torquetum | Entries tagged with vector calculus

Robertson, R.L. (2013). Early vector calculus: A path through multivariable calculus. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 23(2), 133-140.

The author argues for an ordering of topics in a multivariable calculus course which brings the three big theorems as early as possible. The textbook he uses is a standard maths text, with the three big theorems coming last. He lists the topics to be covered before Divergence Theorem can be covered, locating it (by my estimate) a bit less than halfway through the course. Thereafter he covers a few more topics and get to Stokes’ Theorem (probably about 2/3 of the way through the course). Green’s Theorem is presented as a special case of Stokes’ Theorem. The benefits of this approach are argued for convincingly and a few drawbacks are also covered (such as parametrised surfaces before parametrised curves). This is the second paper I have read which recommends Schey’s (2005) Div, Grad, Curl and All That: An Informal Text on Vector Calculus, so I really must track that text down. The practical interpretations of div and curl are emphasised as in so many papers I’m reading. I found this paper intriguing and I also greatly appreciated that the author broke the course down into sufficient detail that I, or someone else, could easily structure a course as he has done.

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings or subjective views.

Dray, T. & Manogue, C.A. (1999). The vector calculus gap: mathematics ≠ physics. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 9(1), 21-28.

Here we have another paper lamenting (justifiably) the difference in the way vector calculus is taught in maths and physics. The authors emphasise how practical applications and situational geometry are far more important in physics (or engineering) than in maths. For example, they discuss how vectors are defined as ordered triples in maths while as arrows in space in physics. Also, div and curl are defined as differential operations on vector fields in maths but in physics are defined first in terms of their physical meaning as represented by Stokes’ Theorem and Divergence Theorem. The coordinates used in maths are almost invariably rectangular coordinates, the authors argue, while physics situations frequently have circular or spherical symmetry and hence use spherical coordinates to simplify the maths. (Some of the paper’s criticisms could apply to my local course, but not all, I think.) The value of the mathematical methods lies in their general applicability, however in physics the types of cases are few and there is an argument for loss of generality in favour of simplification of the common cases. The authors close with an insistence that the relevant departments collaborate closely.

Huang, H., Wang, J., Chen, C. & Zhang, X. (2013). Teaching divergence and curl in an Electromagnetic Field course. International Journal of Electrical Engineering Education, 50(4), 351-357.

The authors teach an electromagnetic field course and recognise that physical interpretation of divergence and curl are a difficulty for students, even though the actual calculations are not. They suggest a teaching method which begins with capturing the students’ interest through fictional and theoretical invisibility cloaks. The maths behind the theory involves Maxwell’s equations and hence divergence and curl. The authors suggest ways of teaching divergence and curl through flux and circulation, beginning with the macro and moving to the micro in logical ways.

Pepper, R.E., Chasteen, S.V., Pollock, S.J. & Perkins, K.K. (2012) Observations on student difficulties with mathematics in upper-division electricity and magnetism. Physical Review Special Topics – Physics Education Research, 8(010111), 1-15.

The authors focus in this paper on the mathematical difficulties experienced by physics and engineering students in an “upper division” electricity and magnetism course. They categorise the difficulties as (p. 2):

“Students have difficulty combining mathematical calculations and physics ideas. This can be seen in student difficulty setting up an appropriate calculation and also in interpreting the results of the calculation in the context of a physics problem. (However, students can generally perform the required calculation.)
Students do not account for the underlying spatial situation when doing a mathematical calculation.
Students do not access an appropriate mathematical tool. Students may instead choose a mathematical tool that will not solve the relevant problem, or may choose a tool that makes the problem too complex for the student to solve.”

Using the troublesome concepts of Gauss’s Law, various vector calculus techniques, and electrical potential, the authors demonstrate these three categories of difficulty with many examples of specific problems and student responses. Of specific interest to me were the vector calculus issues. They find that students struggle with the “vector nature” of a vector field, struggling to think of magnitude and direction simultaneously. Additionally, students can calculate gradient, divergence and curl easily, but struggle with the physical interpretation of these quantities. The authors hypothesise that the way vector calculus is taught in mathematics class, the students fail to see integrals as “sums of little bits of stuff”, which I would like to think is not true in my maths classes, where the sum nature of integrals is repeatedly emphasised. Another vector integration difficulty is found in students struggling to express dA and dV in suitable coordinate systems.

They discuss methods they have used in classroom pedagogy, out of classroom assistance and transformed resources to address these difficulties. Even with all their changes, they find that certain problems remain challenging for the students. They argue that these concepts are hard to understand and that the instructors are not keeping this well enough in mind. They discuss ways of moving forward. I thoroughly enjoyed this paper, found it pertinent to current and future work of mine, and benefited from the thorough literature review.

Burch, K.J. & Choi, Y. (2006). The curl of a vector field: beyond the formula. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 16(3), 275-287.

The authors contextualise curl and Stoke’s Theorem in fluid flow. They begin by considering an infinitesimal rectangle deformed by a vector field. They calculate the angular velocities of two adjacent edges and show how that relates to the concept of viscosity in fluid dynamics and to the familiar curl formula. They show how that means that curl relates to spin about an object's own axis (independent of translation through a fluid) and give some examples. For several given vector fields, they draw the direction field and intuitively predict spin, then they back that up with curl calculations. The authors close with a discussion of the meaning of the Stokes’ Theorem formula and how the curl side of the formula is measuring the viscosity flux of V across S. I found this paper really interesting.

Rees, J.M., Atkin, R.J. & Zimmerman, W.B. (2005). On the use of audio-tapes for teaching vector analysis to engineering undergraduates. International Journal of Mechanical Engineering Education, 33(4), 358-368.

Rees, Atkin and Zimmerman discuss their teaching innovation to replace four vector analysis lectures with audio-tapes and accompanying notes. They open the paper talking about how maths for engineers is best understood through covering fundamental concepts, mathematical techniques and, crucially, applications to engineering. They point out how their course (as with most maths courses) is so packed with content over a short time that there is scant time for applications. I therefore expected that they were using the tapes as replacements for lectures but that the actual lecture time slots would be taken up with examples and applications. This was not the case. The conventional lectures were simply cancelled with the students encouraged to attend the single “examples” class per week.

In the first wave of this initiative, the tapes were poorly used by the students and attendance at the examples classes was poor. In the second wave, the students were given increased ownership over the process, choosing when to receive the tapes and when the replaced lectures would be scheduled. Use of the tapes was better this time and attendance was improved. Also, approval of the initiative was increased, although in both waves many students claimed to prefer conventional lectures to the tape system. I was disappointed with the cancelling of lectures; surely this time could have been spent much more valuably? I was also disappointed that the vector analysis topic coverage was not described. It must be far less than my course since it is only 4 lectures long and mine is closer to 50. Mention is made of scalar and vector operators in the context of fluid flow, but that is all. The use of tapes is outdated now, but this initiative would be analogous to having lectures recorded, for instance on a tablet. I approve of this system, but only in the context of “the flipped classroom” which would mean keeping those lecture time slots active and filling them with worked examples, some sort of workshop, group activities, or whatever is appropriate for deeper learning.

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understandings and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.

Ollerton, R.L., Iskov, G.H. & Shannon, A.G. (2002). Three-dimensional profiles using a spherical cutting bit: problem solving in practice. International Journal of Mathematical Education in Science and Technology, 33(5), 763-769.

This paper is a very interesting application of vector calculus in a real-life context. Computer Numerical Controlled milling machines often use a spherical bit for the milling of high precision 3d surfaces, such as the gripping surfaces of surgical tissue clamps. If we start with a two dimensional curve approximating the undulating surface, we can find three methods of plotting the path of the centre of the spherical bit: basic calculus, Lagrange multipliers and vector calculus. Extending to the 3d surface, the vector calculus method transfers best. In the process of the calculation the concepts of radius of curvature and cycloids come up.

Goulet, J. (2001). An outcomes oriented approach to calculus instruction. Journal of Engineering Education, 90(2), 203-206.

Goulet describes a multivariable calculus course which was outcomes-based in that 15 goals were determined through communication with lecturers of subsequent courses. The students’ assessments were not simply graded as usual, but each question was graded towards achievement of a goal. If goals were not achieved in any particular assessment, a retest had to be taken until all goals were achieved. This way poorly understood concepts would remain current for the students rather than becoming “history”. The students liked “having clear cut objectives stated and pursued, the chance to get better at something if not initially successful, and the means to improve their grade” (p. 205). The response was good, from students and lecturer, although Goulet observes that assessing in this way does add to the lecturer’s workload. I am interested in the role of theory in engineering maths so I find it interesting than none of the goals had anything to do with theorems, proofs or even derivations. Food for thought.

Budny, D., LeBold, W. & Bjedov, G. (1998). Assessment of the impact of freshman engineering courses. Journal of Engineering Education, 87(4), 405-411.

The authors describe the first years of the engineering degree at Purdue University. Students enter a “Freshman College” at first, where they cover calculus, physics, chemistry, English and computers. Once they have achieved a C aggregate in these courses, they proceed to register with one of the engineering disciplines. Entering students write an algebra placement exam. If they fail this exam, they do a pre-calculus semester before beginning calculus. Students also have the opportunity to be fast tracked and proceed straight to second semester calculus. In a quantitative longitudinal study, the authors study the effectiveness of the various courses and the retention and graduation rates of different cohorts of students. At a point in the study, a counsellor-tutorial (CT) program was started, providing “additional services” to make the first-year experience “less lethal” (p. 409). This program was found to be successful, and became even more successful when the program was expanded to take on more students (from 80 to over 400). One concern was that providing support in first-year would merely delay drop out, however that pattern was not apparent in the data.

“There is a relationship between ultimately graduating in engineering and first obtaining a thorough understanding of basic mathematics and science principles. As the academic skills of the average student are honed, this relationship between these skills and the “world of engineering” becomes clearer to the engineering recruit. At Purdue University, we believe that the courses in calculus, chemistry, and physics supply the collegian with these necessary skills. We also believe that if you treat these courses as high-risk and supply additional assistance to those students with a higher probability of failure, then those students will acquire the critical background skills that will make it possible to persist in engineering.” (p. 410)

Zenor, P., Fukai, J., Knight, R., Madsen, N., & Rogers Jr, J. (1995). An interdisciplinary approach to the pre-engineering curriculum. In Frontiers in Education Conference, 1995. Proceedings, (pp. 3c1.18-3c1.22). IEEE.

The first author has already been involved in the design and running of a course integrating calculus and physics. That course has proved to be effective. This paper is on designing a sequence of courses which proceed seamlessly and integrate calculus, other maths topics, physics and several engineering courses such as dynamics, statics, thermodynamics and signals. The maths content will be the heart of this two year programme of study with topics flowing one into the other and physics and engineering topics being included as appropriate for increasing the depth of the mathematics. The idea is that new maths topics will be encountered at a reasonable pace, rather than too fast, and understanding of each will be enriched by immediate application. I am rather daunted by their plan to include vector calculus right from the beginning, but, if I understand correctly, they will start with things like differentiating vector-valued functions and things like Stokes’ Theorem will occur much later in the syllabus. The courses will be team taught and the students will remain together as a cohort for two years. At the time of writing this paper, this project was in its design phase. That was 19 years ago. I wonder if I could find out how it all went. I’ll go scrounging for that at some later date.

Mustoe, L. (1995). Industry expects – but who should determine the curriculum? European Journal of Engineering Education, 20(3), 313-323.

Mustoe asks “How far should an undergraduate education in engineering aim to meet the needs of industry as perceived by industry?” (p. 313). He proceeds to come at this question from multiple directions: the requirements of industry, of engineering academics, of mathematicians, of professional bodies. He finds that the views of what maths is necessary and desirable varies widely across these groups and that it is impossible to meet everyone’s demands. For example, does the engineering mathematics lecturer teach mathematical concepts in context or not? In-context problems at the first-year level are hard to find as real-world problems are messy and ill formed. In addition, the students might not yet be familiar with the engineering context and confusion rather than a sense of relevance will result. Mustoe cites a view that concepts are easier to study in the abstract without confusing contexts, however, many clamour for relevance in the maths curriculum and show how students lose interest due to no clear links between maths class and other courses.

Throughout Mustoe’s literature review of a wide variety of viewpoints, there is an emphasis on basic manipulative skills as well as modelling. “Five general areas of competence worth aiming for are: a ‘feel’ for the magnitudes of quantities, an ability to handle fractions without fear, an appreciation of the use of approximations, the ability to manipulate formulae confidently and the skill to cope with calculus.” (p. 320) Mustoe has little patience for long integration problems: “Other than training in patience, persistence and concentration, there is little to be gained from an exercise in integration, for example, which requires three successive substitutions and the use of a trigonometric identity for good measure.” (p. 321). There is not much mention of the role of proof in engineering mathematics courses within this paper, except for a citation of Craggs (1978): “Craggs was concerned about the level of rigour in the mathematics taught to engineers. His conclusion was that the aims for teaching were to encourage accuracy in manipulation, to state theorems with proof where it was easy and with heuristic justification where it was not, to illustrate the possible gains and losses if work was undertaken outside the domain of these theorems and to respect the additional expertise of the trained mathematician.” (p. 318).

Mustoe concludes that curricula should be dismantled and rebuilt from the ground up with a clear view of what the role of mathematics is in the degree programme. (Note that this is a 1995 paper, so hardly current.) I loved this paper for its variety of views and broad literature review.