## MATH 19: Calculus

Introduction to differential calculus of functions of one variable. Review of elementary functions (including exponentials and logarithms), limits, rates of change, the derivative and its properties, applications of the derivative. Prerequisites: trigonometry, advanced algebra, and analysis of elementary functions (including exponentials and logarithms). You must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.

Terms: Aut, Win, Sum
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Chetard, B. (PI)
;
La, J. (PI)
;
Schaeffer, G. (PI)
;
Falcone, P. (TA)
;
Li, Z. (TA)
;
Marsden, M. (TA)
;
Stavrianidi, A. (TA)
;
Zhou, Y. (TA)

## MATH 20: Calculus

The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus, and the Mean Value Theorem for integrals. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initial-value problems, exponential and logistic models, direction fields, and parametric curves. Prerequisite:
Math 19 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.

Terms: Aut, Win, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Grzegrzolka, P. (PI)
;
Lucianovic, M. (PI)
;
Schaeffer, G. (PI)
...
more instructors for MATH 20 »

Instructors:
Grzegrzolka, P. (PI)
;
Lucianovic, M. (PI)
;
Schaeffer, G. (PI)
;
Chen, S. (TA)
;
Fushida-Hardy, S. (TA)
;
Lim, B. (TA)
;
Mackey, W. (TA)
;
Marsden, M. (TA)
;
McConnell, S. (TA)
;
Ortiz, J. (TA)

## MATH 21: Calculus

Review of limit rules. Sequences, functions, limits at infinity, and comparison of growth of functions. Review of integration rules, integrating rational functions, and improper integrals. Infinite series, special examples, convergence and divergence tests (limit comparison and alternating series tests). Power series and interval of convergence, Taylor polynomials, Taylor series and applications. Prerequisite:
Math 20 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.

Terms: Aut, Win, Spr, Sum
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Datta, I. (PI)
;
Falcone, P. (PI)
;
Grzegrzolka, P. (PI)
...
more instructors for MATH 21 »

Instructors:
Datta, I. (PI)
;
Falcone, P. (PI)
;
Grzegrzolka, P. (PI)
;
Khu, D. (PI)
;
Kim, G. (PI)
;
Lim, B. (PI)
;
Schaeffer, G. (PI)
;
Zhang, S. (PI)
;
Zhou, Z. (PI)

## MATH 21A: Calculus, ACE

Students attend
MATH 21 lectures with different recitation sessions: two hours per week instead of one, emphasizing engineering applications. Prerequisite: application; see
https://web.stanford.edu/dept/soe/osa/ace.fb

Terms: Aut, Win, Spr
| Units: 5

Instructors:
Kim, G. (PI)
;
McConnell, S. (PI)
;
Schaeffer, G. (PI)
...
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Instructors:
Kim, G. (PI)
;
McConnell, S. (PI)
;
Schaeffer, G. (PI)
;
Velcheva, K. (PI)
;
Zhu, B. (PI)
;
Velcheva, K. (TA)

## MATH 51: Linear Algebra, Multivariable Calculus, and Modern Applications

This course provides unified coverage of linear algebra and multivariable differential calculus, and the free course e-text connects the material to many fields. Linear algebra in large dimensions underlies the scientific, data-driven, and computational tasks of the 21st century. The linear algebra portion includes orthogonality, linear independence, matrix algebra, and eigenvalues with applications such as least squares, linear regression, and Markov chains (relevant to population dynamics, molecular chemistry, and PageRank); the singular value decomposition (essential in image compression, topic modeling, and data-intensive work in many fields) is introduced in the final chapter of the e-text. The multivariable calculus portion includes unconstrained optimization via gradients and Hessians (used for energy minimization), constrained optimization (via Lagrange multipliers, crucial in economics), gradient descent and the multivariable Chain Rule (which underlie many machine learning al
more »

This course provides unified coverage of linear algebra and multivariable differential calculus, and the free course e-text connects the material to many fields. Linear algebra in large dimensions underlies the scientific, data-driven, and computational tasks of the 21st century. The linear algebra portion includes orthogonality, linear independence, matrix algebra, and eigenvalues with applications such as least squares, linear regression, and Markov chains (relevant to population dynamics, molecular chemistry, and PageRank); the singular value decomposition (essential in image compression, topic modeling, and data-intensive work in many fields) is introduced in the final chapter of the e-text. The multivariable calculus portion includes unconstrained optimization via gradients and Hessians (used for energy minimization), constrained optimization (via Lagrange multipliers, crucial in economics), gradient descent and the multivariable Chain Rule (which underlie many machine learning algorithms, such as backpropagation), and Newton's method (an ingredient in GPS and robotics). The course emphasizes computations alongside an intuitive understanding of key ideas. The widespread use of computers makes it important for users of math to understand concepts: novel users of quantitative tools in the future will be those who understand ideas and how they fit with examples and applications. This is the only course at Stanford whose syllabus includes nearly all the math background for
CS 229, which is why
CS 229 and
CS 230 specifically recommend it (or other courses resting on it). For frequently asked questions about the differences between
Math 51 and
CME 100, see the FAQ on the placement page on the Math Department website. Prerequisite:
Math 21,
Math 42, or the math placement diagnostic (offered through the Math Department website) in order to register for this course.

Terms: Aut, Win, Spr, Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Cant, D. (PI)
;
Chen, D. (PI)
;
Chetard, B. (PI)
;
Dore, D. (PI)
;
Helfer, J. (PI)
;
Izzo, Z. (PI)
;
Kim, G. (PI)
;
Kraushar, N. (PI)
;
Larson, H. (PI)
;
Lucianovic, M. (PI)
;
Perlman, M. (PI)
;
Sloman, L. (PI)
;
Taylor, C. (PI)
;
Trettel, S. (PI)
;
Wang, G. (PI)
;
White, B. (PI)
;
Wieczorek, W. (PI)
;
Ying, L. (PI)
;
Zavyalov, B. (PI)
;
Arana Herrera, F. (TA)
;
Cant, D. (TA)
;
Helfer, J. (TA)
;
Izzo, Z. (TA)
;
Libkind, S. (TA)
;
Sloman, L. (TA)

## MATH 51A: Linear Algebra, Multivariable Calculus, and Modern Applications, ACE

Students attend
MATH 51 lectures with different recitation sessions: four hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see
https://engineering.stanford.edu/students-academics/engineering-diversity-programs/additional-calculus-engineers-ace

Terms: Aut, Win, Spr
| Units: 6
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Cotner, S. (PI)
;
Lucianovic, M. (PI)
;
Taylor, C. (PI)
...
more instructors for MATH 51A »

Instructors:
Cotner, S. (PI)
;
Lucianovic, M. (PI)
;
Taylor, C. (PI)
;
Wieczorek, W. (PI)
;
Yang, K. (TA)
;
Zachos, E. (TA)

## MATH 52: Integral Calculus of Several Variables

Iterated integrals, line and surface integrals, vector analysis with applications to vector potentials and conservative vector fields, physical interpretations. Divergence theorem and the theorems of Green, Gauss, and Stokes. Prerequisite: 51 or equivalents.

Terms: Aut, Win, Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Dimakis, P. (PI)
;
Grzegrzolka, P. (PI)
;
Nguyen, D. (PI)
...
more instructors for MATH 52 »

Instructors:
Dimakis, P. (PI)
;
Grzegrzolka, P. (PI)
;
Nguyen, D. (PI)
;
Ohrt, C. (PI)
;
Zhang, S. (PI)
;
Nguyen, D. (TA)

## MATH 53: Ordinary Differential Equations with Linear Algebra

Ordinary differential equations and initial value problems, systems of linear differential equations with constant coefficients, applications of second-order equations to oscillations, matrix exponentials, Laplace transforms, stability of non-linear systems and phase plane analysis, numerical methods. Prerequisite: 51 or equivalents.

Terms: Aut, Win, Spr, Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Chodosh, O. (PI)
;
Ottolini, A. (PI)
;
Simper, M. (PI)
...
more instructors for MATH 53 »

Instructors:
Chodosh, O. (PI)
;
Ottolini, A. (PI)
;
Simper, M. (PI)
;
Varolgunes, U. (PI)
;
Wieczorek, W. (PI)

## MATH 61CM: Modern Mathematics: Continuous Methods

This is the first part of a theoretical (i.e., proof-based) sequence in multivariable calculus and linear algebra, providing a unified treatment of these topics. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space as local graphs, integration on Euclidean space, and many examples. The linear algebra content is covered jointly with
Math 61DM. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.

Terms: Aut
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Luk, J. (PI)
;
Dunlap, A. (TA)

## MATH 61DM: Modern Mathematics: Discrete Methods

This is the first part of a theoretical (i.e., proof-based) sequence in discrete mathematics and linear algebra. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. The linear algebra content is covered jointly with
Math 61CM. Students should have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.nnThis sequence is not appropriate for students planning to major in natural sciences, economics, or engineering, but is suitable for majors in any other field (such as MCS ("data science"), computer science, and mathematics).

Terms: Aut
| Units: 5
| UG Reqs: WAY-FR

Instructors:
Vondrak, J. (PI)
;
Wigderson, Y. (TA)

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