Abstract This paper adds to the growing body of research happening in multivariable calculus by examining scalar and vector line integrals. This paper contributes in two ways. First, this paper provides a conceptual analysis for both types of line integrals in terms of how theoretical ways of thinking about definite integrals summarized from the research literature might be applied to understanding line integrals specifically. Second, this paper provides an initial investigation of students’ understandings of line integral expressions, and connects these understanding to the theoretical ways of thinking drawn from the literature. One key finding from the empirical part is that several students appeared to understand individual pieces of the integral expression based on one way of thinking, such as adding up pieces or anti-derivatives, while trying to understand the overall integral expression through a different way of thinking, such as area under a curve.

中文翻译：

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标量和向量线积分：概念分析和学生理解的初步调查

摘要 本文通过检查标量和向量线积分，为多变量微积分中不断增长的研究增添了内容。本文有两个方面的贡献。首先，本文从研究文献中总结的关于定积分的理论思考方式如何应用于具体理解线积分方面，对这两种类型的线积分进行了概念分析。其次，本文对学生对线积分表达式的理解进行了初步调查，并将这些理解与从文献中得出的理论思维方式联系起来。实证部分的一个关键发现是，一些学生似乎基于一种思维方式理解了积分表达式的各个部分，例如加法或反导数，