Results for MATH 20: Calculus

This course explores the global transition to a sustainable global energy system. We will formulate and program simple models for future energy system pathways. We will explore the drivers of global energy demand and carbon emissions, as well as the technologies that can help us meet this demand sustainably. We will consider constraints on the large-scale deployment of technology and difficulties of a transition at large scales and over long time periods. Assignments will focus on building models of key aspects of the energy transition, including global, regional and sectoral energy demand and emissions as well as economics of change. Prerequisites: students should be comfortable with calculus and linear algebra (e.g. Math 20, Math 51) and be familiar with computer programming (e.g. CS106A, CS106B). We will use the Python programming language to build our models.

This course explores the global transition to a sustainable global energy system. We will formulate and program simple models for future energy system pathways. We will explore the drivers of global energy demand and carbon emissions, as well as the technologies that can help us meet this demand sustainably. We will consider constraints on the large-scale deployment of technology and difficulties of a transition at large scales and over long time periods. Assignments will focus on building models of key aspects of the energy transition, including global, regional and sectoral energy demand and emissions as well as economics of change. Prerequisites: students should be comfortable with calculus and linear algebra (e.g. Math 20, Math 51) and be familiar with computer programming (e.g. CS106A, CS106B). We will use the Python programming language to build our models.

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.

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

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.

Additional problem solving session for Math 20 guided by a course assistant. Concurrent enrollment in Math 20 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-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 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.

How are motions of objects in the physical world determined by the 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. The distinction made between fundamental laws of nature and empirical rules that are useful approximations for more complex physics. Problems are 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 the language of mathematics, particularly vector representations and operations, and calculus. Physical understanding is fostered by peer interaction and demonstrations in lecture, and discussion sections based on interactive group problem-solving. 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. Prerequisites: Physics placement diagnostic AND Math 20 or higherCorequisites: Completion of OR co-enrollment of Math 21 or higher. 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.

Physics 41E (Physics 41 Extended) is a 5-unit version of Physics 41 (4 units) for students with little or no high school physics. Course topics and mathematical complexity are similar, but not identical to Physics 41. There is an additional class meeting every week, and attendance at all class sessions is mandatory. The extra classroom time and corresponding extra study time outside of class allows students to engage with concepts and become fluent in mathematical tools that include vector representations and operations, and relevant calculus. There is a strong emphasis on developing problem-solving skills, particularly as applied to real world examples, to leave students prepared for subsequent engineering, physics, or related courses they may take. The course will explore important physical principles in mechanics including: using Newton's Laws and torque to analyze static structures and forces; understanding the equations of kinematics; and utilizing energy in its many forms and applications. Prerequisites: Physics placement diagnostic AND Math 20 or higher. Corequisites: Completion of OR co-enrollment of Math 21 or higher. 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. Priority will be given to students who have had little physics background.

What is temperature? How do the elementary processes of mechanics, which are intrinsically reversible, result in phenomena that are clearly irreversible when applied to a very large number of particles, the ultimate example being life? In thermodynamics, students discover that the approach of classical mechanics is not sufficient to deal with the extremely large number of particles present in a macroscopic amount of gas. The paradigm of thermodynamics leads to a deeper understanding of real-world phenomena such as energy conversion and the performance limits of thermal engines. In optics, students see how a geometrical approach allows the design of optical systems based on reflection and refraction, while the wave nature of light leads to interference phenomena. The two approaches come together in understanding the diffraction limit of microscopes and telescopes. Discussions based on the language of mathematics, particularly calculus. Physical understanding fostered by peer interaction and demonstrations in lecture, and discussion sections based on interactive group problem solving. In order to register for this class students must EITHER have already taken an introductory Physics class (20, 40, or 60 sequence) or have taken the Physics Placement Diagnostic at https://physics.stanford.edu/academics/undergraduate-students/placement-diagnostic. Prerequisite: PHYSICS 41 or equivalent. MATH 21 or MATH 51 or CME 100 or equivalent.

(First in a three-part series: PHYSICS 61, PHYSICS 71, PHYSICS 81.) This course covers Einstein's special theory of relativity and Newtonian mechanics at a level appropriate for students with a strong high school mathematics and physics background, who are contemplating a major in Physics or Engineering Physics or are interested in a rigorous treatment of physics. Postulates of special relativity, simultaneity, time dilation, length contraction, the Lorentz transformation, the space-time invariant, causality, relativistic momentum and energy, and invariant mass. Central forces, friction, contact forces, linear restoring forces. Momentum, work, energy, collisions. Angular momentum, torque, center of mass, moment of inertia, precession. Conserved quantities. Uses the language of vectors and multivariable calculus. Requirements to enroll in the course: Completion of Physics Placement Diagnostic and/or completion of at least one course in PHYSICS 20 or 40 series. Completion of or co-enrollment in MATH 51 or MATH 61CM or MATH 61DM. Prerequisites: mechanics at the level of PHYSICS 41 or score of 5 on AP Physics C Mechanics or equivalent; calculus at the level of MATH 21 or score of 5 on AP Calculus BC or equivalent.

This course will build on and enrich students' fundamental prerequisite skills in foundational mathematics to prepare students for success in Calculus and further mathematics courses at Stanford University. This course is intended for students that will enroll in the Math 19-20-21 sequence, but will broadly be relevant and engaging for success in university-level mathematics courses at Stanford, as well as in other courses at Stanford in other disciplines that rely on these courses as prerequisites. Students will enhance their proficiency with precalculus mathematics, with an emphasis on higher level conceptual understanding and problem-solving. The primary of this course is to help students develop and hone the mathematical skills necessary to successfully transition to university level mathematics at Stanford University. The course will focus on fundamental concepts from algebra, functions and graphs, trigonometry, exponentials and logarithms, and limits.