Learning vector calculus techniques is one of the major hurdles faced by physics undergraduates. However, beginners report various difficulties dealing with the index notation due to its bulkiness. Meanwhile, there have been graphical notations for tensor algebra that are intuitive and effective in calculations and can serve as a quick mnemonic for algebraic identities. Although they have been introduced and applied in vector algebra in the educational context, to the best of our knowledge, there have been no publications that employ the graphical notation to three-dimensional Euclidean vector calculus, involving differentiation and integration of vector fields. Aiming for physics students and educators, we introduce such “graphical vector calculus,” demonstrate its pedagogical advantages, and provide a variety of exercises containing both purely mathematical identities and practical calculations in physics. The graphical notation can readily be utilized in the educational environment to not only lower the barriers in learning and practicing vector calculus but also make students interested and self-motivated to manipulate the vector calculus syntax and, on their own, heuristically comprehend the language of tensors.

中文翻译：

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用图形符号增强向量演算

学习向量微积分技术是物理本科生面临的主要障碍之一。但是，由于索引符号的庞大性，初学者报告了各种困难。同时，张量代数的图形表示法在计算中是直观且有效的，并且可以用作代数恒等式的快速助记符。尽管在教育背景中已经将它们引入矢量代数中并应用到了矢量代数中，但据我们所知，还没有出版物将图形符号用于三维欧几里得向量微积分，涉及矢量场的微分和积分。针对物理学生和教育者，我们介绍了这种“图形矢量演算”，展示了其教学优势，并提供了包含物理上的纯粹数学恒等式和实际计算的各种练习。图形表示法可以很容易地在教育环境中使用，不仅可以降低学习和练习矢量微积分的障碍，而且还可以使学生对学习矢量微积分的语法感兴趣并具有自发性，并且可以启发式地理解张量的语言。 。