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

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors:
Kimport, S. (PI)
;
Ohrt, C. (PI)
;
Wang, Y. (PI)
;
Chaturvedi, S. (TA)
;
Chen, S. (TA)
;
Datta, I. (TA)
;
Mackey, W. (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. Initialvalue 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:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors:
Kemeny, M. (PI)
;
Kimport, S. (PI)
;
Lucianovic, M. (PI)
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Instructors:
Kemeny, M. (PI)
;
Kimport, S. (PI)
;
Lucianovic, M. (PI)
;
Schaeffer, G. (PI)
;
Solis, P. (PI)
;
Chen, D. (TA)
;
Hui, Y. (TA)
;
McConnell, S. (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

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors:
Dore, D. (PI)
;
Falcone, P. (PI)
;
Howe, S. (PI)
;
Kimport, S. (PI)
;
Lim, B. (PI)
;
Schaeffer, G. (PI)
;
Taylor, C. (PI)
;
Wieczorek, W. (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

Grading: Letter or Credit/No Credit
MATH 51: Linear Algebra, Multivariable Calculus, and Modern Applications
This course provides unified coverage of linear algebra and multivariable differential calculus. It discusses applications connecting the material to many quantitative fields. Linear algebra in large dimensions underlies the scientific, datadriven, and computational tasks of the 21st century. The linear algebra portion of the course includes orthogonality, linear independence, matrix algebra, and eigenvalues as well as ubiquitious applications: least squares, linear regression, Markov chains (relevant to population dynamics, molecular chemistry, and PageRank), singular value decomposition (essential in image compression, topic modeling, and dataintensive work in the natural sciences), and more. The multivariable calculus material includes unconstrained optimization via gradients and Hessians (used for energy minimization in physics and chemistry), constrained optimization (via Lagrange multipliers, crucial in economics), gradient descent and the multivariable Chain Rule (which underl
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This course provides unified coverage of linear algebra and multivariable differential calculus. It discusses applications connecting the material to many quantitative fields. Linear algebra in large dimensions underlies the scientific, datadriven, and computational tasks of the 21st century. The linear algebra portion of the course includes orthogonality, linear independence, matrix algebra, and eigenvalues as well as ubiquitious applications: least squares, linear regression, Markov chains (relevant to population dynamics, molecular chemistry, and PageRank), singular value decomposition (essential in image compression, topic modeling, and dataintensive work in the natural sciences), and more. The multivariable calculus material includes unconstrained optimization via gradients and Hessians (used for energy minimization in physics and chemistry), 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 (a crucial part of how GPS works). The course emphasizes computations alongside an intuitive understanding of key ideas, making students wellprepared for further study of mathematics and its applications to other fields. The widespread use of computers makes it more important, not less, for users of math to understand concepts: in all scientific fields, novel users of quantitative tools in the future will be those who understand ideas and how they fit with applications and examples. 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: 21, 42, or the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors:
Church, T. (PI)
;
Conrad, B. (PI)
;
Love, J. (PI)
;
Lucianovic, M. (PI)
;
Mazzeo, R. (PI)
;
Ohrt, C. (PI)
;
Sloman, L. (PI)
;
Taylor, C. (PI)
;
Wei, F. (PI)
;
Wieczorek, W. (PI)
;
Cant, D. (TA)
;
Fayyazuddin Ljungberg, B. (TA)
;
Love, J. (TA)
;
Wei, F. (TA)
;
Zou, J. (TA)
MATH 51A: Linear Algebra, Multivariable Calculus, and Modern Applications, ACE
Students attend
MATH 51 lectures with different recitation sessions: three hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see
https://web.stanford.edu/dept/soe/osa/ace.fb
Terms: Aut, Win, Spr

Units: 6

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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 secondorder equations to oscillations, matrix exponentials, Laplace transforms, stability of nonlinear systems and phase plane analysis, numerical methods. Prerequisite: 51 or equivalents.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 61CM: Modern Mathematics: Continuous Methods
This is the first part of a theoretical (i.e., proofbased) 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, inverse and implicit function theorems, and many examples. The linear algebra content is covered jointly with
Math 61DM. Students should know 1variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BClevel Advanced Placement calculus exam, or consent of the instructor.
Terms: Aut

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors:
Ryzhik, L. (PI)
;
De Groote, C. (TA)
MATH 61DM: Modern Mathematics: Discrete Methods
This is the first part of a theoretical (i.e., proofbased) 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 BClevel 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: WAYFR

Grading: Letter or Credit/No Credit
Instructors:
Fox, J. (PI)
;
Devadas, S. (TA)
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