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In certain chapters of this book we make reference to the Gibbs’ calculus. In Chap. 16 we demonstrate, with the aid of exterior differential forms, that a large class of classical problems in the Newton-Maxwell paradigm is amenable to direct analysis using the operators “div, grad and curl” in arbitrary coordinate charts for Euclidean \(\mathbb {R}^{3}\) with the aid of the dual map , the Hodge map \(\star \) and the exterior derivative d. These act on arbitrary p-forms or vector fields on manifolds of arbitrary dimension. The Hodge operator is defined algebraically with respect to a metric tensor of arbitrary signature whilst the exterior derivative requires only a differential structure on the manifold.

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Gibbs’ Differential Calculus via Exterior Differential Forms

  • 章节

  • 发表日期：

    2025年

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  • Robin W. Tucker,
  • Timothy J. Walton

摘要

In certain chapters of this book we make reference to the Gibbs’ calculus. In Chap. 16 we demonstrate, with the aid of exterior differential forms, that a large class of classical problems in the Newton-Maxwell paradigm is amenable to direct analysis using the operators “div, grad and curl” in arbitrary coordinate charts for Euclidean \(\mathbb {R}^{3}\) with the aid of the dual map , the Hodge map \(\star \) and the exterior derivative d. These act on arbitrary p-forms or vector fields on manifolds of arbitrary dimension. The Hodge operator is defined algebraically with respect to a metric tensor of arbitrary signature whilst the exterior derivative requires only a differential structure on the manifold.