Vector Line Integrals in Mathematics and Physics

Dray, Tevian; Manogue, Corinne A.

doi:10.1007/s40753-022-00206-8

Cited by 6 publications

(1 citation statement)

References 24 publications

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“…As pointed out above, an appropriate visual understanding of differential vector operators and integral theorems is considered to be crucial for approaching fundamental relationships in physics; hence, supporting students in this area turns out to be particularly important. In the aforementioned work, there is consensus that teaching this topic becomes more effective when first introducing it in a purely mathematics context [25,26], and by using explicit instructions and visual representations [12]. These suggestions relate to a general educational notion, that is, the promotion of students' representational competence [27][28][29][30][31].…”

Section: Supporting Students' Visual Understandingmentioning

confidence: 99%

A research-informed graphical tool to visually approach Gauss’ and Stokes’ theorems in vector calculus

Hahn,

Blaue,

Klein

2024

Eur. J. Phys.

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Gauss' and Stokes' theorems are fundamental results in vector calculus and important tools in physics and engineering. When students are asked to describe the meaning of Gauss' divergence theorem, they often use statements like this: "The sum of all sources of a vector field in a region gives the net flux out of the region". In order to raise this description to a mathematically sound level of understanding, we present an educational approach based on the visual interpretation of the vector differential operators, i.e. divergence and curl. As a starting point, we use simple vector field diagrams for a qualitative approach to connect both sides of the integral theorems, and present an interactive graphical tool to support this connection. The tool allows to visualise two-dimensional vector fields, to specify vector decomposition, to evaluate divergence and curl point wise, and to draw rectangles to determine surface and line integrals. From a meta-perspective, we situate this educational approach into learning with (multiple) representations. Based on prior research, the graphical tool addresses various learning difficulties of vector fields that are connected to divergence and curl. The tool was incorporated into the weekly lecture-based recitations of Physics II (electromagnetism) in 2022 and 2023, and we assessed various educational outcome measures. The students overall reported the tool to be intuitive and user-friendly (level of agreement 76%, N=125), considered it helpful for understanding and recommended its use for introductory physics courses (level of agreement 65%, N=65).

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Section: Supporting Students' Visual Understandingmentioning

confidence: 99%

A research-informed graphical tool to visually approach Gauss’ and Stokes’ theorems in vector calculus

Hahn,

Blaue,

Klein

2024

Eur. J. Phys.

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Experts’ intuitive mathematical discourses about integration in complex analysis

Hanke

2024

ZDM Mathematics Education

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Although complex analysis is part of the study programs of many mathematics undergraduates, little research has been done on how individuals interpret basic concepts from complex analysis. To address this gap, this paper investigates how experts individually think about complex path integrals. For this purpose, the commognitive framework is used to conceptualize experts’ interpretations of mathematical concepts discursively, namely in terms of so-called intuitive mathematical discourses. A total of nine interpretations of complex path integrals, so-called discursive images, as well as eight sets of rules governing their construction, so-called discursive frames, are derived from expert interviews. These interpretations range from a rejection of intrinsic meaning to connections with real and vector analysis, mean values, and individual formulations of theorems. The paper also raises questions for the inclusion of the results into teaching and addresses further research.

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Approaches to Integration Based on Quantitative Reasoning: Adding Up Pieces and Accumulation from Rate

Jones

Ely

2022

Int. J. Res. Undergrad. Math. Ed.

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Vector Line Integrals in Mathematics and Physics

Cited by 6 publications

References 24 publications

A research-informed graphical tool to visually approach Gauss’ and Stokes’ theorems in vector calculus

A research-informed graphical tool to visually approach Gauss’ and Stokes’ theorems in vector calculus

Experts’ intuitive mathematical discourses about integration in complex analysis

Approaches to Integration Based on Quantitative Reasoning: Adding Up Pieces and Accumulation from Rate

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