乘积微积分与斯托克斯定理
Product Calculus and Stokes Theorem
AI总结：
本文探讨了加法微积分与乘积微积分中微分形式的类比，通过定义乘积微分，建立了闭形式与恰当形式的乘积导数版本，并推广了斯托克斯定理的乘积积分形式。
摘要：
本文研究了由加法微积分中的积分产生的微分形式与由乘积微积分中的积分产生的形式之间的类比。研究发现，通过适当定义标量乘法和向量加法，可以构造一组向量空间，类似于外微分 calculus 中的情形。定义了一个乘积微分，它允许闭形式和恰当形式的产品导数版本。该乘积微分还允许斯托克斯定理（针对标量函数）的乘积积分版本。
英文摘要：
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar multiplication and vector addition a set of vector spaces can be constructed analogous to what is found in exterior calculus. A product differential is defined which allows for the product derivative version of closed and exact forms. The product differential also allows for a product integral version of Stokes theorem (for scalar functions).
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