printer friendly pageMATH 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:
https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
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
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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. 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:
https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr, Sum
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Amar, S. (PI)
;
Kim, J. (PI)
;
Lee, S. (PI)
;
Lichtman, J. (PI)
;
Wickham, Z. (PI)
;
Wieczorek, W. (PI)
;
Acharya, S. (TA)
;
Chokkhanchitchai, T. (TA)
;
Gangal, A. (TA)
;
Iwasaki, H. (TA)
;
Kim, J. (TA)
;
Lin, A. (TA)
;
Rizk, K. (TA)
;
Shin, H. (TA)
;
Srivastava, E. (TA)
;
Stavrianidi, A. (TA)
;
Woringer, P. (TA)
;
Wu, Y. (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
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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 equivalent (e.g. 5 on the AP Calculus BC test or suitable score on certain international exams:
https://studentservices.stanford.edu/my-academics/earn-my-degree/undergraduate-degree-progress/test-transfer-credit/external-test-2). 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:
https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Math 51 is considered equivalent to
Math 61CM, and credit will not be granted for both courses.
Terms: Aut, Win, Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Li, Z. (PI)
;
Lucianovic, M. (PI)
;
Miller, J. (PI)
;
Morton-Ferguson, C. (PI)
;
Park, J. (PI)
;
Swaminathan, M. (PI)
;
Taylor, C. (PI)
;
Vondrak, J. (PI)
;
Blair, H. (TA)
;
Bonciocat, C. (TA)
;
Chokkhanchitchai, T. (TA)
;
Cholsaipant, P. (TA)
;
Gangal, A. (TA)
;
Hofgard, J. (TA)
;
KAZANIN, S. (TA)
;
Kuelbs, D. (TA)
;
Li, J. (TA)
;
Li, Z. (TA)
;
Mauro, A. (TA)
;
Pandit, N. (TA)
;
Speciel, R. (TA)
;
Yang, H. (TA)
;
Zhang, S. (TA)
;
de la HOGUE MORAN-BOULLOCHE-BR, P. (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.
Math 52 is considered equivalent to
Math 62CM, and credit will not be granted for both courses.
Terms: Win, Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
Instructors:
Chen, Y. (PI)
;
Lucianovic, M. (PI)
;
Ram Sreedharan Nair, A. (PI)
...
more instructors for MATH 52 »
Instructors:
Chen, Y. (PI)
;
Lucianovic, M. (PI)
;
Ram Sreedharan Nair, A. (PI)
;
Pandit, N. (TA)
;
Woringer, P. (TA)
;
Yin, A. (TA)
;
Zhao, F. (TA)
MATH 53: Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications
Ordinary differential equations and initial value problems, linear systems of such equations with an emphasis on second-order constant-coefficient equations, stability analysis for non-linear systems (including phase portraits and the role of eigenvalues), and numerical methods. Partial differential equations and boundary-value problems, Fourier series and initial conditions, and Fourier transform for non-periodic phenomena. Throughout the development we harness insights from linear algebra, and software widgets are used to explore course topics on a computer (no coding background is needed). The free e-text provides motivation from applications across a wide array of fields (biology, chemistry, computer science, economics, engineering, and physics) described in a manner not requiring any area-specific expertise, and it has an appendix on Laplace transforms with many worked examples as a complement to the Fourier transform in the main text. Prerequisite:
Math 21 and
Math 51, or equivalents.
Math 53 is considered equivalent to
Math 63CM, and credit will not be granted for both courses.
Terms: Aut, Win, Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR, WAY-AQR
Instructors:
Anderson, J. (PI)
;
Asserian, L. (PI)
;
Borges Prado, L. (PI)
...
more instructors for MATH 53 »
Instructors:
Anderson, J. (PI)
;
Asserian, L. (PI)
;
Borges Prado, L. (PI)
;
Chen, H. (PI)
;
Lee, J. (PI)
;
Marsden, M. (PI)
;
Parker, G. (PI)
;
Li, Z. (TA)
;
Marsden, M. (TA)
;
Skvortsov, D. (TA)
MATH 56: Proofs and Modern Mathematics
How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but at a pace and level of abstraction not as intense as
Math 61CM/DM, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the course develops some of the main results in that material systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. This course may be paired with
Math 51; though that course is not a pre- or co-requisite.
Terms: Aut, Spr
| Units: 4
| UG Reqs: WAY-FR
Instructors:
Chodosh, O. (PI)
;
Wieczorek, W. (PI)
;
Fazliani, S. (TA)
...
more instructors for MATH 56 »
Instructors:
Chodosh, O. (PI)
;
Wieczorek, W. (PI)
;
Fazliani, S. (TA)
;
Park, J. (TA)
;
Yin, A. (TA)
MATH 63CM: Modern Mathematics: Continuous Methods
A proof-based course on ordinary differential equations. Topics include the inverse and implicit function theorems, implicitly-defined submanifolds of Euclidean space, linear systems of differential equations and necessary tools from linear algebra, stability and asymptotic properties of solutions to linear systems, existence and uniqueness theorems for nonlinear differential equations, behavior of solutions near an equilibrium point, and Sturm-Liouville theory. Prerequisite:
Math 61CM or
Math 61DM.
Math 63CM is considered equivalent to
Math 53, and credit will not be granted for both courses.
Terms: Spr
| Units: 5
| UG Reqs: WAY-FR, GER:DB-Math
Instructors:
Eliashberg, Y. (PI)
;
Kuhrman, J. (TA)
MATH 63DM: Modern Mathematics: Discrete Methods
Third part of a proof-based sequence in discrete mathematics, though independent of the second part (62DM). The first half of the quarter gives a brisk-paced coverage of probability and random processes with an intensive use of generating functions and a rich variety of applications. The second half treats entropy, Bayesian inference, Markov chains, game theory, probabilistic methods in solving non-probabilistic problems. We use continuous calculus, e.g. in handling the Gaussian, but anything needed will be reviewed in a self-contained manner. Prerequisite:
Math 61DM or 61CM
Terms: Spr
| Units: 5
| UG Reqs: WAY-FR
Instructors:
Tokieda, T. (PI)
;
Serio, C. (TA)
MATH 77Q: Probability and gambling
One of the earliest probabilistic discussions was in 1654 between two French mathematicians, Pascal and Fermat, on the following question: 'If a pair of six-sided dice is thrown 24 times, should you bet even money on the occurrence of at least one `double six'?' Shortly after the discussion, Huygens, a Dutch scientist, published De Ratiociniis in Ludo Aleae (The Value of all Chances in Games of Fortune) in 1657; this is considered to be the first treatise on probability. Due to the inherent appeal of games of chance, probability theory soon became popular, and the subject underwent rapid development in the 18th century with contributions from mathematical giants, such as Bernoulli, de Moivre, and Laplace. There are two fairly different lines of thought associated with applications of probability: the solution of betting/gambling and the analysis of statistical data related to quantitative subjects such as mortality tables and insurance rates. In this Introsem, we will discuss poker and
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One of the earliest probabilistic discussions was in 1654 between two French mathematicians, Pascal and Fermat, on the following question: 'If a pair of six-sided dice is thrown 24 times, should you bet even money on the occurrence of at least one `double six'?' Shortly after the discussion, Huygens, a Dutch scientist, published De Ratiociniis in Ludo Aleae (The Value of all Chances in Games of Fortune) in 1657; this is considered to be the first treatise on probability. Due to the inherent appeal of games of chance, probability theory soon became popular, and the subject underwent rapid development in the 18th century with contributions from mathematical giants, such as Bernoulli, de Moivre, and Laplace. There are two fairly different lines of thought associated with applications of probability: the solution of betting/gambling and the analysis of statistical data related to quantitative subjects such as mortality tables and insurance rates. In this Introsem, we will discuss poker and other games of chance, such as daily fantasy sports, from the perspective of risk analysis. This Introsem does not require any programming knowledge, but some experience with Excel, MATLAB, R, and/or Python will enhance your experience in our discussion of daily fantasy sports. Students should be familiar with all material from
Math 51. No prior knowledge of sports and games of chance is required. Students must apply through the IntroSem application process.
Terms: Win, Spr
| Units: 3
| UG Reqs: WAY-FR
Instructors:
Kim, G. (PI)
MATH 87Q: Mathematics of Knots, Braids, Links, and Tangles
Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics.
Terms: Spr
| Units: 3
| UG Reqs: WAY-FR
Instructors:
Wieczorek, W. (PI)
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