An Illustrative Guide To Multivariable And Vector Calculus

[Custodio Groves]

Thistextbook focuses on one of the most valuable skills in multivariable and vector calculus: visualization. With over one hundred carefully drawn color images, students who have long struggled picturing, for example, level sets or vector fields will find these abstract concepts rendered with clarity and ingenuity. This illustrative approach to the material covered in standard multivariable and vector calculus textbooks will serve as a much-needed and highly useful companion.

An Illustrative Guide to Multivariable and Vector Calculus will appeal to multivariable and vector calculus students and instructors around the world who seek an accessible, visual approach to this subject. Higher-level students, called upon to apply these concepts across science and engineering, will also find this a valuable and concise resource.

This book originated as a set of lectures prepared for courses given by me atthe University of Linkping in Sweden and at the University of SouthAustralia in Australia. At Linkping University the material (apart fromSection 3.E) was delivered in a second year, single semester course (14 weeks,2 two-hour lectures per week) to engineering students, with the first halffocused on the differential calculus of real-valued multivariable functions,while the second half was divided between integral calculus and vector calculus. At the University of South Australia the subject was delivered in two separate semester courses (12 weeks, 2 two-hour lectures per week), the first ofwhich was offered to second year engineering, science and mathematics students and featured differential and integral calculus, including an introduction to partial differential equations. The second course, taken mostly by third year mathematics and science students, dealt with vector calculus, although onlythe first five weeks of that course was covered by the material in this book.

The lectures generally were so well-received by students that it was thoughtthe material might appeal to a wider audience. Having taken the decision toconvert my notes into a book, I aimed for a document of manageable sizerather than generate yet another bulky tome on calculus. The result is a bookthat students can carry easily to and from class, can take out and leaf throughon the library lawn, or in a booth of a pub, or while lying on the banks of ariver waiting for the fish to bite.

Dirichlet process mixtures (DPM) are a popular class of models widely used in the estimation of probability densities and clustering. In this talk, a tutorial on Dirichlet process will be given. Time varying Dirichlet process mixtures which are proposed to estimate the densities evolving over time will also be introduced. The application of the model to the problem of multi-target tracking via sequential Monte Carlo methods will be demonstrated.

Aircraft trajectory prediction is of paramount importance for the safety and capacity of the airspace. It provides air traffic controllers with an instrument to assist in their search for efficient and collision free flight paths. However, meteorological predictions - mainly wind velocity - have inaccuracies that may result in large deviations from the expected path of an aircraft. Fortunately, the position of the aircraft, during its flight, incorporates the effect of the wind on its dynamics. Additionally, wind uncertainty (or else the error of weather forecasts) is not entirely random but is correlated in time and space. We can employ position measurements (from a ground based radar) for multiple aircraft flying in the same airspace, in order to predict their trajectory and estimate the wind in space-time. Finally, by using direct wind measurements, from the aircraft flying in a given sector, and transmit those to the ground, we can further improve our results.

Attitude estimation is a classical problem with a rich and fascinating history still holding a forefront position as the subject of intensive research. Attitude observers based only on the rotation kinematics are of special interest for applications using inertial sensors and attitude aiding devices, with application to autonomous vehicles. In this seminar, the problem of attitude estimation is motivated by presenting some autonomous robotic platforms under development at DSOR, and a nonlinear attitude observer defined on SO(3), based on inertial and vector measurements, is proposed. The observer has an almost globally asymptotically stable equilibrium point at the origin and convergences exponentially fast for any initial condition in an explicit region. The feedback terms, derived constructively using the Lyapunov's stability theory, are an explicit function of the available vector observations and biased angular velocity measurements. Topological limitations for achieving global stabilization on the SO(3) manifold provide important guidelines and are discussed and illustrated in the present observer. Exponential convergence and convergence bounds are obtained, resorting to the recent results for parameterized linear time-varying (LTV) systems. The properties of the observer are illustrated in simulation for inertial sensor characteristics and initial alignment errors commonly found in practical setups.

Abstract, part 1: This will be a short presentation of the content of an article with the same title. We will motivate the need for generalizing the determinant minimization criterion, which is encountered in reduced rank regression problems. We will also present the neat solution and make some geometric interpretations with respect to signal properties.

Abstract, part 2: The second part will be an introduction on tensors and low (multilinear) rank tensor approximation. Tensors in this context are multidimensional data arrays. The objective function of the approximation problem is defined on a product of Grassmann manifolds.

We will outline the key ideas of optimization methods (Newton and quasi-Newton algorithms) that solve this problem.

The term Neuro Mechanical Network (NMN) refers to a system in which the basic structural unit is a multi-functional element. Whereas in most traditional systems the basic structural units provide perhaps one or two functions each, e.g. stiffness, the multi-functional element can also provide mechanical actuation, sensing, signal processing and other functions. The biological inspiration is obvious, one may for instance think of muscle cells in the heart.

Apart from biology, the theoretical framework takes inspiration and knowledge from fields such as smart structures, neural networks, and structural optimization. The present formulation consists of a coupled system of equations describing a mechanical truss with a neural network superimposed onto it. Methods from structural optimization are employed in order to create structures with adaptive stiffness. Possible applications include fixtures which can adapt to various load conditions, and in general situations where stiffness and low weight are conflicting demands.

In this talk we will address the problem of synchronization and coordination of multi-agent systems from the perspective of passivity based control. Such systems arise in a number of emerging application areas, such as sensor networks,autonomous flying and swimming vehicles, networked communication and control systems, and computer networks. We will discuss how synchronization naturally arises in both natural and engineered multi-agent systems and how feedback in such systems leads to interesting emergent behaviors. Examples will be given in bilateral teleoperation and in attitude synchronization of multiple robots.

The functionality of modern automotive vehicles is becoming increasingly dependent on control systems. Active safety is an area in which control systems play a pivotal role. Currently, rule-based control algorithms are widespread throughout the automotive industry. In order to improve performance and reduce development time, model-based methods may be employed. The primary contribution of this thesis is the development of a vehicle dynamics controller for rollover mitigation. A central part of this work has been the investigation of control allocation methods, which are used to transform high-level controller commands to actuator inputs in the presence of numerous constraints. Quadratic programming is used to solve a static optimization problem in each sample. An investigation of the numerical methods used to solve such problems was carried out, leading to the development of a modified active set algorithm. Vehicle dynamics control systems typically require input from a number of supporting systems, including observers and estimation algorithms. A key parameter for virtually all VDC systems is the friction coefficient. Model-based friction estimation based on the physically-derived brush model is investigated.

Information provides a quantitative metric for describing the value of individual systems components in autonomous systems tasks such as tracking, mapping and navigation, search and exploration; tasks in which the objective is information gain in some form. An information model is an abstraction of system capabilities in an anonymous form which allows a priori reasoning on the system itself. By construction, information measures have properties of composability and additivity and thus provides a natural means of modelling and describing large scale systems of systems.

This talk will begin by describing how information measures arise naturally in autonomous tracking, mapping and navigation, search and exploration tasks. It is then demonstrated that the performance of individual sensors and platforms can be modelled using these information measures and that system-level performance metrics can be computed. These ideas are illustrated in a series of tasks involving mixed air and ground autonomous systems. These include flight-tests of cooperative UAVs engaged in tracking and navigation tasks, mixed UAV, ground vehicles and human operatives, engaged in mapping and picture compilation operations, and operations involving multi-platform search in constrained environments. In each, it is shown how information provides both a performance metric and design objective underpinning large-scale systems of systems operation. Finally current work in using these ideas to design and manage large-scale autonomous systems in applications such as cargo handling and mining will be described.

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