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

Instructors:
Khayms, V. (PI)
;
Le, H. (PI)
;
Ali, F. (TA)
;
Brink, T. (TA)
;
Chaudhari, N. (TA)
;
De Sota, R. (TA)
;
Hoyt, C. (TA)
;
LABROGERE, A. (TA)
;
Vasudevan, V. (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
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

## ENGR 15: Dynamics

The application of Newton's Laws to solve 2-D and 3-D static and dynamic problems, particle and rigid body dynamics, freebody diagrams, and equations of motion, with application to mechanical, biomechanical, and aerospace systems. Computer numerical solution and dynamic response. Prerequisites: Calculus (differentiation and integration) such as
Math 19, 20; and
ENGR 14 (statics and strength) or a mechanics course in physics such as
PHYSICS 41.

Terms: Aut, Win
| Units: 3
| UG Reqs: WAY-SMA, GER:DB-EngrAppSci

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

Instructors:
Khayms, V. (PI)
;
Le, H. (PI)
;
Ali, F. (TA)
;
Brink, T. (TA)
;
Chaudhari, N. (TA)
;
De Sota, R. (TA)
;
Hoyt, C. (TA)
;
LABROGERE, A. (TA)
;
Vasudevan, V. (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: WAY-FR, GER:DB-Math

## ENGR 199A: Additional Calculus for Engineers

Additional problem solving practice for the calculus courses. Sections are designed to allow students to acquire a deeper understanding of calculus and its applications, work collaboratively, and develop a mastery of the material. Limited enrollment, permission of instructor required. Concurrent enrollment in
MATH 19, 20, 52, or 53 required

Terms: Win, Spr
| Units: 1
| Repeatable
for credit

Instructors:
Andrade, L. (PI)

## 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:DB-Math, WAY-FR

Instructors:
Asserian, L. (PI)
;
Jack, T. (PI)
;
Sun, W. (PI)
;
Chu, Y. (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://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus.

Terms: Aut, Win
| Units: 1

Instructors:
Har, M. (TA)

## 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)
;
Dore, D. (TA)
;
Qin, Q. (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
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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. Autumn 2021-22: Class will be taught remote synchronously in active learning format with much of the learning in smaller breakout rooms. The class will not be recorded. Please enroll in a section that you can attend regularly. In order to register for this class students who have never taken an introductory Physics course at Stanford 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 19 (or equivalent high school calculus if sufficiently rigorous). Minimum co-requisite:
MATH 20 or equivalent (if possible, taking
Math 20 as a prerequisite and
Math 21 as a co-requisite is recommended). Since high school math classes vary widely, it is recommended that you take at least one math class at Stanford before or concurrently with
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, Win
| Units: 4
| UG Reqs: GER: DB-NatSci, WAY-SMA

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

Instructors:
Graham, P. (PI)
;
Nanni, E. (PI)
;
Tompkins, L. (PI)
;
Ames, D. (TA)
;
Dinc, F. (TA)
;
Jiang, J. (TA)
;
Kuenstner, S. (TA)
;
Simon, O. (TA)
;
Trbalic, B. (TA)
;
Valenzuela Lombera, I. (TA)
;
Zamora, A. (TA)

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