论文标题
标量基本上是可集成的局部凸价值张量场。 Stokes定理
Scalarly Essentially Integrable Locally Convex Vector Valued Tensor Fields. Stokes Theorem
论文作者
论文摘要
此注释是对即将上映的工作\ cite {sil}的建议。在这里，我们开发了该论文所需的术语和结果。 More specifically we introduce the concept of scalarly essentially integrable locally convex vector-valued tensor fields on a smooth manifold, generalize on them the usual operations, in case the manifold is oriented define the weak integral of scalarly essentially integrable locally convex vector-valued maximal forms and finally establish the extension of Stokes theorem for smooth locally convex vector-valued forms.在我们看来，这种对标量基本上可集成和平滑的局部凸电量张量场的基本理论的方法似乎是新的。特别是新的（1）定义标量基本上可以集成的本地凸vector矢量价值张量字段，为$ \ MATHCAL {a}（u）$ - tensor产品，尽管是由通常的平稳且实用值的上下文产生的动机； （2）$ \ MATHCAL {a}（u）$ - 线性化$ \ mathcal {a}（u）$ - 双线性地图的过程，以扩展通常的操作，尤其是楔形产品； (3) the exploitation of the uniqueness decomposition of the $\mathcal{A}(U)$-tensor product with a free module in order to define not only (a) the exterior differential of smooth locally convex vector-valued forms, but also (b) the weak integral of scalarly essentially integrable locally convex vector-valued maximal forms; （4）使用投影拓扑张量产品理论来定义楔形产品。
This note is propaedeutic to the forthcoming work \cite{sil}; here we develop the terminology and results required by that paper. More specifically we introduce the concept of scalarly essentially integrable locally convex vector-valued tensor fields on a smooth manifold, generalize on them the usual operations, in case the manifold is oriented define the weak integral of scalarly essentially integrable locally convex vector-valued maximal forms and finally establish the extension of Stokes theorem for smooth locally convex vector-valued forms. This approach to the basic theory of scalarly essentially integrable and smooth locally convex vector-valued tensor fields seems to us to be new. Specifically are new (1) the definition of the space of scalarly essentially integrable locally convex vector-valued tensor fields as a $\mathcal{A}(U)$-tensor product, although motivated by a result in the usual smooth and real-valued context; (2) the procedure of $\mathcal{A}(U)$-linearizing $\mathcal{A}(U)$-bilinear maps in order to extend the usual operations especially the wedge product; (3) the exploitation of the uniqueness decomposition of the $\mathcal{A}(U)$-tensor product with a free module in order to define not only (a) the exterior differential of smooth locally convex vector-valued forms, but also (b) the weak integral of scalarly essentially integrable locally convex vector-valued maximal forms; (4) the use of the projective topological tensor product theory to define the wedge product.
