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# 1 - 10 of 18 results for: MATH 21: Calculus

## 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: Math 21 (preferred), or equivalent (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)
Terms: Aut, Spr | Units: 5 | UG Reqs: WAY-FR, GER:DB-Math

## 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: Math 21 (preferred), or equivalent (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)
Terms: Aut, Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

## CME 215:Machine Learning and the Physical Sciences (GEOPHYS 148, GEOPHYS 248)

This course provides a survey of the rapidly growing field of machine learning in the physical sciences. It covers various areas such as inverse problems, emulating physical processes, model discovery given data, and solution discovery given equations. It both introduces the background knowledge required to implement physics-informed deep learning and provides practical in-class coding exercises. Students have the opportunity to apply this emerging methodology to their own research interests across all fields of the physical sciences, including geophysics, climate, fluids, or other systems where the same technique applies. Students develop individual projects throughout the semester. Recommended Prerequisite: Calculus (e.g. Math 21), Differential Equations (e.g. MATH 53 or PHYSICS 111) or equivalents.
Terms: Spr | Units: 3

## 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: Math 21 (preferred), or equivalent (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)
Terms: Aut, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

## 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: Math 21 (preferred), or equivalent (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)
Terms: Aut, Win | Units: 5 | UG Reqs: WAY-FR, GER:DB-Math

## GEOPHYS 148:Machine Learning and the Physical Sciences (CME 215, GEOPHYS 248)

This course provides a survey of the rapidly growing field of machine learning in the physical sciences. It covers various areas such as inverse problems, emulating physical processes, model discovery given data, and solution discovery given equations. It both introduces the background knowledge required to implement physics-informed deep learning and provides practical in-class coding exercises. Students have the opportunity to apply this emerging methodology to their own research interests across all fields of the physical sciences, including geophysics, climate, fluids, or other systems where the same technique applies. Students develop individual projects throughout the semester. Recommended Prerequisite: Calculus (e.g. Math 21), Differential Equations (e.g. MATH 53 or PHYSICS 111) or equivalents.
Terms: Spr | Units: 3 | UG Reqs: WAY-AQR, WAY-SMA
Instructors: Lai, C. (PI)

## GEOPHYS 248:Machine Learning and the Physical Sciences (CME 215, GEOPHYS 148)

This course provides a survey of the rapidly growing field of machine learning in the physical sciences. It covers various areas such as inverse problems, emulating physical processes, model discovery given data, and solution discovery given equations. It both introduces the background knowledge required to implement physics-informed deep learning and provides practical in-class coding exercises. Students have the opportunity to apply this emerging methodology to their own research interests across all fields of the physical sciences, including geophysics, climate, fluids, or other systems where the same technique applies. Students develop individual projects throughout the semester. Recommended Prerequisite: Calculus (e.g. Math 21), Differential Equations (e.g. MATH 53 or PHYSICS 111) or equivalents.
Terms: Spr | Units: 3
Instructors: Lai, C. (PI)

## 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. 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 | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

## MATH 21ACE:Calculus, ACE

Terms: Aut | Units: 1

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