Calculus in 3D: Geometry, Vectors, and Multivariate Calculus

<div class="abstract" data-abstract-type="normal"> In the 1999 film <span class='italic'>The Matrix</span>, there is the following interchange between Neo and Trinity. <dispQuote> Neo: <span class='italic'>What is the Matrix?</span> Trinity: <span class='italic'>The answer is out there, Neo, and it's looking for you, and it will find you if you want it to</span>. </dispQuote> As an applied mathematician specializing in linear algebra, I see many applications of linear systems in the world. From computer graphics to data analytics, linear algebra is useful and powerful. So, to me, a matrix, connected to an application, is indeed out there–waiting, in a certain sense, to be formed. When you find it, you can create compelling computer graphics or gain insight on real world phenomenon. This book is meant to engage high schoolers through professors of mathematics in applications of linear algebra. The book teaches enough linear algebra to dive into applications. If someone wanted to know more, a Google search will turn up a lot of information. If more information is desired, a course or textbook in linear algebra would be the next step. This book can either serve as a fun way to step into linear algebra or as a supplementary resource for a class. My hope is that this book will ignite ideas and fuel innovation in the reader. From trying other datasets for data analytics to variations on the ideas in computer graphics, there is plenty of room for your personal discovery. So, this book is a stepping stone, intended to springboard you into exploring your own ideas. This book is connected to a Massive Open Online Course (also known as a MOOC) that will launch in February of 2015 through Davidson College and edX. The book and online course are not dependent on each other. Participants of the MOOC do not need this book, and readers of this book do not need to participate in the MOOC. The experiences are distinct, but for those interested, I believe the book and online course will complement each other. </div>

Chapter 4.1 - Maxwell’s equations and numerical electromagnetic modeling in the context of the theory of differential forms

This article presents a study in which applications were integrated in the Multivariable Calculus course at the Technion in the framework of supplementary tutorials. The purpose of the study was to test the opportunity of extending the conventional curriculum by optional applied problem-solving activities and get initial evidence on the possible impact of the tutorials on students’ beliefs about the value of learning mathematics with applications. The study lasted three semesters and consisted of three experiments in which supplementary tutorials were offered in different forms: a weekly evening meeting for interested students, a weekly extra hour added to the conventional calculus class, and a workshop which introduces mathematics concepts from the application perspective. The study reveals: (1) significant positive effect of the tutorials on the students’ beliefs (in all three experiments); (2) statistically significant advantage of the group involved in the tutorials in relation to the group that learned in the conventional way (second experiment); (3) students’ positive evaluation of the workshops for better understanding the course lectures (third experiment). Grounding on the study experience, we propose for further discussion a stage model of the applied problem solving cycle.

This study aims to determine students' perceptions of the use of multimedia learning software GeoGebra in Multivariable Calculus course. Students' perceptions of the use of media or multimedia learning are closely related to their achievement. Media or multimedia learning in accordance with the desire of students will make them motivated. This research is qualitative data analysis was done descriptively. The data obtained in this study comes from researchers as the main instrument, the data questionnaire and interview data is unstructured. The results are students' perceptions of the use of multimedia learning software GeoGebra in Multivariable Calculus courses are in the positive category.

Richard A. Brualdi is Professor of Mathematics at the Univer? sity of Wisconsin in Madison, where he has taught since 1965. He is the author of the text Introductory Combinatorics (Eisevier Science, second edition 1991) and author with H. J. Ryser of the monograph Combinatorial Matrix Theory (Cambridge University Press, 1991). He serves on the editorial boards of several journals, and is editor-in-chief of Linear Algebra and Its Applica? tions. His research interests lie primarily in the interface of combinatorics/graph theory and linear algebra.

In order to develop courses related to artificial intelligence and mathematics, this study provides plans for the composition and operation of courses related to artificial intelligence and mathematics in college general mathematics by analyzing the mathematics curriculum, artificial intelligence and mathematics-related research, and selecting learning elements and contents. The content needed to deal with the principles of artificial intelligence in mathematics is mainly related to calculus, probability and statistics, and linear algebra. Such mathematical learning content is one of the functions of artificial intelligence. Since it is necessary for analysis and expression, neural networks, clustering, unsupervised learning, and optimization, these contents must be included in artificial intelligence and mathematics-related courses in college general mathematics. In addition, in courses related to artificial intelligence and mathematics, it is difficult for students to learn various topics such as linear algebra and multi-variable calculus, which are essential as basic mathematics for understanding machine learning, all in one semester. Thus, in this study, I propose that we divide the course into two subjects, &lt;Artificial Intelligence Mathematics 1&gt; and &lt;Artificial Intelligence Mathematics 2&gt;, so that college students can select and take the courses according to their abilities and needs. Lastly, while theory-oriented classes are essential for understanding the principles of artificial intelligence in artificial intelligence and mathematics-related courses, applied learning is also crucial. This includes learning the basics of Python, a widely used programming language, for data analysis, data organization, and preprocessing. Additionally, applying these skills in real-life projects involving data analysis, artificial intelligence, and deep learning is necessary for comprehensive education.

This note contains a linear algebra attack on elementary complex analysis which leads to a three‐fold unification between linear algebra, multivariable calculus and complex analysis.

This research examined the effects of a problem-based learning (PBL) chemistry laboratory course for second-year college students (N = 20) on students’ scientific attitudes with an emphasis on their creative thinking abilities. The findings were contrasted with a traditional laboratory course (N = 26) to elucidate any differences in the influence of the courses. Only female students participated in the study which was conducted in a private university for women in Korea. A 20-item Scientific Attitudes Questionnaire administered to both groups as a pretest and a posttest revealed that there were significant changes in criticism, cooperativeness, and creativity at the end of instruction only among students in the PBL course. The posttest scores of the Torrance Tests of Creative Thinking to gauge students’ creative thinking ability were significantly higher for the students in the PBL course on all three dimensions. The research suggests that PBL laboratory courses in chemistry have great potential to positively change students’ scientific attitudes towards learning chemistry and enhance their creative thinking abilities.

problem was a routine calculation of curvature, requiring the memorization of a particular formula and the mechanics of differentiation. Being conscientious in my duties, I devised a grading scheme whereby a mistake in algebra would cost a few points and a mistakenly computed derivative a few more points, with the largest point deduction reserved for an incorrect formula for curvature. Not only had the material been reduced to rote calculation, but that method of instruction, at least for the calculus sequence, was being promulgated to the next generation of instructors. Possibly the difficulty lies in the subject itself-how can a critical understanding of curvature be tested on a one-hour exam, especially when curvature is merely one of several topics to be tested? Should the students be asked to explain the definition of curvature as the magnitude of the rate of change of the unit tangent with respect to arclength? In subsequent courses I have spent more time motivating that particular definition than actually computing curvature, but find the definition rather far removed from an intuitive idea of curvature. Should the students be asked to sketch the osculating circle to a given curve at a given point, and then notice that as the point of contact changes, the radius of the circle is inversely proportional to the measure of curvature? This seems more intuitive, and remains quite a viable introduction to curvature, although the concept of an osculating circle requires some explanation. Why would anyone have bothered to construct such a circle, our students and teaching assistants may wonder. Perhaps the most complete explanation of curvature lies in its history and offers the best understanding of the subject. Modem calculus textbooks, however, do not present the subject via its history, but opt for opaque definitions, slick formulas, or routine calculations. Although many mathematicians are familiar with the historical development of curvature, it remains unclear how often this is used to present curvature in a calculus course. This may, in part, be due to a lack of appropriate curricular material for calculus students. In this article we suggest a remedy for the problem. 2. TWO PROJECTS. What follows are descriptions and the actual text of two written projects, based on key developments in the history of curvature, that I used recently in a multivariable calculus course. For each project, the students were given about two weeks to write a detailed solution to a sequence of guided questions that begins with an event of historical mathematical significance and culminates with a key observation about curvature. Each project took the place of an in-class examination. first, The Radius of Curvature According to Huygens, opens with the Dutch scholar's discovery of the isochronous pendulum [5], [7]. In the project students learn firsthand from an English translation of Huygens's Horologium oscillatorium (The Pendulum Clock) how the radius of curvature, used in the construction of the pendulum, can be described in terms of ratios of segments. goal of the project is then to reconcile Huygens's very geometric description, typical of mathematics in the seventeenth century, in terms of the modem equation of curvature. An engaging problem that shows how the radius

We are concerned about a view in undergraduate computer science education, especially in the early courses, that it's okay to be math-phobic and still prepare oneself to become a computer scientist. Our view is the contrary: that any serious study of computer science requires students to achieve mathematical maturity (especially in discrete mathematics) early in their undergraduate studies, thus becoming well-prepared to integrate mathematical ideas, notations, and methodologies throughout their study of computer science. A major curricular implication of this theme is that the prerequisite expectations and conceptual level of the first discrete mathematics course should be the same as it is for the first calculus course --- secondary school pre-calculus and trigonometry. Ultimately, calculus, linear algebra, and statistics are also essential for computer science majors, but none should occur earlier than discrete mathematics. This paper explains our concerns and outlines our response as a series of examples and recommendations for future action.

On a possible approximation of discontinuous dynamical systems

This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x + k ẋ + g(x,t) = ε(t) where k is a constant; g is continuous, continuously differentiable with respect to x, and is periodic of period P in the variable t; ε(t) is continuous and periodic of period P, and when ∂g/∂x satisfies some additional boundedness conditions. This means that there exist initial values x(0) = α* and ẋ (0) = β* so that the solution to the corresponding initial value problem is periodic of period P and is unique (up to a translation of the time variable) with this property. The proof of this result is constructive, so that starting with any initial conditions x(0) = α and ẋ(0) = β, a path in the phase plane can be produced, starting at (α, β) and terminating at (α*, β*). Both the theoretical proof and a constructive proof are discussed and a Mathematica implementation developed which yields an algorithm in the form of a Mathematica notebook (which is posted on the webpage http://pax.st.usm.edu/downloads). The algorithm is robust and can be used on differential equations whose terms do not satisfy Li and Shen's hypotheses. The ideas used reinforce concepts from beginning courses in ordinary differential equations, linear algebra, and numerical analysis.

This book is designed for senior undergraduate and graduate students pursuing courses in mathematics, physics, engineering and biology. The text begins with a study of ordinary differential equations. The concepts of first- and second-order equations are covered initially. It moves further to linear systems, series solutions, regular Sturm–Liouville theory, boundary value problems and qualitative theory. Thereafter, partial differential equations are explored. Topics such as first-order partial differential equations, classification of partial differential equations and Laplace and Poisson equations are also discussed in detail. The book concludes with heat equation, one-dimensional wave equation and wave equation in higher dimensions. It highlights the importance of analysis, linear algebra and geometry in the study of differential equations. It provides sufficient theoretical material at the beginning of each chapter, which will enable students to better understand the concepts and begin solving problems straightaway.

(1987). Differential Equations with Linear Algebra. By Zbigniew H. Nitecki and Martin M. Guterman/ Introduction to Differential Equations. By Richard K. Miller/ Fundamentals of Differential Equations. By R. Kent Nagle and Edward B. Saff/ Introduction to Differential Equations with Applications. By Fred Brauer and John A. Nohel and Ordinary Differential Equations. By Morris Tenenbaum and Harry Pollard. The American Mathematical Monthly: Vol. 94, No. 10, pp. 1022-1024.

Chapter 9 - Differential Equations: Introduction