## 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:
Asserian, L. (PI)
;
Jack, T. (PI)
;
Kotas, J. (PI)
;
Sun, W. (PI)
;
Chen, D. (TA)
;
Chu, Y. (TA)
;
Ho, W. (TA)
;
KimLee, H. (TA)
;
McDonald, V. (TA)
;
Mistele, J. (TA)
;
Serio, C. (TA)

## MATH 19A: Calculus, ACE

Additional problem solving session for
Math 19 guided by a course assistant. Concurrent enrollment in
Math 19 required. Application required:
https://forms.gle/ruykWBk6zJMgXRB49

Terms: Aut, Win
| Units: 1

## MATH 20: Calculus

The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus. 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:
Dai, I. (PI)
;
Jack, T. (PI)
;
Li, Z. (PI)
;
Wieczorek, W. (PI)
;
Chaudhry, M. (TA)
;
Dore, D. (TA)
;
Ho, W. (TA)
;
Iwasaki, H. (TA)
;
Jajoo, A. (TA)
;
Li, H. (TA)
;
Mistele, J. (TA)
;
Qin, Q. (TA)
;
lou, s. (TA)

## MATH 20A: Calculus, ACE

Additional problem solving session for
Math 20 guided by a course assistant. Concurrent enrollment in
Math 20 required. Application required:
https://forms.gle/ruykWBk6zJMgXRB49

Terms: Aut, Win, Spr
| Units: 1

Instructors:
Dai, I. (PI)
;
Li, Z. (PI)
;
Wieczorek, W. (PI)
;
Har, M. (TA)
;
Panda, A. (TA)
;
Rodriguez, A. (TA)

## MATH 21: Calculus

This course addresses a variety of topics centered around the theme of "calculus with infinite processes", largely the content of BC-level AP Calculus that isn't in the AB-level syllabus. It is needed throughout probability and statistics at all levels, as well as to understand approximation procedures that arise in all quantitative fields (including economics and computer graphics). After an initial review of limit rules, the course goes on to discuss sequences of numbers and of functions, as well as limits "at infinity" for each (needed for any sensible discussion of long-term behavior of a numerical process, such as: iterative procedures and complexity in computer science, dynamic models throughout economics, and repeated trials with data in any field). Integration is discussed for rational functions (a loose end from
Math 20) and especially (improper) integrals for unbounded functions and "to infinity": this shows up in contexts as diverse as escape velocity for a rocket, the pres
more »

This course addresses a variety of topics centered around the theme of "calculus with infinite processes", largely the content of BC-level AP Calculus that isn't in the AB-level syllabus. It is needed throughout probability and statistics at all levels, as well as to understand approximation procedures that arise in all quantitative fields (including economics and computer graphics). After an initial review of limit rules, the course goes on to discuss sequences of numbers and of functions, as well as limits "at infinity" for each (needed for any sensible discussion of long-term behavior of a numerical process, such as: iterative procedures and complexity in computer science, dynamic models throughout economics, and repeated trials with data in any field). Integration is discussed for rational functions (a loose end from
Math 20) and especially (improper) integrals for unbounded functions and "to infinity": this shows up in contexts as diverse as escape velocity for a rocket, the present value of a perpetual yield asset, and important calculations in probability (including the famous "bell curve" and to understand why many statistical tests work as they do). The course then turns to infinite series (how to "sum" an infinite collection of numbers), some useful convergence and divergence rests for these, and the associated killer app: power series and their properties, as well as Taylor approximations, all of which provide the framework that underlies virtually all mathematical models used in any quantitative field.

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

Instructors:
Dowlin, N. (PI)
;
Ho, W. (PI)
;
Jack, T. (PI)
;
Lai, Y. (PI)
;
Ma, C. (PI)
;
Wieczorek, W. (PI)
;
Ahmed, N. (TA)
;
Ho, W. (TA)
;
King, M. (TA)
;
Nuti, P. (TA)
;
Pagadora, J. (TA)
;
Ryzhik, A. (TA)

## MATH 21A: Calculus, ACE

Students attend one of the regular
MATH 21 lectures with a longer discussion section of two hours per week instead of one. Active mode: students in small groups discuss and work on problems, with a TA providing guidance and answering questions. Application required:
https://forms.gle/ruykWBk6zJMgXRB49

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

Instructors:
Lai, Y. (PI)
;
Ma, C. (PI)
;
Wieczorek, W. (PI)
;
Khu, D. (TA)
;
Nuti, P. (TA)
;
Taylor, L. (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 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:
Asserian, L. (PI)
;
Dowlin, N. (PI)
;
Khu, D. (PI)
;
Kim, G. (PI)
;
Kraushar, N. (PI)
;
Lucianovic, M. (PI)
;
Ma, C. (PI)
;
Park, J. (PI)
;
Taylor, C. (PI)
;
Aboumrad, G. (TA)
;
Angelo, R. (TA)
;
Bayrooti, J. (TA)
;
Chen, D. (TA)
;
Cowan, N. (TA)
;
Dimakis, P. (TA)
;
Falcone, P. (TA)
;
Foster, B. (TA)
;
Godoy, F. (TA)
;
Khu, D. (TA)
;
Kilgore, E. (TA)
;
King, M. (TA)
;
Kraushar, N. (TA)
;
Li, Z. (TA)
;
Li, Z. (TA)
;
Liu, Y. (TA)
;
Madden, I. (TA)
;
Miagkov, K. (TA)
;
Pandit, N. (TA)
;
Ryzhik, A. (TA)
;
Sanghi, S. (TA)
;
Wu, Y. (TA)
;
Zhu, J. (TA)
;
lou, s. (TA)

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

Students attend one of the regular
MATH 51 lectures with a longer discussion section of four hours per week instead of two. Active mode: students in small groups discuss and work on problems from a worksheet distributed 2 or 3 days in advance, with a TA providing guidance and answering questions. Application required:
https://forms.gle/ruykWBk6zJMgXRB49

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

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

Instructors:
Kim, G. (PI)
;
Lucianovic, M. (PI)
;
Taylor, C. (PI)
;
Falcone, P. (TA)
;
Khu, D. (TA)
;
Libkind, S. (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:
Math 21 and
Math 51 or equivalents.

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

Instructors:
Asserian, L. (PI)
;
Li, Z. (PI)
;
Wieczorek, W. (PI)
...
more instructors for MATH 52 »

Instructors:
Asserian, L. (PI)
;
Li, Z. (PI)
;
Wieczorek, W. (PI)
;
Lolas, P. (TA)
;
Marsden, M. (TA)
;
Pagadora, J. (TA)
;
Qian, L. (TA)

## MATH 52A: Integral Calculus of Several Variables, ACE

Additional problem solving session for
Math 52 guided by a course assistant. Concurrent enrollment in
Math 52 required. Application required:
https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus

Terms: Spr
| Units: 1

Instructors:
Asserian, L. (PI)
;
Shaib, G. (TA)

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