The divergence theorem is an important mathematical tool used in many areas of physics, most notably, electricity and magnetism. Students have nominally encountered the divergence theorem in their math courses; however, its application in physics often differs meaningfully from its application in a pure math context. In this paper, we investigate physics students' understanding and interpretation of the divergence theorem through written responses to a brief two-part prompt which was designed to elicit separate responses for students' literal and qualitative interpretations of the divergence theorem. This differentiated between understanding the mathematical statement and being able to interpret its fundamental meaning. We found that students provided moderately successful literal interpretations but provided fewer meaningful qualitative interpretations. We also found that the ability to accurately translate the mathematical statement of the divergence theorem appeared to be a necessary, but not sufficient condition for being able to provide a meaningful
interpretation.