# Advanced Mathematical Methods With Maple

Environmental engineering often involves biochemical processes. Biochemical processes are important in chemical analysis, molecular interactions, toxicology, cellular microbiology, and industrial design. With the development of modern biochemical techniques, advanced environmental and biochemical devices, equipment, programs, and software have been created for effective and efficient process control. These biochemical tools are based on appropriate materials, electrical automation, and system control, where mathematical analysis is a major part of data processing. Therefore, using advanced mathematical tools and methods to improve mathematical analysis is an important way to design environmental and biochemical processes and related devices, equipment, programs, and software.

## Advanced mathematical methods with Maple

In order to simulate textural colors of leaves, the Phong lighting model with a diffuse component derived from leaf pigments is adopted to directly compute the reflections on the surfaces of leaves [9]. Other methods use the technique of texture mapping to produce the leaves' appearances, and the textures can be changed to reflect the appearance changes of leaves [10]. In our method, we apply multiple textures to represent appearance changing of leaves in different seasons.

The leaves gradually become withered and curled up during the transitions of different seasons. The deformation of geometric shapes of leaves is very important to simulate the seasonal changes. The 3D deformation algorithms are mainly classified into two categories, which are free-form-based deformation methods [19] and physically based deformation methods [20]. Free-form-based deformation methods are widely used in the field of computer animation and geometric modeling [21]. These kinds of methods embed the objects into a local coordinate space and transform the local space to make the objects deformed. There are two common physically based deformation methods: skeleton-based method and mass-spring-based method. The deformation method based on skeleton is relatively simple [7] and produces more realistic deformation results of leaves. However, it requires much human interaction. Mass-spring model is more frequently used in fabric deformation [22]. Tang and Yang [23] adopt the mass-spring model to generate the deformation of leaves, in which the mesh of the leaf is not optimized, and the deformation effects are relatively unnatural and difficult to control. Double mass-spring model proposed by Chi et al. [8] is capable of simulating the changes of leaves more realistically. However, it is complex and difficult to be implemented.

The parameters of time, temperature, and humidness are set by users. Taking the maple leaves in Figure 8, for example, we use three specific combinations of textures and shapes for each season. For instance, three main states are used to represent leaves in summer, which are texture 2 in Figure 8 combined with the first deformation in Figure 4, texture 3 combined with the second deformation, and texture 4 combined with the third deformation.

The Department of Mathematical Sciences offers undergraduate and graduate degree programs in mathematics for students with various interests and career goals. Students may pursue the standard program or a program focused on actuarial mathematics, applied mathematics, mathematics education, or mathematical statistics. Students may complement other interests by taking a double major in mathematics and a related field, such as chemistry, economics, physics, computer science, or engineering.

The department manages the Math Tutoring Center, which offers free tutoring for first- and second-year math courses. Tutoring is given by advanced mathematics students and is available on a drop-in basis with daytime and evening hours throughout the semester.

We describe our extensive collective experience in the conduct of courses based on the use of advanced mathematical software to enhance the capabilities of students of chemistry to solve problems with a mathematical component. These courses have been based on an interactive electronic textbook Mathematics for Chemistry with Symbolic Computation that has been concurrently developed and expanded. The admirable performance of the students who have benefited from our courses leaves no doubt that mathematical software is an invaluable tool for the teaching, learning and practice of mathematics in a chemical context.

Chemistry is the natural and physical science that treats matter and its transformations. As a physical science, chemistry has a strong quantitative mathematical component, from the stoichiometric proportions applied in chemical reactions to the physical models in its theoretical bases (thermodynamics, kinetics, molecular interactions and spectra), passing through molecular structure (bond distances and angles as geometric attributes), to quantum theories and statistical thermodynamics. In most approaches to the teaching of chemistry, students learn how to handle chemical calculations performed with paper and pencil, some treatment of data using general software such as a spreadsheet, and perhaps also the use of black-box software for structural calculations. Those approaches are effective for introducing chemical concepts and for preparing the students for the future practice of their chemical profession, but chemical curricula in universities around the world typically lack the development of powerful mathematical capabilities using mathematical software on digital computers, in particular the capability of symbolic computation. We specify here symbolic computation because it allows the solution of problems arising in chemistry stated in the most natural way, beginning from basic equations applied to a particular situation. Harnessing the power of mathematical software in the form of computer algebra enables the students to become capable of solving complicated problems unfeasible by manual means or even with merely arithmetical computation. In this article we discuss the benefits of an interactive electronic textbook, of title Mathematics for Chemistry with Symbolic Computation, based on a particular mathematical program, Maple, to teach the mathematical skills that are applicable to chemistry.

Of two parts of the textbook Mathematics for Chemistry, the first is designed to encompass all mathematics that an instructor of chemistry might wish that his students would have learned in courses typically, but not exclusively, offered in departments of mathematics as a service to science students; the second part comprises chapters each of which is devoted to a particular area of mathematics, with its chemical applications, that might be taught within advanced courses in chemistry, including group theory, graph theory and dynamic chemical equilibrium. After a summary of various useful Maple commands, Part I hence has two chapters before the introduction of calculus, covering arithmetic in various forms and scientific notation, elementary functions, algebra, plotting, geometry, trigonometry, series and introductory complex analysis. Differential and integral infinitesimal calculus of one variable precedes multivariate calculus, linear algebra including matrices, vectors, eigenvalues, vector calculus and tensors, before differential and integral equations, and statistical aspects including probability, distribution and univariate statistics for the treatment of laboratory data, linear and non-linear regression and optimization.

Likewise for integral calculus, our explanation of the principle comprises a definition in words accompanied by an animated display of the area under a curve as the number of rectangular subdivisions of the axis increases; the total area of the rectangles is simultaneously exhibited at each stage. When we authors were students before access to mathematical software, the advice given to us in the course on integral calculus was to learn standard methods, such as change of variables, integration by parts and trigonometric substitutions, and to memorize a list of common integrations; if such manual methods failed to yield a convincing answer in a particular application, we should find a similar expression in a table of integrals, but then we had to undertake whatever transformation was required into the particular variables in the problem at hand. With powerful mathematical software such as Maple that contains more knowledge about integrals than any single printed table ever, one need not be concerned about typographical or other errors that have been prevalent in static printed tables; in any case one should directly differentiate the result of an unfamiliar indefinite integration to prove its correctness. The emphasis in our course is to encourage the student to think about the solution of a problem, and then to let the software, according to the subset of total commands and instructions included in the course, undertake the tedious algebraic manipulations or prepare an illuminating graph. Some 70 years ago in advanced countries, pupils in schools were trained how to extract manually a square root of a number, with no explanation of the principle (actually a binomial expansion, but that was never stated); the flood of pocket calculators more than 40 years ago terminated that drudgery, and now even most professors of mathematics are unable to extract a square root of a number by hand. The use of computer software for symbolic mathematics is a natural progression that is consistent with the use of computers for manifold other routine or boring purposes. To replace the learning of mechanical sequences of manual operations, an enhanced understanding of the principles is a valuable dividend of symbolic computation, as incorporated in the design of our interactive electronic textbook and as proven in tests over nearly two decades.

Among the benefits acquired by students with skills in mathematical computation is notably the loss of fear about facing elaborate calculations. In many situations a complicated physical or chemical model is based on a statement of systems of equations. Each such equation is typically not difficult to set based on a knowledge of the underlying physics and chemistry; the difficult part of solving many problems is finding the solution for the mathematical system (algebraic or differential), because that task is tedious and commonly difficult to perform with merely paper and pencil. In other cases a problem can be correctly stated equation by equation, but the knowledge to solve the problem lies beyond the expected mathematical preparation of a student of chemistry. That barrier disappears on learning how to use software for symbolic and numeric computation: the student does the thinking and makes the statement, and the computer undertakes the slave labour that yields the answer. In this respect we recall the superb performance of a student in the physical-chemistry laboratory who undertook a special experimental project on the chemical kinetics of oscillating reactions; this student was able to model his experimental data on solving the system of differential equations that described the variation of concentration of each involved substance with time; the demonstration of his results inspired awe in the other students watching the presentation. Another instance was solving the Schroedinger equation for the hydrogen atom, which can be readily effected in several coordinate systems simply on translating the laplacian operator from one system to another. 041b061a72