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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.IEEE TRANSACTIONS ON EDUCATION1Geometrical Approach to Vector Analysisin Electromagnetics EducationBranislav M. Notaroš, Senior Member, IEEEAbstract—A geometrical approach to teaching and learningvector calculus and analysis as applied to electromagnetic fieldsis proposed for junior-level undergraduate electromagneticseducation. For undergraduate students, electromagnetics is typically the most challenging subject in the Electrical Engineeringcurriculum, and the most challenging component of the subject isthe application of vector analysis to electromagnetic field theoryand problem solving. According to the geometrical approach, thestudents are taught to “read” the figure and to “translate” this toequations at all times throughout the computation or derivation,instead of their “crunching” the formulas and numbers withouteven visualizing the structure. In performing vector manipulations, integrals, and derivatives, the students are taught toalways deal with real geometrical entities and quantities (arrows,lengths, angles, points, lines, surfaces, volumes, etc.). They learnto “translate” the geometry and the electromagnetic physicsattached to it into mathematical models (equations and symbolicor numerical values) using “first mathematical principles” insteadof just “black-box” formulas as a computer would do. As opposedto the traditional formal algebraic approach to vector analysisin electromagnetics, which is very general but also very abstractand dry, the geometrical approach is problem-dependent but alsomuch more intuitive and visual, and as such can do a great dealto increase students’ understanding and appreciation of vectoranalysis and its application to electromagnetic theory and problemsolving. This is confirmed by preliminary class testing and assessment of student learning, success, and satisfaction in the coursesElectromagnetic Fields I and II at Colorado State University.Index Terms—Electromagnetics education, geometrical andvisual approach, undergraduate fields courses, vector analysis.I. INTRODUCTIONELECTROMAGNETIC theory or the theory of electromagnetic fields and waves is a fundamental underpinningof technical education; at the same time, it is one of the mostdifficult subjects for students to master. To undergraduatestudents, electromagnetics courses are typically the mostchallenging and demanding courses in the Electrical Engineering (EE) curriculum. This material is extremely abstractand mathematically rigorous and intensive, and students findit rather difficult to grasp. This is not unique to any particularschool, department, country, or geographical region. It is wellknown and established internationally that the electromagnetictheory or fields course (or course sequence), as it is usuallyManuscript received February 03, 2012; revised August 11, 2012; acceptedOctober 19, 2012.The author is with the Department of Electrical and Computer Engineering,Colorado State University, Fort Collins, CO 80523-1373 USA (e-mail:notaros@colostate.edu).Digital Object Identifier 10.1109/TE.2012.2227745referred to, is always, averaged over all students in a class, themost challenging EE subject in the undergraduate curriculum.Electromagnetics courses are taught primarily in the junioryear in EE, Electrical and Computer Engineering (ECE),Physics, and similar departments and schools, typically covering some or all of the following major topics: electrostaticfields, steady electric currents, magnetostatic fields, slowlytime-varying (low-frequency) electromagnetic fields, rapidlytime-varying (high-frequency) electromagnetic fields, uniformplane electromagnetic waves, transmission lines, waveguidesand cavity resonators, and antennas and radiation. The importance of electromagnetic theory, as a fundamental scienceand engineering discipline and a foundation of electrical andcomputer engineering as a whole, to ECE education can hardlybe overstated. In addition, electromagnetics has immediateimpact on a great variety of cutting-edge technologies andapplications in practically all ECE areas, and a comprehensiveknowledge and firm grasp of electromagnetic fundamentals isessential for students in a number of other undergraduate andgraduate courses, as well as for ECE graduates as they join theworkforce, now and in the future.Perhaps the best illustration of a great struggle of educatorsand scholars worldwide to find an “ideal” or at least satisfactoryway of teaching and learning electromagnetics, and a proofthat such a way has not yet been found and established, isthe fact that there are is extremely large number of quite different textbooks for undergraduate electromagnetics availableand “active” (about 30 books published in North Americaonly)—probably more than for any other discipline in scienceand engineering. Some initiatives to advance undergraduateelectromagnetics education and surveys and experiences inteaching/learning electromagnetic fields and waves are presented in [1]–[6].Generally, there is a great diversity in the teaching ofundergraduate electromagnetics courses, in content, scope,and pedagogical philosophy. Some electromagnetics coursesimplement the direct or chronological order of topics, whichcan briefly be characterized as: first teaching static and thendynamic topics, or first teaching fields (static, quasistatic, andrapidly time-varying) and then waves (uniform plane waves,transmission lines, waveguides, and antennas)—e.g., [7]and [8]. Some courses follow the inverse (nonchronological)order of topics in teaching/learning electromagnetics: start withgeneral Maxwell’s equations and then teach everything elseas applications of these equations, namely, teach all types offields as special cases of the general high-frequency electromagnetic field—e.g., [9] and [10]. Some instructors employthe transmission-lines-first approach to teaching the course:0018-9359/ 31.00 2012 IEEE

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.2start with the analysis of transmission lines, based on only purecircuit-theory concepts, with per-unit-length characteristics(distributed parameters) of the lines assumed to be known, andthen continue with the chronological order of topics—e.g., [11]and [12].Regardless of the ordering of the material, there is also agreat nonuniformity in balancing the coverage and emphasis incourses in terms of general questions like: more fields versusmore waves, more static versus more dynamic topics, morebreadth versus more depth, more fundamentals versus more applications, etc. In addition, engineering educators are exploringvarious innovative ways of electromagnetics class delivery andare far from reaching consensus on pedagogical dilemmas like:traditional lecturing versus interactive in-class explorations anddiscussions (active teaching and learning), inclusion of teamwork and peer instruction (collaborative teaching/learning), allanalysis versus considerable design component, more conceptsversus more computation, etc. There are also new challengesrelated to a great increase in demand for distance learning, online courses, and other forms of nontraditional course delivery.Not less importantly, about half the classes worldwide arenow offered as a single mandatory fields and waves course, withthe other half still being offered as a mandatory sequence of twoelectromagnetics courses. Another fact that must be taken intoaccount when considering the state of the art in electromagneticsteaching is that in a very large number of programs, worldwide,fields and waves courses are taught by instructors who are notelectromagnetics-trained and are teaching electromagnetics outside their personal expertise area. Finally, there is an evidentdecline (on average) of the mathematical and problem-solvingpreparedness of students taking these courses.Two general approaches have been pursued to overcome theproblems and challenges outlined above in the methodology andpractice of electromagnetics teaching, with respect to the diversity and nonuniformity of curricular contents, teaching methods,pedagogical goals, instructor expertise, areas of emphasis, desired outcomes of the course or sequence of courses, the timeavailable, and the decline in the average student’s preparedness and interest and motivation for fields and waves courses.According to the first approach, the class material is very significantly reduced (many educators say almost trivialized) bysimply skipping or skimming the challenging topics, concepts,examples, problems, derivations, and applications in order toattract students’ attention and to remedy (bypass) any deficiencies in their mathematical and problem-solving background, aswell as to save class time. Hence, the material is mostly covered only as an itemized list of final facts and selected formulas, and the examples and problems are of the pure formulaic(plug-and-chug) type. The second approach provides a rigorousand complete (as far as possible, given the available time in thecourse) treatment of the material and presents it to students ina consistent and pedagogically sound manner with enough detail, derivations, and explanations to be fully understandable andappreciable. The examples and problems emphasize physicalconceptual reasoning, mathematical synthesis of solutions, andrealistic engineering context, providing opportunities for students to develop their conceptual understanding of the materialand true electromagnetic problem-solving skills. By its generalIEEE TRANSACTIONS ON EDUCATIONphilosophy and goals, this paper belongs to the second approachto electromagnetics teaching and learning.However, whatever the coverage, emphasis, and ordering ofthe material in a course or courses, the curricular context, levelof breadth and depth, or the teaching method and pedagogicalapproach, the most problematic and most important componentof electromagnetics teaching and learning is vector calculus andanalysis as applied to electromagnetic field computation andproblem solving. This is integral to all class topics and is met inpractically all lectures, recitations (problem-solving sessions),homework assignments, and tests. It is a consensus of electromagnetics educators and scholars that any improvement in thepedagogy of vector analysis in electromagnetics would be welcomed by students and instructors alike.Vector analysis in electromagnetic fields courses, if presentedin a traditional manner, is extremely poorly received and notappreciated by students, primarily because of its abstract, dry,and overcomplicated pure mathematical formalisms, includingmultiple integrations, multivariable vector calculations, andcurvilinear coordinate systems. While doing their best to solveproblems, understand derivations, and perform studies, studentswill very often admit that with so many mathematical conceptsand degrees of freedom appearing in equations, they, in fact,have little or no idea what is actually going on in their analysisor computation. Because of the lack of understanding, theysoon lose confidence, then they lose motivation, and the wholelearning process is sooner or later reduced to their franticallypaging through the textbook in a quest for a suitable finalformula or set of formulas that look applicable and that will beapplied in a nearly random fashion.This paper proposes a geometrical approach to teaching andlearning vector analysis, including vector algebra, integral multivariable calculus, and differential vector calculus, as applied toelectromagnetics. It is based on geometrical visualizations andemphasizes the geometry of the problem, rather than formal algebraic algorithms and brute force algebraic computation. Thestudents are taught to “read” the figure and to “translate” itto equations, rather than to “crunch” the formulas and numbers without even visualizing the structure with which they aredealing. In the existing electromagnetics textbooks [7]–[21] andteaching/learning practices, vector algebra and calculus are usedin topics on electromagnetic fields and waves in a traditional,formal, purely “algebraic” way. Briefly, the formal algebraic approach is very general, but also very abstract and dry and analogous to the way a programmer actually “instructs” a computer todo vector analysis in computer programs. The geometrical approach, on the other hand, is problem-dependent, but also muchmore intuitive and visual, and as such can do a great deal to increase students’ understanding and appreciation of vector analysis and its application to electromagnetics.There are hundreds of examples to illustrate this approach, afew of which are presented in this paper, but overall, the students are taught to link the equations to the picture of the realstructure that is under consideration, at all times throughout thecomputation or derivation. Hence, in vector analysis of electromagnetic problems, including all sorts of vector manipulations,integrals, and derivatives, the students are, unlike the computerprogram, taught to always visualize the structure and deal with

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.NOTAROŠ: GEOMETRICAL APPROACH TO VECTOR ANALYSIS IN ELECTROMAGNETICS EDUCATIONreal geometrical entities and quantities (arrows, lengths, angles,points, lines, surfaces, volumes, etc.) and then to just “translate” the geometry and the electromagnetic physics attached toit into mathematical models (equations and symbolic or numerical values) using the “mathematical first principles” (and notgeneral black-box formulas). Ultimately, in a fundamental electromagnetic course, the main objective is always to help the students really understand a theoretical statement or derivation, ora solution to a practical problem, and to develop ways of “electromagnetic thinking,” rather than to offer the computationallymost efficient and most generic toolboxes for different classesof electromagnetic situations and problems.The proposed geometrical and visual approach to teachingand learning vector calculus and analysis as applied toelectromagnetic field computation and problem solvinghas been implemented in the new undergraduate textbookElectromagnetics [22]. However, the book does not explicitlyrefer to the approach as such, never identifying it or mentioningit on the many occasions when it is applied in the book’stheoretical derivations and problem solving, nor does it explainand discuss it, either from the pedagogical point of view or todraw a comparison to the traditional formal algebraic approachto vector analysis in electromagnetics. In addition, the geometrical approach to vector analysis can be used in conjunctionwith any other textbook on electromagnetic fields and waves. Itcan also be applied to teaching and learning electromagneticsat any level.This paper is organized as follows. Section II presents andexplains the geometrical and visual approach to teaching andlearning vector analysis in electromagnetics. In Section III, theproposed approach is illustrated and discussed in several characteristic examples of fields class topics including vector algebra,application of Maxwell’s integral equations, and spatial differential vector operators. Section IV discusses class testing andassessment of student learning, success, and satisfaction in classdelivery using the proposed geometrical and visual approach inthe Electromagnetic Fields I and II courses in the ECE Department at Colorado State University, Fort Collins. Section V summarizes the main conclusions of the paper and puts them in abroader perspective of current and future electromagnetics education research.II. GEOMETRICAL AND VISUAL APPROACH TO TEACHING ANDLEARNING VECTOR ANALYSIS IN ELECTROMAGNETICSIn manipulations with vectors, the proposed geometricalapproach to teaching and learning electromagnetic fields andwaves always emphasizes that vectors are real arrows in spaceand not only triplets of numbers. The magnitude of a vectoris primarily appreciated and used as the geometrical lengthof the arrow in the context of other geometrical quantities inthe figure; a component of the vector is therefore just anotherarrow in the figure, whose length is found as the real lengthof the vector arrow multiplied by the cosine or sine of a realangle identified in the figure (and not found by using abstractgeneral formulas). For example, if the components of a givenvector in an adopted (e.g., cylindrical) coordinate system ina structure are needed, they are obtained geometrically as the3corresponding projections of the vector in the real picture ofthe structure, and not formally using analytical transformations(which are not only unintuitive, but often lead to incorrectanswers).In addition, line, surface, and volume integrals in multivariable integral vector calculus are always viewed and solved, according to the proposed geometrical approach, as integrals alonga real line, over a real surface, and throughout a real volume,respectively, and not formally as single, double, and triple integrals with respect to one, two, or three coordinates in an appropriate coordinate system using coordinate transformations andgeneral algebraic-type formulas.As an example, according to the geometrical approach,the electric flux through a Gaussian spherical surface in aproblem with spherical symmetry solved applying Gauss’law (Maxwell’s third equation) in integral form is understoodsimply as the vector magnitude times the area of the spheresurface , so as, with being the sphere radius; in contrast, the formal approach would perform a doubleintegration with respect to the angles and in a sphericalcoordinate system, with an elemental surface areaobtainedusing coordinate transformations. Even in cases whereisneeded in the integral, it is much better to obtain it geometrically, as the area of a “patch” (a surface element of the sphere)with sides equal in length to the corresponding elemental arcsin the and directions (lengths of arcs are computed simplyas the corresponding radius times the angle and are multipliedtogether for the patch area) than to use formal coordinate transformations. Similarly, in a volume integral of a functionover the volume of a sphere, the elemental volume is taken tobe that of a thin spherical shell of radius and thickness ,so the volume element. This can be visualizedand obtained (with no differential calculus) as the volumeof a thin flat slab (a “flattened” spherical shell) of the samethicknessand the same surface area, soas(surface area times thickness of the slab). Theintegration is performed only with respect to , rather thanformally performing threefold integration. Similar geometricalvisualizations, rather than algebraic algorithms and formulas,are used with spatial derivatives, including gradient, divergence, and curl, so that these important operators really cometo life and their physical meaning becomes very obvious andnatural.As a part of the geometrical approach to electromagneticseducation, a general strategy for solving volume and surfaceintegrals arising in electromagnetic analysis is also employed.This strategy basically solves an integral of a function overa volume or a surface by adopting as large a volume element , or surface elements, as possible, the only restriction being the condition that is constant inor[22]. Inother words, the larger the volume or surface element for integration, the simpler the integration; it is seldom necessary to usestandard elements that are differentially small in all dimensionsor along all (curvilinear) coordinates. This simple strategy is extremely useful; it is used extensively throughout the fields andwaves course(s) in all sorts of volume and surface integrals thatpossess some kin

electromagnetics education and surveys and experiences in teaching/learning electromagnetic ﬁelds and waves are pre-sented in [1]–[6]. Generally, there is a great diversity in the teaching of undergraduate electromagnetics courses, in content, scope, and pedagogical philosophy. Some electromagnetics courses

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