The exterior derivative and the mean value equality in ℝ^n
Daniel Fadel, Henrique N. Sá Earp, Tomás S. R. Silva
本调查通过探索外部衍生物的广义视角,重新审视了矢量微积分和分析中的经典结果,将其解释为“无限通量”的度量。 这种观点导致了平均价值定理的更高维度的类似物,适用于差分k形式,并提供了斯托克斯定理的自然公式,反映了微积分基本定理的确切假设 - 不需要差分形式的完全 C^1 平滑度。 作为一个数值应用程序,我们提出了一种在R^n中外部分化的算法,该算法仅依赖于对差分形式的黑箱访问,提供了一个实用的计算工具,而不需要网格离散化或显式符号表达式。
This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional analogue of the Mean Value Theorem, valid for differential k-forms, and provides a natural formulation of Stokes' theorem that mirrors the exact hypotheses of the Fundamental Theorem of Calculus – without requiring full C^1 smoothness of the differential form. As...