH 121 (W) | Galois Theory | 4 |

MATH 131P (W) | Partial Differential Equations | 4 |

MS&E 121 (W) | Intro to Stochastic Modeling | 4 |

or more advanced Mathematics courses via approval of petition to deviate | ||

Statistics & Probability* | ||

CME 106 (W,Sum) | Intro to Probability & Statistics for Engineers (same as ENGR 155C | 4 |

STATS 60/160 (A,W,S,Sum) | Introduction to Statistical Methods: Precalculus | 5 |

STATS 110 (A) | Statistical Methods in Engineering and the Physical Sciences | 5 |

STATS 116 (A) | Theory of Probability | 5 |

CS 109 (A,W,S,Sum) | Introduction to Probability for Computer Scientists (prereqs: CS 106B or X, CS 103; multivariable calculus) | 5 |

EE 178 (A,S) | Probabilistic Systems Analysis | 4 |

MATH 151 (W) | Intro to Probability Theory | 4 |

MS&E 120 (A) | Probabilistic Analysis | 4 |

MS&E 125 (S) | Introduction to Applied Statistics | 4 |

CEE 203 (A) | Probabilistic Models in Civil Engineering | 4 |

or more advanced Statistics courses numbered over 100 via approval of petition to deviate |

* Some major programs allow only specific courses or allow/require courses in addition to those listed above; check your major program sheet footnotes to see what specific courses can be applied toward each major.

** If MATH 51 and CME 100 are both taken, only 8 units of credit will be allowed toward the SoE major program due to overlapping material.

*** Many majors (e.g. BioE) need MATLAB. CME 192 is a 4-week, 1-unit MATLAB course