## CME 100: Vector Calculus for Engineers (ENGR 154)

Computation and visualization using MATLAB. Differential vector calculus: vector-valued functions, analytic geometry in space, functions of several variables, partial derivatives, gradient, linearization, unconstrained maxima and minima, Lagrange multipliers and applications to trajectory simulation, least squares, and numerical optimization. Introduction to linear algebra: matrix operations, systems of algebraic equations with applications to coordinate transformations and equilibrium problems. Integral vector calculus: multiple integrals in Cartesian, cylindrical, and spherical coordinates, line integrals, scalar potential, surface integrals, Green's, divergence, and Stokes' theorems. Numerous examples and applications drawn from classical mechanics, fluid dynamics and electromagnetism. Prerequisites: knowledge of single-variable calculus equivalent to the content of
Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with
Math 21, 5 on Calc AB with
Math 21). Placement diagnostic (recommendation non-binding) at:
https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.

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

Instructors:
Khayms, V. (PI)
;
Le, H. (PI)
;
Burugupalli, P. (TA)
...
more instructors for CME 100 »

Instructors:
Khayms, V. (PI)
;
Le, H. (PI)
;
Burugupalli, P. (TA)
;
Giovanardi, D. (TA)
;
Nervo, G. (TA)
;
Rowley, J. (TA)

## CME 102: Ordinary Differential Equations for Engineers (ENGR 155A)

Analytical and numerical methods for solving ordinary differential equations arising in engineering applications are presented. For analytical methods students learn to solve linear and non-linear first order ODEs; linear second order ODEs; and Laplace transforms. Numerical methods using MATLAB programming tool kit are also introduced to solve various types of ODEs including: first and second order ODEs, higher order ODEs, systems of ODEs, initial and boundary value problems, finite differences, and multi-step methods. This also includes accuracy and linear stability analyses of various numerical algorithms which are essential tools for the modern engineer. This class is foundational for professional careers in engineering and as a preparation for more advanced classes at the undergraduate and graduate levels. Prerequisites: knowledge of single-variable calculus equivalent to the content of
Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with
Math 21, 5 on Calc AB with
Math 21). Placement diagnostic (recommendation non-binding) at:
https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.

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

## ENGR 154: Vector Calculus for Engineers (CME 100)

Computation and visualization using MATLAB. Differential vector calculus: vector-valued functions, analytic geometry in space, functions of several variables, partial derivatives, gradient, linearization, unconstrained maxima and minima, Lagrange multipliers and applications to trajectory simulation, least squares, and numerical optimization. Introduction to linear algebra: matrix operations, systems of algebraic equations with applications to coordinate transformations and equilibrium problems. Integral vector calculus: multiple integrals in Cartesian, cylindrical, and spherical coordinates, line integrals, scalar potential, surface integrals, Green's, divergence, and Stokes' theorems. Numerous examples and applications drawn from classical mechanics, fluid dynamics and electromagnetism. Prerequisites: knowledge of single-variable calculus equivalent to the content of
Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with
Math 21, 5 on Calc AB with
Math 21). Placement diagnostic (recommendation non-binding) at:
https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.

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

Instructors:
Khayms, V. (PI)
;
Le, H. (PI)
;
Burugupalli, P. (TA)
...
more instructors for ENGR 154 »

Instructors:
Khayms, V. (PI)
;
Le, H. (PI)
;
Burugupalli, P. (TA)
;
Giovanardi, D. (TA)
;
Nervo, G. (TA)
;
Rowley, J. (TA)

## ENGR 155A: Ordinary Differential Equations for Engineers (CME 102)

Analytical and numerical methods for solving ordinary differential equations arising in engineering applications are presented. For analytical methods students learn to solve linear and non-linear first order ODEs; linear second order ODEs; and Laplace transforms. Numerical methods using MATLAB programming tool kit are also introduced to solve various types of ODEs including: first and second order ODEs, higher order ODEs, systems of ODEs, initial and boundary value problems, finite differences, and multi-step methods. This also includes accuracy and linear stability analyses of various numerical algorithms which are essential tools for the modern engineer. This class is foundational for professional careers in engineering and as a preparation for more advanced classes at the undergraduate and graduate levels. Prerequisites: knowledge of single-variable calculus equivalent to the content of
Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with
Math 21, 5 on Calc AB with
Math 21). Placement diagnostic (recommendation non-binding) at:
https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.

Terms: Aut, Win, Spr
| Units: 5
| 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
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, Sum
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Kim, G. (PI)
;
Lee, S. (PI)
;
Ma, C. (PI)
;
Wieczorek, W. (PI)
;
Cant, D. (TA)
;
Chaturvedi, S. (TA)
;
Dore, D. (TA)
;
Fushida-Hardy, S. (TA)
;
Xu, M. (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/BZJqJTawa5PUqe9E7.

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

Instructors:
Kim, G. (PI)
;
Ma, C. (PI)
;
Wieczorek, W. (PI)
;
Helfer, J. (TA)
;
Libkind, S. (TA)
;
Nuti, P. (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:
Kim, G. (PI)
;
Lucianovic, M. (PI)
;
Spink, H. (PI)
;
Taylor, C. (PI)
;
Venkatesh, S. (PI)
;
Arana Herrera, F. (TA)
;
Chen, D. (TA)
;
Chen, S. (TA)
;
Devadas, S. (TA)
;
Falcone, P. (TA)
;
He, J. (TA)
;
Kraushar, N. (TA)
;
Love, J. (TA)
;
McConnell, S. (TA)
;
Nguyen, D. (TA)
;
Ortiz, J. (TA)
;
Perlman, M. (TA)
;
Raksit, A. (TA)
;
Stavrianidi, A. (TA)
;
Tam, K. (TA)
;
Truong Vu, N. (TA)
;
Zavyalov, B. (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, Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

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

Instructors:
Grzegrzolka, P. (PI)
;
Li, Z. (PI)
;
Wieczorek, W. (PI)
;
Angelo, R. (TA)
;
Qian, L. (TA)
;
Zhou, Z. (TA)

## PHYSICS 41: Mechanics

How are motions of objects in the physical world determined by laws of physics? Students learn to describe the motion of objects (kinematics) and then understand why motions have the form they do (dynamics). Emphasis on how the important physical principles in mechanics, such as conservation of momentum and energy for translational and rotational motion, follow from just three laws of nature: Newton's laws of motion. Distinction made between fundamental laws of nature and empirical rules that are useful approximations for more complex physics. Problems drawn from examples of mechanics in everyday life. Skills developed in verifying that derived results satisfy criteria for correctness, such as dimensional consistency and expected behavior in limiting cases. Discussions based on language of mathematics, particularly vector representations and operations, and calculus. Physical understanding fostered by peer interaction and demonstrations in lecture, and discussion sections based on inte
more »

How are motions of objects in the physical world determined by laws of physics? Students learn to describe the motion of objects (kinematics) and then understand why motions have the form they do (dynamics). Emphasis on how the important physical principles in mechanics, such as conservation of momentum and energy for translational and rotational motion, follow from just three laws of nature: Newton's laws of motion. Distinction made between fundamental laws of nature and empirical rules that are useful approximations for more complex physics. Problems drawn from examples of mechanics in everyday life. Skills developed in verifying that derived results satisfy criteria for correctness, such as dimensional consistency and expected behavior in limiting cases. Discussions based on language of mathematics, particularly vector representations and operations, and calculus. Physical understanding fostered by peer interaction and demonstrations in lecture, and discussion sections based on interactive group problem solving. Spring 2020-21: Class will be taught remote synchronously in active learning format with much of the learning in smaller breakout rooms so class will not be recorded. Please enroll in a section that you can attend regularly. In order to register for this class all FROSH must complete the Physics Placement Diagnostic at
https://physics.stanford.edu/academics/undergraduate-students/placement-diagnostic. Students who complete the Physics Placement Diagnostic by 3 PM (Pacific) on Friday will have their hold lifted over the weekend. Minimum prerequisites: High school physics and
MATH 20 (or high school calculus if sufficiently rigorous). Minimum co-requisite:
MATH 21 or equivalent. Since high school math classes vary widely, it is recommended that you take at least one math class at Stanford before or concurrently with taking
Physics 41. In addition, it is recommended that you take
Math 51 or
CME 100 before taking the next course in the
Physics 40 series,
Physics 43.

Terms: Aut, Spr
| Units: 4
| UG Reqs: GER: DB-NatSci, WAY-SMA

Instructors:
Graham, P. (PI)
;
Nanavati, C. (PI)
;
Tompkins, L. (PI)
...
more instructors for PHYSICS 41 »

Instructors:
Graham, P. (PI)
;
Nanavati, C. (PI)
;
Tompkins, L. (PI)
;
Allen, J. (TA)
;
Arevalo, C. (TA)
;
Haidu Gerea, P. (TA)
;
Hing, D. (TA)
;
Jonay, C. (TA)
;
Kometter, C. (TA)
;
Li, X. (TA)
;
Liu, A. (TA)
;
Nowak, J. (TA)
;
Prabhu, A. (TA)
;
San Miguel, J. (TA)

## PHYSICS 43: Electricity and Magnetism

What is electricity? What is magnetism? How are they related? How do these phenomena manifest themselves in the physical world? The theory of electricity and magnetism, as codified by Maxwell's equations, underlies much of the observable universe. Students develop both conceptual and quantitative knowledge of this theory. Topics include: electrostatics; magnetostatics; simple AC and DC circuits involving capacitors, inductors, and resistors; integral form of Maxwell's equations; electromagnetic waves. Principles illustrated in the context of modern technologies. Broader scientific questions addressed include: How do physical theories evolve? What is the interplay between basic physical theories and associated technologies? Discussions based on the language of mathematics, particularly differential and integral calculus, and vectors. Physical understanding fostered by peer interaction and demonstrations in lecture, and discussion sections based on interactive group problem solving. Prerequisite:
PHYSICS 41 or equivalent.
MATH 21 or
MATH 51 or
CME 100 or equivalent. Recommended corequisite:
MATH 52 or
CME 102.

Terms: Win, Sum
| Units: 4
| UG Reqs: GER: DB-NatSci, WAY-SMA

Instructors:
Cabrera, B. (PI)
;
Abe, M. (TA)
;
Corbin, J. (TA)
...
more instructors for PHYSICS 43 »

Instructors:
Cabrera, B. (PI)
;
Abe, M. (TA)
;
Corbin, J. (TA)
;
Lee, K. (TA)
;
Liu, A. (TA)
;
Moreau, G. (TA)
;
Mousatov, A. (TA)
;
Ren, X. (TA)
;
Rodenbach, L. (TA)
;
Thompson, J. (TA)

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