Students' difficulties with vector calculus in electrodynamics

Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell's equations in electrodynamics.

Understanding Maxwell's equations in differential form is a prerequisite to study the electrodynamic phenomena that are discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell's equations in electrodynamics.

I. INTRODUCTION
It is difficult to overestimate the importance of Maxwell's equations in the study of electricity and magnetism. Together with the Lorentz force law, these equations provide the foundations of classical electrodynamics. They can be used to solve problems concerning electromagnetic phenomena including those occurring in electric circuits and wave optics. Moreover, Maxwell's equations are the first example of a gauge theory (which is commonly used in particle physics) and are the starting point for Einstein's theory of special relativity, both historically and in many curricula. It is therefore desirable that students have a profound understanding of these equations.
For physics education researchers, an electrodynamics course is an ideal 'laboratory' to explore the link between mathematics and physics since students have acquired knowledge of the physical concepts in an introductory electricity and magnetism course, and learned the necessary mathematical techniques during instruction on calculus. To describe more complex electromagnetic phenomena, they will need to apply this advanced mathematics in the description of the physical reality. We are therefore interested in problems students encounter with both physics and mathematics when using Maxwell's equations.
Maxwell's equations can be formulated in differential or in integral form. In differential form, the four laws are written in the language of vector calculus that includes the differential operators divergence and curl. These are typically expressed using the nabla ∇ symbol to denote the del operator. The divergence of a vector field (∇ · A) is a scalar quantity that measures the magnitude of a source or sink of the field at a given point. The curl of a vector field (∇ × A) results in a vector that describes the infinitesimal rotation at a certain spot. Both quantities are defined locally: they only describe the characteris-tics of a vector field at a single point. This is the most important distinction with Maxwell's equations in integral form, which describe the electromagnetic field in a region of space. For physics majors, the two formulations are equally important since they both have certain advantages and limitations in specific contexts. The research described in this paper focuses on students' understanding of Maxwell's equations in differential form and includes their knowledge and skills concerning vector calculus.
In Section II we provide an overview of the related literature, including some work on the link between mathematics and physics, research on electromagnetism, vector calculus and the combination of the two. This leads to our research goals, which are described in Section III. To formulate our research questions, we distinguish four kinds of skills that need to be acquired by the students: 1. structural understanding 1,2 of the concepts of gradient, divergence and curl 2. the interpretation of these operators in the context of a graphical representation 3. doing calculations that involve vector operators 4. conceptual understanding of Maxwell's equations in differential form.
The educational context and methodology are described in Section IV, followed by the results of our study in Section V. This section contains a discussion of our students' skills and difficulties concerning the four topics that are listed above. The most important findings and possible implications for teaching are summarized in Section VI.

II. RELATED LITERATURE
The majority of physics education research (PER) at university level to date concerns introductory courses (examples can be found in the summary of McDermott and Redish 3 ). These studies have, among other things, yielded an extensive inventory of conceptual problems in physics and the finding that many students struggle with the application of their mathematical knowledge in a physical context. 4-6 Manogue and Dray went so far as to state that physicists and mathematicians speak a different language, but use the same vocabulary . 7 The role of mathematics in physics education is therefore an important topic in recent PER projects. 6,8-13 A tendency for students to focus on equations and calculations rather than on the physical meaning behind the symbols has been identified as a recurring issue. 14-16 Another source of difficulties is the use of multiple representations in physics and mathematics. Students have severe difficulties combining the information in texts, equations, symbols, graphs and figures into a single unambiguous story. 17-22 Our work is related to the investigation of these issues in the context of an intermediate electrodynamics course.
Introductory electricity and magnetism (E&M) courses are a popular setting to learn about student misconceptions. [23][24][25][26][27][28][29][30][31][32][33][34][35] A fair amount of research has also been carried out on the use of integrals in E&M. [36][37][38][39][40][41][42][43][44] This is of interest for the work we present here, since we have adopted some of the ideas and methodologies in these papers as a starting point for our own research. At Dublin City University students ideas about integrals were investigated using an approach based on the idea of the concept image. 36 This is a collection of all the mental processes that are activated when students are encountered with a certain concept (e.g. an integral of a vector operator). It is unique for every person, and therefore differs from the (formal) concept definition, which is a description that is accepted by the wider community . 45 One aspect of our study concerns our students' concept image of vector operators, which we relate to the difficulties they encounter when applying Maxwell's equations in differential form.
In advanced courses (often called E&M2 or electrodynamics) vector calculus plays an important role. It is known that operations with vectors and vector fields (e.g. vector addition and the dot product) provide students with a lot of problems. 46-51 Furthermore students struggle with the use of vectors in different coordinate systems and the application of appropriate unit vectors. 46,[51][52][53] However, little is known about situations where the del operator is applied to scalar or vector fields. Gire and Price discussed the option to use more graphical representations when teaching about vector fields and vector calculus. Based on their experience with different types of in-class activities, they argued that algebraic representations are useful since they can easily be manipulated, but students gain more insight in the differences between components and coordinates when using a graphical approach. Moreover they expect students will benefit from being able to translate one representation to another. 54 Singh and Maries report that about half of their graduate students before instruction, and one out of three after instruction, are unable to determine where the divergence and/or curl is (non)zero when provided with a graphical representation of a vector field. They argue that physics courses are often a missed learning opportunity because they strongly focus on mathematics but fail to develop a functional understanding of the underlying concepts. 55 In the previous paragraph, we discussed the physics and mathematics education research on vector calculus. In the context of electrodynamics, the amount of research on vector calculus is limited. The educational setting in this context however is different. Manogue and Dray pointed out that in mathematics the gradient, divergence and curl are used in a general and abstract way, while in physics they are mostly used in certain symmetries (Cartesian, cylindrical or spherical). 56 A research group of the University of Colorado showed that problems arise when asking students to determine where the divergence of an electric field vanishes, given the charge distribution. This type of question can be solved with the differential form of Gauss's law in a straightforward way. However, the authors report that only 26% of their students were able to give a correct answer. 57 Baily and Astolfi found that what students from St Andrews learned about the divergence in one context (Gauss's law) often did not translate to their understanding in other contexts (e.g. continuity equation). 58 In summary, the literature reviewed here shows that students struggle when they have to use their knowledge from mathematics in a physical context. Clearly, this also applies to the specific case of applying vector calculus in electrodynamics. However, there is still a lot of research to do on this subject. In the next section, the contribution of our study is formulated in terms of goals and research questions.

III. RESEARCH DESIGN
This paper gives an account of an exploratory study of students' strengths and weaknesses in using vector calculus in mathematical and physical contexts. The research extends the previous findings mentioned in Section II by adopting a broader approach to ascertain the knowledge, skills and understanding our students have acquired. The goal of this study is twofold: it aims to provide both researchers and teachers with insights into the learning results of traditional instruction of electrodynamics, and it is the first stage of a large scale investigation of students' understanding of Maxwell's equations in differential form.
In this first stage of the investigation we aim to gain insight into the difficulties students encounter with vector calculus in a pure mathematical or physical context. To this end we distinguish four different kinds of skills and competencies students need to acquire: structural understanding 1,2 of divergence and curl, graphical interpretation of vector fields, calculation of divergence and curl, and conceptual understanding of Maxwell's equations in differential form. Based on their instruction prior to the electrodynamics course and results from the literature, we expect our students to have reasonable facility with the mathematical techniques needed to carry out calculations while lacking experience with interpreting graphical representations of vector fields. We have investigated our students' attainment at the start of the electrodynamics course and have tried to establish to what extent their understanding of Maxwell's equations in differential form changes while taking the course. We have focussed on the following research questions: • Did the students acquire a structural understanding 1,2 of gradient, divergence and curl from their introductory and intermediate mathematics courses?
-What is their concept image 45 of gradient, divergence and curl?
-How do they describe the meaning of the vector operators?
• Can students interpret a graphical representation of a vector field in terms of its divergence and curl?
-Can they deduce where the divergence and curl of vector fields are (non)zero in a purely mathematical context?
-Can they deduce where the divergence and curl of electromagnetic fields are (non)zero?
-What strategies do they use to interpret these representations?
• Did our students acquire the necessary mathematical techniques to perform calculations involving vector operators with and without a physical context?
-What technical difficulties do they encounter?
-Do different kinds of coordinate systems (Cartesian, cylindrical, spherical) present different challenges?
• Do students conceptually understand Maxwell's equations in differential form?
-Are they able to correctly deduce whether the divergence and curl of an electromagnetic field are zero or non-zero in a given situation?

IV. EDUCATIONAL CONTEXT AND METHODOLOGY
To answer the research questions, we gave written paper-and-pencil questions to second year university students in a traditional thirteen week intermediate electrodynamics course. The students major in physics or mathematics at the KU Leuven. They use Griffiths' textbook 59 and are instructed in one two-hour lecture and one two-hour problem solving session per week, in which they discuss typical end of chapter problems from the textbook. The students have already completed an introductory electromagnetism course using the textbook of Serway and Jewett 60 that leads up to Maxwell's equations in integral form, and at least two calculus courses 61 that include a chapter on vector calculus. Therefore they have encountered the necessary mathematical tools and physical situations presented in the electrodynamics course.
To identify the prior knowledge of our students they were given a pretest before the first lecture in the advanced electromagnetism course based on Griffiths' textbook. 59 Since these students had encountered vector calculus mostly in a mathematical setting, the questions on the pretest do not contain any physical context. To encourage students to write down their reasoning, calculations and thinking process, all questions were openended. A post-test was given after instruction on chapters 1-7 of Griffiths' textbook 59 , during a lecture about halfway through the semester. The post-test assignments are mostly similar to those on the pretest; however, a physical context is introduced in some cases to investigate whether information on the physical situation helps students to interpret the divergence and curl of the (electromagnetic) vector fields. The post-test also comprises questions that evaluate students' understanding of Maxwell's equations. There were no time constraints for the students to complete the pre-and post-test.
The analysis focuses on the solution method and thinking process rather than the result. To describe and explain the variation in students' conceptions, ideas of phenomenography are used. Phenomenography is an empirical approach that aims to identify and categorize the different qualitative ways in which different people perceive and understand phenomena. 28,62 The categories used in the analysis of our data emerged from the answers students gave, the strategies they used and the mistakes they made. Since the amount of students is limited (N = 30 on the pretest and N = 19 on the post-test), the numbers and percentages should be generalized with care. Nevertheless they should give a clear view of the limitations in our students' understanding of vector calculus in electrodynamics.

V. RESULTS AND DISCUSSION
In this section the results of the pre-and post-test are presented and discussed. The questions can be found in Appendices A and B.

A. Pretest
The pretest shown in Appendix A was given to all 30 students at the beginning of the course to probe their knowledge and understanding acquired in previous courses. The first part of the pretest identifies students' concept images 45 of the operators grad, div, and curl. In the second part, the students' calculational skills and their ability to interpret graphical representations of vector fields are tested. For this part only, some useful formulas were attached to the questions (Appendix C).

Concept image of grad, div and curl
The concept image question serves to get a better understanding of what students associate with the gradient, divergence and curl in a very general sense. Expressions for grad, div, and curl are given to the students, and they are asked to write down everything they think of. From this, we can make some statements about the students' concept image 36,45,63 of these operators. The students' responses to the questions are described qualitatively in Table I. We distinguished three important emerging categories in the students' answers: information about the structural meaning of the vector operators, the scalar or vector character of the expression, and the name and symbolic expression that students wrote down. Obviously, a student can give more than one interpretation and therefore the percentages sum to more than 100%. We do not suggest that students do not know something they did not write, but we do think the question reveals what is cued first and foremost. Only a few students gave a description of the operators we deemed conceptual. Some provided a more or less correct description that resembles the concept definition: "The curl tells you how strong and which way the vector field A rotates." Others had very incomplete or incorrect conceptual ideas: "The divergence is a measure for how the field is changing." "The gradient of A is the vector normal to the plane." We also observed that some students misidentified the vector or scalar character of the expression. More than half of the students mentioned which operations result in a vector field, and which produce a scalar field. Sometimes they explicitly wrote it, in some other cases it could be derived from their notation. It is interesting to see that 2/20 (10%) of the students who indicate the vector character explicitly or implicitly made it clear that in their opinion the gradient of a scalar is again a scalar. One of these students, together with another student, seemed to think the divergence of a vector field is again a vector field. We did not observe a single misidentification of the vector character of the curl of a vector field. This corroborates the findings of Barniol and Zavala, who showed that students have significantly more problems with the dot product than with the vector product. 51 Since the students correctly described A as being a scalar (field) and A as being a vector (field), we have no indications that there was a problem with the notation in the question. About one out of three students did not include the vector or scalar character in their answer (e.g. they just named ∇ · A "divergence", without any explanation), so it is possible that a larger fraction of our students is unsure which quantities are vectors and which are scalars.
The category 'Naming' contains all students who wrote down the name of the expression. No students remembered names incorrectly or mixed up the terms gradient, divergence and curl. Nevertheless, half of the students did not explicitly identify ∇ · A as the divergence and ∇ × A as the curl of a vector field. It seems that most students recognized these expressions, but it appears that many did not remember the name, or did not think it was important. It is also interesting that only 17% (5) of our students wrote that ∇ is the nabla symbol, and one student called it the del symbol. 20% (6) however called this symbol the gradient and 7% (2) called it the Laplacian. Some students used more than one of these names for the same symbol. 30% (9) students wrote that ∇ is an operator and used this term correctly. Two students thought the entire expression is an operator: " ∇A is called the gradient operator ; ∇ · A is called the divergence operator." While the ability to correctly name operations and operators was not a main concern of this study, it seems clear that not being able to do this could be an impediment to further learning. In some cases it was not possible to determine whether students had the concepts confused, or the names, or both: " ∇ × A is the vector product between the gradient and A." In the second stage of our research we will interview students to gain further insight into the meaning of this kind of response.
Furthermore, about half of the students tried to write down a formula, even though some of them did not know it by heart and made a mistake. Students seemed to feel these vector operators are a tool to evaluate something. This focus on evaluation was also seen in students' concept image of integration. In this particular case, students rather tried to evaluate an integral that was impossible to calculate than to describe it as an area under a curve or a sum of infinitesimal parts. 36 In the category 'other' there were some correct statements that explained the link with Stokes' law, the divergence theorem or conservative fields. These students seem to have made connections with the integral form of Maxwell's equations.
In summary our students focused on evaluation but seem to lack a structural understanding 1,2 of the gradient, divergence and curl. The second part of the pretest is designed to investigate their ability to interpret graphical representations of vector fields and their skill at doing calculations in vector calculus.

Graphical interpretation of vector fields
We gave our students a two-dimensional representation of four different vector fields and asked them to indicate where the divergence and curl are (non-)zero. The assignments can be found in Appendix A. The fields are numbered from left to right. The divergence is non-zero everywhere in the first one and the curl is non-zero everywhere in the second and third field. The last field is less obvious: it clearly has zero curl. The field we sketched has 1/s dependence, so that the divergence is non-zero only at the center of the field. This could for example be the electric field of a charged wire pointing in the zdirection. However, if students saw a general dependence on s and stated that they could not decide whether the divergence was zero, we deemed their answer correct.
First the students' answers were checked for correctness. Table II shows that our students have severe difficulties with these graphical representations. For the first field, half of the students gave the correct answer, but for the other three vector fields less than one out of four explained correctly where the divergence and curl are non-zero. In the last field, just two students could determine that the divergence is non-zero only at the center of the field. About 30% of the students made at least one statement that pointed toward the typical error 57 of confusing the derivative of the field with its value (e.g. the derivative is zero when the field is zero). Our students are easily misdirected, and very inconsistent in their reasoning. Moreover, a significant number of students did not answer the question (about 20-30% for the divergence and 30-40% for the curl). It is likely that these students did not know how to solve these problems, since they did answer the other pretest questions.
Secondly, we looked at the strategy students used to obtain their answer. The prevalence of the strategies is shown in Table III. We distinguish categories: • Concept based strategy: This category includes explanations from students that show a good under- standing of the underlying concepts. Their answers are based on drawings together with the definition and potentially some derived formulas that link the differential and integral form. Typically the change in flux per unit area is determined by drawing a small box around a point: if there is no net flux in the area bounded by the box, the divergence is zero as there is no source/sink in this area. To obtain the curl, a virtual paddle wheel is placed in the field. If it rotates, the curl is non-zero at that particular location.
• Formula based strategy: The student mostly relies on "the formula" for divergence and curl, and uses the derivatives of x and y to get an answer. Some students even (try to) obtain an algebraic expression for the vector field and then apply the definition of the operator (typically in Cartesian coordinates).
• Description based strategy: The students use a (correct/incorrect/incomplete) qualitative description of the divergence and curl to obtain an answer. Typically the student relies on the common English definition of the words 'divergence' and 'curl' and links this in naive way to the graphical representation of a vector field. This is illustrated by some examples for the third vector field: -False descriptions: " ∇ · A = 0 because the length of the arrows increase." " ∇ × A = 0 because the field is not rotating." -More or less correct descriptions: " ∇ · A = 0 since nothing is added to the field anywhere." " ∇ × A = 0 because the field is rotating locally." It is of course possible that students who wrote down these descriptions have some conceptual insights as well, but their answers provided no evidence for this.
• Unclear : The reasoning is not explained or it is unclear.
• No answer : The student did not answer the question.
Students generally used just one of these strategies to solve the question. However, some used a different approach to determine the divergence and the curl of a field. In the last four columns of Table III, the success rate using a certain strategy is given. The student who used the concept based technique was very successful in determining the divergence and curl of the fields. The formula based technique is useful if the vector function can be found and calculations are carried out correctly, which may cause problems for complex cases (e.g. the fourth vector field). Students who use a description based strategy or give little or no explanation seem to have a low chance of being successful in determining the divergence and curl of a graphical representation of a vector field. There are some exceptions to these generalizations, like a student who determined the divergence and curl correctly for every field, but gave no explanation whatsoever. However, his answers were probably well considered, as he gave a fairly accurate description of the divergence and curl in the first part of the pretest: "The divergence of A is a scalar field that tells you how much is added to the vector field A" "The curl of A is a vector field that tells you how strong and in which way the field A turns" This leads us to believe that this particular student used his conceptual insight in the divergence and curl to tackle problems concerning the graphical representation of vector fields.

Calculation of divergence and curl
In the last set of questions on the pretest we asked our students to calculate the divergence and curl of three vector fields (see Appendix A). Two fields were given in Cartesian coordinates, the third in cylindrical coordinates. For each of the 6 calculations, we split answers into four categories: complete and correct calculations, calculations with minor mistakes or omissions (e.g. a forgotten minus sign or an expression is left unsimplified), calculations with major mistakes (e.g. an error in the use of the formula for div/curl, an error when taking the derivative or inappropriate use of unit vectors), or no answers. The results are presented in Table IV.
Exercise (a) required students to make a quite straightforward calculation. Nevertheless, only 60% of the students were able to calculate the divergence correctly, and allowing for minor errors about three-quarters calculated the curl correctly. One student even calculated the Laplacian instead of the curl. He did not give an answer for the other parts.
Exercise (b) was more difficult than the first one, since some challenging algebra is required to solve the equation in Cartesian coordinates. This explains the higher number of students who make minor errors. When allowing for minor errors, again about 60% of students gave correct answers. These are mostly students that could also correctly calculate the divergence and curl in the first exercise. Two students were smart enough to convert the equation to polar coordinates and then calculate the divergence and curl. One of them knew the formula for the divergence in 2 dimensions 64 (only the formulas for 3 dimensions were given to them), the other student made a mistake at this point. Both of them noticed the curl is zero without any calculation at all. Two other students did not calculate the curl in this part, because they argued a vector product is only defined in 3 dimensions. It may not have occurred to them that the vector field could be considered three-dimensional with zero z-component. While almost all students attempted to calculate the divergence, over one-third of the students did not attempt to calculate the curl. They may have been discouraged by difficulties they had when calculating the divergence.
The calculation required in Exercise (c) is as straightforward as that of Exercise (a), and a similar fraction of students calculated the divergence and curl correctly. Again, this is more or less the same group of students that could do the calculations in the first two exercises. However, many more students did not give an answer at all. It is unlikely that they did not know how to calculate the divergence and curl in cylindrical coordinates, since expressions were given to them.
In general, we can see that approximately 60% of the students are able to calculate the divergence and the curl of given vector fields, independent of the level of difficulty and the coordinate system used, if we allow minor errors. The major errors can be classified in three subcategories: 10 times an error was made when taking the derivative, 11 students used the expressions incorrectly and in 10 cases unit vectors were used inappropriately (e.g. unit vectors were appended to terms in the divergence on a vector field). Of course a single student could make multiple errors during one calculation. Concerning the use of unit vectors, it was striking to see that students used them very inconsistent during the test.
When we compare the results of the calculations to the number of correct answers in the graphical representation question, the prior knowledge of these students clearly shows. They seem to have some difficulties calculating the divergence or curl, but struggle way more with exercises that ask for more insight. This confirms what we observed in the concept image of the students: most of them lack a conceptual understanding of the divergence and curl, and focus on evaluation.

B. Post-test
After instruction up to Chapter 7 in Griffiths' textbook 59 , we gave the students a post-test questionnaire. It comprises two questions concerning graphical interpre-  tation of vector fields (one with and one without physics context), two questions where students have to calculate the divergence and curl after imposing a condition and two conceptual questions in which they have to use the differential form of Maxwell's equations to interpret a series of physical situations. These questions can be found in Appendix B. All 19 participants also took the pretest.

Graphical interpretation of (EM) vector fields
a. Vector fields without physical context This question is similar to the Question 1 of part 2 on the pretest. The fields on the post-test are analogous to the third and first fields on the pretest respectively (however, the 'view' is changed a bit). The results are summarized in Table V and are compared to the answers on the pretest.
It seems that students did better with the first vector field of the post-test, but made a more or less equal number of mistakes when interpreting the second vector field. This was analyzed more profoundly by looking at how many students' answers improved (incorrect at pretest; correct in post-test), disimproved (correct at pretest; incorrect at post-test) and stayed the same. For the first field, three students could correctly determine divergence and curl in the post-test, but not in the pretest. Not a single student made a 'new' mistake. For the second field, four students improved their answers, but two went from answering correctly to answering incorrectly. This means that most students stick to their answers: there is a slight increase in correct answers, but still only about 50% can determine the divergence and curl from a graphical representation of a simple vector field. Note that these percentages are similar to the results of the graduate students (before instruction) from the article of Singh and Maries. 55 However, four students made incorrect statements like "The divergence is zero in the x direction, but not in the y direction." This kind of reasoning was not seen in the graphical pretest questions, though it did emerge in the calculational pretest questions. We intend to explore this issue further in the future; it illustrates in any case that many students still struggle with divergence. Looking at the strategies the students used to determine the divergence and curl (Table VI) we see an increase in concept based reasoning and formula based reasoning. The number of unclear answers decreased and every student at least tried to give an answer this time. This effect might be due to instruction: in lectures they were told that it is possible to use a paddle wheel to determine the curl, for example. However, students were TABLE V. Number of correct students' answers for the context-free graphical representation of vector fields on the post-test. Vector field 1 on the post-test should be compared to field c on the pretest, and field 2 to field a.

b. Electromagnetic fields
The post-test questions on graphical representations of electromagnetic fields ask similar questions in context. Furthermore, the fields are a bit more 'difficult' in the sense that they have a cylindrical instead of a Cartesian symmetry. The number of correct answers is very small: only a few students could correctly determine both the divergence and the curl (Table VII). To solve this question, students could use the same strategies as before, but could also use Maxwell's equations (physics based reasoning). Because some students used such an argument to confirm their answer based on another strategy, it is now possible that they are entered in multiple categories. The answers are summarised in Table VIII. TABLE VIII. Strategies students used to interpret the divergence and curl of graphical representations of electromagnetic fields in the post-test. The first part shows how many students used a certain approach. (Some students used a generic approach together with a physics based strategy, so the total exceeds 19.) The second part shows the number of correct answers, and the success rate when students use a certain strategy.

Field a Field b Div
Curl Div Curl Concept based 100% (1) 0% (0) --Physics based 100% (8) 50% (1) 50% (2) 100% (5) Formula based 67% (2) 0% (0) 0% (0) -Description based 67% (2) 0% (0) 0% (0) 100% (3) Unclear 17% (1) 0% (0) 10% (1) 91% (10) Students tend to change their strategy over the questions (this is why both questions are treated separately) and fail to correctly apply the strategies they use. They seem especially unsure about the use of Maxwell's equations. To illustrate this, we analyze the answer of Marc for the magnetic field question: Concerning the divergence, Marc states correctly it is zero: "∇ · B = 0 → there is no magnetic monopole", which is indeed always true. When looking at the curl, he writes "∇ × B = µ 0 I" (note it should be the current density J instead of the current I). But then he crosses out the right hand side, and simply writes "non-zero". In the end, it is unclear what argument he used to obtain this incorrect result.
Many students do similar things: they try to use Ampère's law, but fail to apply it correctly. They do not appear to understand that the curl of a magnetic field is only non-zero where a current flows, and that it varies from point to point. Furthermore, they misinterpret the use of a paddle wheel: they seem to think it rotates everywhere, but it does so only in the center of the field (everywhere else it translates in a circle around the current carrying wire). Similar mistakes were made in the case of the electric field: no more than three students understood that the divergence is non-zero only where charges are present. Formula based reasoning is less effective here, because the students struggle with the use of cylindrical coordinates or try to set up the equation of the field in Cartesian coordinates.

Calculation of divergence and curl of EM fields
These questions are intended to check if students can do calculations in an electromagnetism context, which takes the form of imposing a simple condition. To determine which field could be a magnetic field, students should check if ∇ · B = 0 applies; to check whether a field could be an electrostatic field, they need to verify that ∇ × E = 0.
Recognizing and imposing this condition proved to problematic for our students: 74% (14) students were able to do this for the magnetic field and only 53% (10) for the electric field. One student did not give answers to any of these questions. Others used qualitative reasoning that contained something about the radial or zdependence. A few students calculated the divergence of the potentially electrostatic fields, but then struggled to interpret the result.
Almost all students who obtained the correct condition calculated the divergence and curl correctly. Some even did not need to calculate a full expression, but could determine whether the divergence and curl were (non-)zero by sight. However, a few errors were still made. It seems that the higher-achieving students were able to get the correct condition, and they were already quite proficient at doing calculations before instruction. Nevertheless, a slight progression could be observed. This may be explained by the huge emphasis on calculations in Griffiths' textbook 59 and the exercise sessions.

Conceptual understanding of Maxwell's equations
The last set of post-test questions intends to investigate students' conceptual understanding of and insight into Maxwell's equations in differential form. Therefore they have to determine whether the curl and divergence are zero or not in four electric and five magnetic fields. 65 When Maxwell's equations in differential form are applied correctly in every situation, one obtain the answers: ∇·E = 0 always except for the first situation, ∇ × E = 0 in situation a, c, and d, ∇ · B = 0 always and ∇ × B = 0 in the last two situations. We did not give the students a list of Maxwell's equations to avoid pointing students in a particular direction. In our opinion students who understand these laws will be able to reproduce at least the causality between the fields and sources. However, as shown in Table IX, students encountered tremendous difficulties solving this question. Only one student did not make a single mistake, while all other 18 students made at least three errors. We could not find a correlation between the errors, but some patterns did emerge: • Gauss's law, which states that the divergence of the electric field is non-zero only if charge is present, or more conceptually still, that the source (sink) of an electric field is a positive (negative) electric charge, elicited most errors. Only one student could correctly determine where for all five situations. This confirms some of the findings from the University of Colorado: students have difficulties applying the divergence in an electromagnetic context 57 .
• At least one mistake was made in the application of Faraday's law by 53% (10) students. It is remarkable that 9 students did not appear to know that the curl of an electric field is non-zero when there is a changing magnetic field, as stated explicitly in situation b.
• Despite the elaborate discussion on the nonexistence of magnetic monopoles during instruction, 53% (10) students did not check every box under ∇ · B. This is alarming, since it is a truly elementary law that is easy to apply.
• Only 21% (4) students were able to correctly evaluate Ampère-Maxwell's law in every situation.
Strikingly, fewer than half of the students could interpret situation c, which is the classic textbook example to show Maxwell's correction to Ampère's law. It was both discussed during the lectures and in Griffiths' textbook 59 .
These results show that our students did not profoundly understand Maxwell's equations in differential form.

VI. CONCLUSIONS AND IMPLICATIONS FOR TEACHING
We have investigated students' understanding of divergence and curl in mathematical and physical contexts. Concerning their initial concept image, we found that they focused on evaluation, and paid little attention to the conceptual meaning of the vector operators. Furthermore, their conceptual descriptions often were incomplete and contained incorrect information. Some students were confused about the vector or scalar character of the operators, and used incorrect terminology.
Interpreting graphical representations of vector fields is a difficult exercise for students. Even after instruction only half of the students were able to determine where the divergence and curl of a simple vector field are (non)-zero. When more complex and realistic electromagnetic fields had to be considered, only a few students succeeded in solving the question correctly. Moreover, many students used various strategies inconsistently. This suggests they lack a structural understanding of the mathematical concepts, and on top of this they are unable to use their acquired skills in a physical context.
Since we (and many others 54,55,57 ) believe that these graphical representations are helpful when trying to conceptualize the abstract mathematical structures in vector calculus, we think it would be advisable to put more effort in this kind of exercises in both physics and mathematics instruction. In our opinion it would help students to understand the physical meaning of Maxwell's equations and provide them with more insight to tackle the subsequent problems concerning electromagnetic radiation, gauge theory and the introduction to special relativity.
In Griffiths' textbook 59 a lot of exercises focus on complex calculations. We found that students are reasonably comfortable with the required algebra, but have problems when they need to interpret the context of a calculation. One out of four students was unable to come up with the condition a vector field should satisfy in order to be a realistic magnetic field, and only half of them knew this condition for an electrostatic field. Since understanding and explaining electrodynamic phenomena is one of the main objectives of this course, we suggest more attention should be paid to the interpretation of the equations and setting up the problem at the expense of doing algebraic manipulations.
When we investigated the competences students show when they encounter situations that can be solved using Maxwell's equations in differential form, we observed that students have tremendous difficulties with the application of all four laws. This calls for an instruction that puts more effort in linking mathematics and physics and uses a more qualitative approach. Some great ideas can be found in the article by Huang et al 66 , although we think even more graphical and conceptual examples are needed in order to fully show students the power and usefulness of Maxwell's equations in differential form.
In future work we are planning to conduct student interviews to gain more insight in the thinking process of students when they solve problems linked to the differential form of Maxwell's equations. This will help us understand how graphical representation and a better structural understanding of the mathematical concepts can help students to apply their skills in a physical context. In a later stage the results of these interviews will be used to create new or improved questions on the pre-and post-test, and to iteratively design a tutorial that aims to help students understanding Maxwell's equations in differential form.

ACKNOWLEDGMENTS
We gratefully acknowledge the fruitful discussions with Charles Baily and thank him for his valuable comments on the paper. We greatly appreciate the cooperation with Wojciech De Roeck and the students who made this study possible.

Appendix A: Pretest questions
The first part was given to the students without any expressions for div, grad and curl. After they finished and turned in Part 1, they were given Part 2 which contained the expressions given in Appendix C. Some space was left blank for the students to answer after each question. Figures are displayed smaller than in the actual test.  1. Indicate where the divergence and/or curl is (non)zero for the next vector fields in the x, y-plane. The zcomponent is zero everywhere. Explain and show your work.
x y x y x y x y 2. Calculate the divergence and curl of the following vector fields.

Appendix B: Post-test questions
The expressions of Appendix C were appended to these questions. Some space was left blank for the students to answer after each question. Figures are displayed smaller than in the actual test.
1. Indicate where the divergence and/or curl is (non-)zero for the following vector fields in the (x, y) plane. The z-component is zero everywhere. Explain and show your work.
x y x y 2. For the following physical situations, explain where the divergence and/or curl of the field are (non-)zero. The z-component of the fields is zero everywhere. Show your work.
(a) The magnetic field of an infinite current carrying wire along the z-axis.
x y (b) The electric field of a charged infinitely long cylinder with radius R. In the figure, the crosssection in the x, y plane is given. ∇ · E = 0 ∇ × E = 0 (a) The electric field at a distance r < R from the center of a uniformly charged sphere with radius R.
(b) The electric field generated by a changing magnetic field.
(c) The electric field at a distance r from a pure electric dipole.
(d) The electric field inside a charged conductor.