Данный курс поможет обучающимся преодолеть трудности, вызванные недостаточным знанием математики и использовать свои знания для решения физических задач. Он поможет создать базу для успешного освоения дисциплин, в которых необходимы знания основ дифференциального и интегрального исчисления. Без этих навыков невозможна успешная деятельность инженеров и специалистов любого профиля. Физические явления и законы так или иначе описываются математическими формулами. Важнейшие открытия и изобретения в мире не обходятся без математики. Для решения широкого круга задач необходимо использовать математический аппарат, связанный с дифференциальным и интегральным исчислением функций одной вещественной переменной. Освоив этот курс, вы сможете применять свои знания не только в физике для расчета, например, траекторий движения космических ракет и спутников, для прогнозирования работы ядерных реакторов, но и в геологии, биологии, экономике и др. для прогнозирования различных динамических процессов. Курс включает в себя тесты и набор заданий, формирующий основные навыки, которыми несколько облегчат изучение общей физики, так как позволят сконцентрироваться на физической сути явлений. Освоившие курс учащиеся смогут решать следующие физические задачи: • находить кинематические характеристики движения тел; • исследовать характер движения тел при заданном законе движения; • определять качественное поведение рассматриваемых физических величин в предельных случаях; • определять действующие на тело силы при заданном законе движения и т. п. Также прошедшие обучение смогут: • строить уравнение касательной к графику функции в заданной точке; • находить среднее значение функции на заданном отрезке; • находить площади фигур, ограниченных заданными кривыми. Для успешного освоения курса слушателю желательно знать основы математики и физики в объеме школьной программы.

In the first part of this course you will explore methods to compute an approximate solution to an inconsistent system of equations that have no solutions. Our overall approach is to center our algorithms on the concept of distance. To this end, you will first tackle the ideas of distance and orthogonality in a vector space. You will then apply orthogonality to identify the point within a subspace that is nearest to a point outside of it. This has a central role in the understanding of solutions to inconsistent systems. By taking the subspace to be the column space of a matrix, you will develop a method for producing approximate (“least-squares”) solutions for inconsistent systems. You will then explore another application of orthogonal projections: creating a matrix factorization widely used in practical applications of linear algebra. The remaining sections examine some of the many least-squares problems that arise in applications, including the least squares procedure with more general polynomials and functions. This course then turns to symmetric matrices. arise more often in applications, in one way or another, than any other major class of matrices. You will construct the diagonalization of a symmetric matrix, which gives a basis for the remainder of the course.

Vous voulez apprendre l'algèbre linéaire, un précieux outil complémentaire à vos connaissances acquises durant vos études en économie, ingénierie, physique, ou statistique? Ou simplement pour la beauté de la matière? Alors ce cours est fait pour vous! Outre remplir le rôle d'outil dans les différentes branches mentionnées ci-dessus (permettant la résolution de problèmes concrets), l'algèbre linéaire, qui capture l'essence des mathématiques -à savoir, l'algèbre et la géométrie- vous introduira au monde plus abstrait des mathématiques. Proposé comme complément de cours aux ingénieurs de première année à l'Ecole Polytechnique Fédérale de Lausanne, ce MOOC (composé de trois parties) n'en est pas moins un cours à part entière et peut être considéré comme une base solide d'algèbre linéaire pour tout étudiant intéressé par l'apprentissage de cette matière. Bien que les vidéos constituent le coeur du cours, des exercices de type QCM (Questions à choix multiples) ainsi que des séries au format PDF seront disponibles chaque semaine, ainsi que des corrigés appropriés. Plus précisément, les séries d'exercices seront accompagnées d'un corrigé au format PDF et certains problèmes bénéficieront d'une correction détaillée en vidéo, dans laquelle l'un des enseignants présentera la solution, étape par étape. Finalement, chaque vidéo de cours sera suivie d'un quiz, dont le but est de tester le degré d’assimilation des connaissances acquises. Le cours est organisé en dix chapitres dans lesquels une approche très détaillée des concepts théoriques est proposée, ainsi que de multiples exemples illustratifs : 1) Systèmes d'équations linéaires. 2) Algèbre matricielle. 3) Espaces vectoriels. 4) Bases et dimensions. 5) Applications linéaires. 6) Matrices et applications linéaires. 7) Déterminants. 8) Vecteurs propres, valeurs propres, diagonalisation. 9) Produits scalaires et espaces euclidiens. 10) Matrices orthogonales et matrices symétriques. Cette première partie du cours sera dévouée à l'étude des quatre premiers chapitres cités plus haut. Aucune connaissance particulière n’est requise pour comprendre les concepts abordés dans ce MOOC, mais il est conseillé de travailler régulièrement et de manière assidue, de façon à ne pas prendre de retard lors de l'apprentissage de la matière.

In this course, we go beyond the calculus textbook, working with practitioners in social, life and physical sciences to understand how calculus and mathematical models play a role in their work. Through a series of case studies, you’ll learn: How standardized test makers use functions to analyze the difficulty of test questions; How economists model interaction of price and demand using rates of change, in a historical case of subway ridership; How an x-ray is different from a CT-scan, and what this has to do with integrals; How biologists use differential equation models to predict when populations will experience dramatic changes, such as extinction or outbreaks; How the Lotka-Volterra predator-prey model was created to answer a biological puzzle; How statisticians use functions to model data, like income distributions, and how integrals measure chance; How Einstein’s Energy Equation, E=mc2 is an approximation to a more complicated equation. With real practitioners as your guide, you’ll explore these situations in a hands-on way: looking at data and graphs, writing equations, doing calculus computations, and making educated guesses and predictions. This course provides a unique supplement to a course in single-variable calculus. Key topics include application of derivatives, integrals and differential equations, mathematical models and parameters. This course is for anyone who has completed or is currently taking a single-variable calculus course (differential and integral), at the high school (AP or IB) or college/university level. You will need to be familiar with the basics of derivatives, integrals, and differential equations, as well as functions involving polynomials, exponentials, and logarithms. This is a course to learn applications of calculus to other fields, and NOT a course to learn the basics of calculus. Whether you’re a student who has just finished an introductory Calculus course or a teacher looking for more authentic examples for your classroom, there is something for you to learn here, and we hope you’ll join us!

Matrix Algebra underlies many of the current tools for experimental design and the analysis of high-dimensional data. In this introductory online course in data analysis, we will use matrix algebra to represent the linear models that commonly used to model differences between experimental units. We perform statistical inference on these differences. Throughout the course we will use the R programming language to perform matrix operations. Given the diversity in educational background of our students we have divided the series into seven parts. You can take the entire series or individual courses that interest you. If you are a statistician you should consider skipping the first two or three courses, similarly, if you are biologists you should consider skipping some of the introductory biology lectures. Note that the statistics and programming aspects of the class ramp up in difficulty relatively quickly across the first three courses. You will need to know some basic stats for this course. By the third course will be teaching advanced statistical concepts such as hierarchical models and by the fourth advanced software engineering skills, such as parallel computing and reproducible research concepts. These courses make up two Professional Certificates and are self-paced: Data Analysis for Life Sciences: PH525.1x: Statistics and R for the Life Sciences PH525.2x: Introduction to Linear Models and Matrix Algebra PH525.3x: Statistical Inference and Modeling for High-throughput Experiments PH525.4x: High-Dimensional Data Analysis Genomics Data Analysis: PH525.5x: Introduction to Bioconductor PH525.6x: Case Studies in Functional Genomics PH525.7x: Advanced Bioconductor This class was supported in part by NIH grant R25GM114818.

Introduction to the mathematical concept of networks, and to two important optimization problems on networks: the transshipment problem and the shortest path problem. Short introduction to the modeling power of discrete optimization, with reference to classical problems. Introduction to the branch and bound algorithm, and the concept of cuts.

Differential equations are the mathematical language we use to describe the world around us. Most phenomena can be modeled not by single differential equations, but by systems of interacting differential equations. These systems may consist of many equations. In this course, we will learn how to use linear algebra to solve systems of more than 2 differential equations. We will also learn to use MATLAB to assist us. We will use systems of equations and matrices to explore: The original page ranking systems used by Google, Balancing chemical reaction equations, Tuned mass dampers and other coupled oscillators, Threeor more species competing for resources in an ecosystem, The trajectory of a rider on a zipline. The five modules in this seriesare being offered as an XSeries on edX. Please visit the Differential EquationsXSeries Program Page to learn more and to enroll in the modules. *Zipline photo by teanitiki on Flickr (CC BY-SA 2.0)

More than 2000 years ago, long before rockets were launched into orbit or explorers sailed around the globe, a Greek mathematician measured the size of the Earth using nothing more than a few facts about lines, angles, and circles. This course will start at the very beginnings of geometry, answering questions like "How big is an angle?" and "What are parallel lines?" and proceed up through advanced theorems and proofs about 2D and 3D shapes. Along the way, you'll learn a few different ways to find the area of a triangle, you'll discover a shortcut for counting the number of stones in the Great Pyramid of Giza, and you'll even come up with your own estimate for the size of the Earth. In this course, you'll be able to choose your own path within each lesson, and you can jump between lessons to quickly review earlier material. GeometryX covers a standard curriculum in high school geometry, and CCSS (common core) alignment is indicated where applicable. Learn more about our High School and AP* Exam Preparation Courses This course was funded in part by the Wertheimer Fund.

Preparing for the AP Calculus AB exam requires a deep understanding of many different topics in calculus as well as an understanding of the AP exam and the types of questions it asks. This course is Part 1 of our XSeries: AP Calculus AB and it is designed to prepare you for the AP exam. In Part 2, you will use and apply the meaning and interpretations of derivatives from Part 1 to the integral, antiderivatives and differential equations. You will learn some applications of integrals including finding volumes of solids and solids of revolution, volumes with known cross sections and applications to Velocity-Time graphs. We will close with an introduction to differential equations and see how they are used. As you work through this course, you will find lecture videos taught by expert AP calculus teachers, practice multiple choice questions and free response questions that are similar to what you will encounter on the AP exam and tutorial videos that show you step-by-step how to solve problems. By the end of the course, you should be ready to take on the AP exam!

Vector Calculus for Engineers covers both basic theory and applications. In the first week we learn about scalar and vector fields, in the second week about differentiating fields, in the third week about multidimensional integration and curvilinear coordinate systems. The fourth week covers line and surface integrals, and the fifth week covers the fundamental theorems of vector calculus, including the gradient theorem, the divergence theorem and Stokes’ theorem. These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics. Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3. Two semesters of single variable calculus (differentiation and integration) are a prerequisite. The course is organized into 53 short lecture videos, with a few problems to solve following each video. And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of five weeks to the course, and at the end of each week there is an assessed quiz. Download the lecture notes: http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf Watch the promotional video: https://youtu.be/qUseabHb6Vk