Observations on student difficulties with mathematics in upper-division electricity and magnetism

We discuss common difficulties in upper-division electricity and magnetism (E) in the areas of Gauss's law, vector calculus, and electric potential using both quantitative and qualitative evidence. We also show that many of these topical difficulties may be tied to student difficulties with mathematics. At the junior level, some students struggle to combine mathematical calculations and physics ideas, to account for the underlying spatial situation when doing a mathematical calculation, and to access appropriate mathematical tools. We discuss the implications of our findings for E instruction at the junior level.