论文标题
作为代数可操纵实体的总和部分差异
Total and Partial Differentials as Algebraically Manipulable Entities
论文作者
论文摘要
差分运算符通常会导致衍生物表示为差分的比率。对于除最简单的衍生物以外的所有人，这些比率通常不是在代数上可以操纵的，但必须将其作为一个单位将其固定在一起，以防止矛盾。但是，这主要是一个符号和概念问题。亚伯拉罕·罗宾逊（Abraham Robinson）的作品表明，无限差分孤立地操作的概念没有任何矛盾。为了使该系统扩展到所有微积分，需要对标准的微积分表示法进行一些调整。以这种方式了解差异实际上为学生提供了对所有微积分的更直接的理解，并最大程度地减少了学生需要记住的专业定理数量，因为所有术语都可以自由地通过代数自由操纵。
Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to prevent contradictions. However, this is primarily a notational and conceptual problem. The work of Abraham Robinson has shown that there is nothing contradictory about the concept of an infinitesimal differential operating in isolation. In order to make this system extend to all of calculus, however, some tweaks to standard calculus notation are required. Understanding differentials in this way actually provides a more straightforward understanding of all of calculus for students, and minimizes the number of specialized theorems students need to remember, since all terms can be freely manipulated algebraically.
